sethuraman sankaran and nicholas zabaras
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Sethuraman Sankaran and Nicholas Zabaras
Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering
188 Frank H. T. Rhodes HallCornell University
Ithaca, NY 14853-3801
Email: ss524@cornell.edu, zabaras@cornell.eduURL: http://mpdc.mae.cornell.edu/
Maximum entropy approach for statistical modeling of three-dimensional polycrystal
microstructures
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Research Sponsors
U.S. AIR FORCE PARTNERS
Materials Process Design Branch, AFRL
Computational Mathematics Program, AFOSR
CORNELL THEORY CENTER
ARMY RESEARCH OFFICE
Mechanical Behavior of Materials Program
NATIONAL SCIENCE FOUNDATION (NSF)
Design and Integration Engineering Program
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Why do we need a statistical model?
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?
Different statistical samples of the manufactured specimen
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Development of a mathematical model
Compute a PDF of microstructures
Grain size features
Orientation Distribution
functions
Grain sizeO
DF
(a fu
nctio
n of
145
rand
om p
aram
eter
s)Assign
microstructures to the macro specimen after
sampling from the PDF
Random variable 1(scalar or vector)
Random variable 2:High dimensions
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
The main idea
Extract features of the microstructure Geometrical: grain size
Texture: ODFs
Phase field simulations
Experimental microstructures
Compute a PDF of microstructuresMAXENT
Compute bounds on macro properties
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Generating input microstructures: The phase field Generating input microstructures: The phase field modelmodel
Define order parameters:
q {1,2,..., }q Q where Q is the total number of orientations possible
Define free energy function (Allen/Cahn 1979, Fan/Chen 1997) :
21 2
1
( ) [ ( ( , ), ( , ),..., ( , )) ( ) ]2
o Q qq
F t f r t r t r t dr
Non-zero only near grain boundaries
2 2 2 2 2
1 1 1
({ }) ( , ) ( ) ( )2 4 2
Q Q Q Q
o q q q q sq q q s q
f r t
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Physics of phase field methodPhysics of phase field method
Driving force for grain growth:
Reduction in free energy: thermodynamic driving force to eliminate grain boundary area (Ginzburg-Landau equations)
3 2 2( , ) ( , ) 2 ( , ) ( , ) ( , )q s
q q q q qs q
L r t r t r t r t r tt
qL kinetic rate coefficients related to the mobility of grain boundaries
Assumption: Grain boundary mobilties are constant
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Phase Field – Problem parametersPhase Field – Problem parameters
• Isotropic mobility (L=1)Isotropic mobility (L=1)
• Discretization :Discretization :
problem size : 75x75x75problem size : 75x75x75
Order parameters:Order parameters:
Q=20Q=20
•
• Timesteps = 1000Timesteps = 1000
• First nearest neighbor approx.First nearest neighbor approx.
1; 2
0.1t 2x
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Input microstructural samplesInput microstructural samples
3D microstructural samples
2D microstructural samples
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
The main idea
Extract features of the microstructureGeometrical: grain sizeTexture: ODFs
Phase field simulations
Experimental microstructures
Compute a PDF of microstructuresMAXENT
Compute bounds on macro properties
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
3D microstructures
Grain size is computed from the volumes of individual grains
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Cubic crystal
Microstructural feature : ODF
RODRIGUES’ REPRESENTATIONRODRIGUES’ REPRESENTATIONFCC FUNDAMENTAL REGIONFCC FUNDAMENTAL REGION
Crystal/lattice
reference frame
e2^
Sample reference
frame
e1^ e’1
^
e’2^
crystalcrystal
e’3^
e3^
ORIENTATION SPACEEuler angles – symmetries
Neo Eulerian representation
n
Rodrigues’ Rodrigues’ parametrizationparametrization
Orientation Distribution Function
Volume fraction of crystals with a specific orientation
Particular crystal
orientation
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
The main idea
Extract features of the microstructureGeometrical: grain sizeTexture: ODFs
Phase field simulations
Experimental microstructures
Compute a PDF of microstructuresMAXENTTool for
microstructure modeling
Compute bounds on macro properties
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Review
Grain size
OD
F (a
func
tion
of 1
45
rand
om p
aram
eter
s)
Know microstructures at some points
Given: Microstructures at some pointsObtain: PDF of microstructures
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
The MAXENT principle
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.
E.T. Jaynes 1957
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.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Subject to
Lagrange Multiplier optimization
Lagrange Multiplier optimization
feature constraints
features of image I
MAXENT as an optimization problem
Partition Function
Find
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Gradient EvaluationGradient Evaluation
• Objective function and its gradients: Objective function and its gradients:
• Infeasible to compute at all points in one conjugate gradient iterationInfeasible to compute at all points in one conjugate gradient iteration
• Use sampling techniques to sample from the distribution evaluated Use sampling techniques to sample from the distribution evaluated at the previous point. (Gibbs Sampler)at the previous point. (Gibbs Sampler)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
The main idea
Extract features of the microstructureGeometrical: grain sizeTexture: ODFs
Phase field simulations
Experimental microstructures
Compute a PDF of microstructuresMAXENT
Compute bounds on macroscopic properties
Tool for microstructure
modeling
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Microstructure modeling : the Voronoi structure
Division of n-D space into non-overlapping regions such that the union of all regions fill the entire space.
Voronoi cell tessellation :
{p1,p2,…,pk} : generator points.
1, 2{ ,..., } kn nS p p p
{ : , ( , ) ( , )}ki i jC x j i d x p d x p
Division of into subdivisions so that for each point, pi
there is an associated convex cell,
kCell division of k-dimensional space :
Voronoi tessellation of 3d space. Each cell is a microstructural grain.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Stochastic modeling of microstructuresStochastic 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.
0 5 10 15 20 25 300
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Mean Grain size
Pro
ba
bili
ty
Weakly consistent scheme
0 5 10 15 20 25 3000.020.040.060.080.10.120.140.160.180.2
Grain size
Pro
ba
bili
tyStrongly consistent scheme
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Heuristic algorithm for generating voronoi centers
Generate sample points on a uniform grid from Sobel
sequence
0 5 10 15 20 25 3000.020.040.060.080.10.120.140.160.180.2
Mean Grain size
Pro
ba
bili
ty
( , ) ( ) ( ) ( , ) if ( ) ( ) ( , )
( , ) 0 otherwise
F i j r i r j dis i j r i r j dis i j
F i j
Forcing function
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.
Rcorr(y,d)>0.95?
No
Yesstop
0 50 100 150 200 2500
0.05
0.1
0.15
0.2
0.25
Grain size ( m)
Pro
ba
bili
ty m
as
s f
un
cti
on
CorrCoef=0.8689KL=0.0015
Given: grain size distributionConstruct: a microstructure which matches the given distribution
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
The main idea
Extract features of the microstructureGeometrical: grain sizeTexture: ODFs
Phase field simulations
Experimental microstructures
Compute a PDF of microstructuresMAXENT
Compute bounds on macroscopic properties
Tool for microstructure
modeling
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
(First order) homogenization scheme(First order) homogenization scheme
(a) (b)
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)
0
10
20
30
40
50
60
0.000 0.010 0.020 0.030 0.040 0.050 0.060
Equivalent plastic strain
Equ
ival
ent s
tres
s (M
Pa)
Simple shear
Plane strain compression
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Numerical Example: Strong sampling
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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.
MAXENT tool
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.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 200000
0.05
0.1
0.15
0.2
0.25
Grain volume (voxels)
Pro
babi
lity
mas
s fu
nctio
n
Grain volume distribution
using phase field simulations
pmf reconstructed using MaxEnt
K.L.Divergence=0.0672 nats
Grain size distribution computed using MaxEnt
Comparison of MaxEnt grain size distribution
with the distribution of a phase field
microstructure
K.L( ; ) log( )iii i
pp q p
q
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
0 5000 10000 15000 20000 250000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Grain volume (voxels)
Pro
ba
bili
ty m
ass
fu
nct
ion
Rcorr
=0.9644
KL=0.0383
Reconstructing strongly consistent microstructures
Computing microstructures using the Sobel sequence method
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
0 5000 10000 15000 20000 250000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Grain volume (voxels)
Pro
babi
lity
mas
s fu
nctio
n
Rcorr=0.9830
KL=0.05
Reconstructing strongly consistent microstructures (contd..)
Computing microstructures using the Sobel sequence method
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Input ODF
Reconstructed samples using
MAXENT
ODF reconstruction using MAXENT
Representation in Frank-
Rodrigues space
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Input ODF
Expected property of reconstructed samples of
microstructures
Ensemble properties
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
0 1 2 3
x 10-4
0
10
20
30
40
50
60
Equivalent strain
Equ
ival
ent s
tres
s (M
Pa)
Mean stress-strain curve
Mean std
Statistical variation of properties
Statistical variation of Statistical variation of homogenized stress-homogenized stress-
strain curves. strain curves.
Aluminium polycrystal Aluminium polycrystal with rate-independent with rate-independent strain hardening. Pure strain hardening. Pure tensile test.tensile test.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Numerical Example: Weak sampling
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
A grain boundary network of one microstructural sample
3D microstructures: Grain boundary topology network
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Grain volume (voxels)
Pro
ba
bili
ty m
as
s f
un
cti
on
Two grain size moments
Three grain size momentsFour grain size moments
Distribution of microstructures computed using MaxEnt technique using mean grain size as a microstructural feature
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Grain volume (voxels)
Pro
ba
bili
ty m
as
s f
un
cti
on
Two grain size moments
Three grain size momentsFour grain size moments
Samples of microstructures computed at different points of the PDF
Microstructures computed using the mean grain sizes, which are sampled from the PDF
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Randomness in texture
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.
Expected ODF distribution that is given as a constraint to the MaxEnt
algorithm
Samples of the reconstructed ODF function
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Meshing microstructure samples using hexahedral elements (Cubit TM)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Equivalent strain
Equ
ival
ent
stre
ss (
MP
a)
0 0.5 1 1.5 2 2.5 3 3.5
x 10- 3
0
10
20
30
40
50
60
Equivalent strain
Equ
ival
ent
stre
ss (
MP
a)
Bounds on plastic
properties
0 0.5 1 1.5 2 2.5 3 3.5
x 10- 3
0
10
20
30
40
50
60
Equivalent strain
Equ
ival
ent
stre
ss (
MP
a)
Bounds on plastic
properties
Statistical variation of Statistical variation of homogenized stress-homogenized stress-
strain curves. strain curves.
Aluminium polycrystal with Aluminium polycrystal with rate-independent strain rate-independent strain hardening. Pure tensile hardening. Pure tensile test.test.
Extremal bounds of homogenized stress-strain properties
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Limited set of input microstructures computed using
phase field technique
Statistical samples of microstructure at certain collocation points computed using
maximum entropy technique
Diffusivity properties in a statistical class of microstructures
Future work: Diffusion in microstructures induced by topological uncertainty
Diffusion coefficient
Pro
ba
bili
ty
Variability of effective diffusion
coefficient of microstructure
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
InformationInformation
RELEVANT PUBLICATIONSRELEVANT PUBLICATIONS
S.Sankaran and N. Zabaras, Maximum entropy method for statistical modeling of S.Sankaran and N. Zabaras, Maximum entropy method for statistical modeling of microstructures, Acta Materialia, 2006microstructures, Acta Materialia, 2006
Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering
188 Frank H. T. Rhodes HallCornell University
Ithaca, NY 14853-3801Email: zabaras@cornell.edu
URL: http://mpdc.mae.cornell.edu/
Prof. Nicholas Zabaras
CONTACT INFORMATIONCONTACT INFORMATION
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