Transcript MAXENT distribution of grain sizes

An information-theoretic approach for
property prediction of
random microstructures
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: [email protected]
URL: http://mpdc.mae.cornell.edu/
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NEED FOR UNCERTAINTY ANALYSIS
 Uncertainty is everywhere
From NIST
Porous
media
From Intel website
Silicon
wafer
From GE-AE website
Aircraft
engines
From DOE
Material
process
 Variation in properties, constitutive relations
 Imprecise knowledge of governing physics, surroundings
 Simulation based uncertainties (irreducible)
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UNCERTAINTY AND MULTISCALING
 Uncertainties introduced across various length scales have
a non-trivial interaction
 Current sophistications – resolve macro uncertainties
Micro
 Physical
properties, structure
follow a statistical
description
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Meso
Macro
 Use micro
averaged
models for
resolving
physical scales
 Imprecise
boundary conditions
 Initial
perturbations
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UNCERTAINTY IN METAL FORMING PROCESSES
Material
Process
Model
Forging rate
Stereology/Grain texture
Yield surface changes
Die/Billet shape
Dynamic recrystallization
Friction
Phase transformation
Isotropic/Kinematic
hardening
Cooling rate
Phase separation
Stroke length
Internal fracture
Billet temperature
Other heterogeneities
Softening laws
Rate sensitivity
Internal state variables
Dependance
Nature and degree
of correlation
Forging velocity
Small change in preform shape
could lead to underfill
Die shape
Die/workpiece friction
Initial preform
shape
Material
properties/models
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Texture, grain sizes
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RANDOM VARIABLES = FUNCTIONS ?
 Math: Probability space (W, F, P)
Sample space
Probability measure
Sigma-algebra
 Random variable
F
W
W

W : Random variable
W : (W)
 A stochastic process is a random field with variations
across space and time
X : ( x, t , W)
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SPECTRAL STOCHASTIC REPRESENTATION
 A stochastic process = spatially, temporally varying random
function
X : ( x, t , W)
CHOOSE APPROPRIATE
BASIS FOR THE
PROBABILITY SPACE
HYPERGEOMETRIC ASKEY POLYNOMIALS
GENERALIZED POLYNOMIAL
CHAOS EXPANSION
SUPPORT-SPACE
REPRESENTATION
PIECEWISE POLYNOMIALS (FE TYPE)
SPECTRAL DECOMPOSITION
KARHUNEN-LOÈVE
EXPANSION
COLLOCATION, MC (DELTA FUNCTIONS)
SMOLYAK QUADRATURE,
CUBATURE, LH
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KARHUNEN-LOEVE EXPANSION

X ( x, t ,  )  X ( x, t )   X i ( x, t )i ( )
i 1
ON random variables
Deterministic functions
Stochastic Mean
process function
 Deterministic functions ~ eigen-values , eigenvectors of
the covariance function
 Orthonormal random variables ~ type of stochastic
process
 In practice, we truncate (KL) to first N terms
X ( x, t ,  )  fn( x, t , 1 ,
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,N )
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GENERALIZED POLYNOMIAL CHAOS
 Generalized polynomial chaos expansion is used to
represent the stochastic output in terms of the input
X ( x, t ,  )  fn( x, t , 1 ,

,N )
Stochastic input
Z ( x, t ,  )   Z i ( x, t ) i (ξ( ))
i 0
Stochastic
output
Askey polynomials in input
Deterministic functions
 Askey polynomials ~ type of input stochastic process
 Usually, Hermite, Legendre, Jacobi etc.
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SUPPORT-SPACE REPRESENTATION
 Any function of the inputs, thus can be represented as a
function defined over the support-space
A  ξ  (1 , ,  N ) : f (ξ)  0
FINITE ELEMENT GRID REFINED
IN HIGH-DENSITY REGIONS
X  Xˆ
L2



2
ˆ
  ( X (ξ )  X (ξ )) f (ξ )dξ 
A

 Ch q 1
JOINT PDF OF A
TWO RANDOM
VARIABLE INPUT
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– SMOLYAK
QUADRATURE
OUTPUT REPRESENTED ALONG
SPECIAL COLLOCATION POINTS
– IMPORTANCE
MONTE CARLO
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UNCERTAINTY DUE TO MATERIAL HETEROGENEITY
State variable based power law model.
State variable – Measure of deformation resistance- mesoscale property
Material heterogeneity in the state variable assumed to be a second order
random process with an exponential covariance kernel.
Eigen decomposition of the kernel using KLE.
Initial and mean
deformed config.
Eigenvectors
 
f  0  
s
n
 r 

b 

2
s( p)  s0 (1   i i vi ( p))
R ( p1 , 0, p2 , 0)   2 exp 
i 1
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V1
0.409396
0.395813
0.38223
0.368646
0.355063
0.341479
0.327896
0.314313
0.300729
0.287146
V2
0.339819
0.239033
0.138247
0.0374605
-0.0633257
-0.164112
-0.264898
-0.365684
-0.466471
-0.567257
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UNCERTAINTY DUE TO MATERIAL HETEROGENEITY
Dominant effect of
material heterogeneity on
response statistics
Load vs Displacement
SD Load vs Displacement
14
1.6
12
1.4
1.2
8
SD Load (N)
10
Load (N)
Homogeneous material
Heterogeneous material
Mean
6
1
0.8
0.6
4
0.4
2
0.2
0
0
0.1
0.2
0.3
0.4
0.5
Displacement (mm)
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0.6
0.7
0.8
0
0
0.1
0.2
0.3
0.4
0.5
Displacement (mm)
0.6
0.7
0.8
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NISG - FORMULATION
Parameters of interest in stochastic analysis are the moment information (mean,
standard deviation, kurtosis etc.) and the PDF.
For a stochastic process
g ( x, t ,  )  g ( x , t ,  )
x  X , t T ,   W
Definition of moments
M p   ( g ( x, t ,  )) p f ( )d
W
NISG - Random space W discretized using finite elements to W h
M
h
p

nel
n
h
p h
(
g
(
x
,
t
,

))
f
(

)
d


w
(
g
(
x
,
t
,

))
f i ( ie )

i
e
ie

h
Wh
nel nint
p
h
M h p   wi ( g eih ( x, t )) p f eih
e 1 i 1
e 1 i 1
Deterministic evaluations
at fixed points  ie
Output PDF computed using local least squares interpolation from function
evaluations at integration points.
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NISG - DETAILS
Finite element representation of the support
space.
True PDF
Interpolant
Inherits properties of FEM – piece wise
representations, allows discontinuous
functions, quadrature based integration rules,
local support.
Provides complete response statistics.
Decoupled function evaluations at element
integration points.
FE Grid
Convergence rate identical to usual finite elements, depends on order of
interpolation, mesh size (h , p versions).
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EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR
Material heterogeneity induced by random distribution of micro-voids modeled
using KLE and an exponential kernel. Gurson type model for damage evolution
2
fˆ ( p)  fˆ0 (1   i n vi ( p))
Initial
i 1
Final
Mean
Uniform 0.02
SD Eq. strain
SD-Void fraction
0.268086
0.0098
0.242507
0.0096
0.216928
0.0094
0.191349
0.0092
0.16577
0.0091
0.140191
0.0089
0.114612
0.0087
0.0890327
0.0634537
0.0378747
Void fraction
0.0419
0.0388
0.0357
0.0325
0.0294
0.0263
0.0231
SD-Void fraction
0.0186
0.0172
0.0158
0.0143
0.0129
0.0115
0.0101
Using 6x6 uniform support space grid
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EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR
Load displacement curves
Mean
0.7
6
0.6
SD Load (N)
Load (N)
5
4
3
2
0.5
0.4
0.3
0.2
Mean +/- SD
0.1
1
0.1
0.2
0.3
Displacement (mm)
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0.4
0
0.1
0.2
0.3
0.4
Displacement (mm)
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PROCESS UNCERTAINTY
Random ?
friction
Random ?
Shape
Axisymmetric cylinder upsetting
– 60% height reduction (Initial
height 1.5 mm)
Random initial radius – 10%
variation about mean (1 mm)–
uniformly distributed
Random die workpiece friction
U[0.1,0.5]
Power law constitutive model
Using 10x10 support space grid
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Void fraction: 0.002 0.004 0.007 0.009 0.011 0.013 0.016
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PROCESS STATISTICS
Force
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SD Force
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PROCESS STATISTICS
Final force statistics
Parameter
Monte Carlo Support
(20000 LHS space 10x10
samples)
Mean
2.2859e3
2.2863e6
SD
297.912
299.59
m3
-8.156e6
-9.545e6
m4
1.850e10
1.979e10
3.50E-04
3.00E-04
Convergence study
Relative Error
Relative error
2.50E-04
2.00E-04
1.50E-04
1.00E-04
5.00E-05
0.00E+00
0
2
4
6
8
10
12
14
Grid resolution (Number of elements per dimension)
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CURSE OF DIMENSIONALITY
As the number of random variables increases, problem size rises exponentially.
1E+20
Function evaluations
1E+16
1E+12
1E+08
10000
1
0
5
10
15
20
25
No. of variables
(assume 10 evaluations per random dimension)
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PROPOSED SOLUTIONS
ADAPTIVE DISCRETIZATION BASED ON OUTPUT
STOCHASTIC FIELD
•
Refine/Coarsen input support space grid based on output
defined control parameter (Gradients, standard deviations etc.)
•
Applicable using standard h,p adaptive schemes.
Support-space of input
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Importance spaced grid
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PROPOSED SOLUTIONS
DIMENSION ADAPTIVE QUADRATURE (Gerstner et. al. 2003)
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1
Full grid Scheme
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1
Sparse grid Scheme
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1 -0.8-0.6 -0.4-0.2 0 0.2 0.4 0.6 0.8 1
Dimension adaptive
Scheme
Very popular in computational finance applications.
Has been used in as high as 256 dimensions.
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Idea Behind Information Theoretic Approach
Basic Questions:
1. Microstructures are realizations of a random field. Is there a principle by which the
underlying pdf itself can be obtained.
2. If so, how can the known information about microstructure be incorporated in the solution.
3. How do we obtain actual statistics of properties of the microstructure characterized at macro
scale.
Information
Theory
Statistical
Mechanics
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Rigorously quantifying
and modeling
uncertainty, linking
scales using criterion
derived from
information theory, and
use information
theoretic tools to predict
parameters in the face
of incomplete
Information etc
Linkage?
Information Theory
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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 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 gi ( I ) is
i
Parameters of the distribution
gi ( I ) 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
No information provided
(unconstrained optimiz.)
The uniform distribution
Mean, variance given
Mean provided
1-parameter exponential family
(similar to Poisson distribution)
Gaussian distribution
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
-2
0
2
4
6
8
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10
12
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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.
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Microstructural feature : ODF
e^ 3
Crystal/lattice
reference
frame
^
e’
3
 CRYSTAL SYMMETRIES?
Sample
reference
frame
Same axis of rotation => planes
^
e’
crystal
e^ 1
^
e’
1
e^ 2
2
Each symmetry reduces the
space by a pair of planes
RODRIGUES’ REPRESENTATION
FCC FUNDAMENTAL REGION
n
 ORIENTATION SPACE
Euler angles – symmetries
Neo Eulerian representation
Rodrigues’
parametrization
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Particular
crystal
orientation
Cubic crystal
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MAXENT as an optimization problem
Find
feature constraints
Subject to
features of image I
Lagrange Multiplier optimization
Lagrange Multiplier optimization
Partition Function
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Equivalent log-linear model
Equivalent log-likelihood problem
Find
that minimizes
Kuhn-Tucker theorem: The
that minimizes the dual function L also
maximizes the system entropy and satisfies the constraints posed by
the problem
A
comparison
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Direct models
Log-linear models
Concave
Concave
Constrained (simplex)
Unconstrained
“Count and normalize”
(closed form solution)
Gradient based methods
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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
Start from a random
microstructure.
Each processor
goes through only
a subset of the
grains.
…
Processor 1
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Go through each grain of
the microstructure and
sample an ODF according
to the conditional probability
distribution (conditioned on
the other grains)
Processor r
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Optimization Schemes
Convergence analysis with stabilization
Convergence analysis w/o stabilization
Noise in function evaluation increases as step size for the next minima increases. This ensures that the
impact on the next evaluation is reduced.
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Voronoi structure
Voronoi cell tessellation :
Division of n-D space into non-overlapping regions such that the union
of all regions fill the entire space.
Sn  { p1, p2 ,..., pn }  k
{p1,p2,…,pk} : generator points.
Cell division of k-dimensional space :
Division of  k into subdivisions so that for each point, pi
there is an associated convex cell,
Voronoi tessellation of 3d
space. Each cell is a
microstructural grain.
Ci  {x k : j  i, d ( x, pi )  d ( x, p j )}
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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.
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:
1.The face name, which is a string.
2.The boundary of the face, which is a list of faces of
one lower dimension.
3.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
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Volumes need to be hulled to
obtain consistent
representation with
commercial packages
Convex hulling to obtain a
triangulation of
surfaces/grain boundaries
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Preprocessing: stage 1
Growth of big grains to accommodate small grains entrenched in-between
 Compute volumes of all grains
 Adjust vertices of neighboring grains so that the new voronoi
tessellation fills the volume of initial grain
 Recompute surfaces and planes of the new geometry
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Preprocessing: stage 2
Steps
 Obtain input voronoi representation in OFF format.
 Obtain the convex hull of the volumes/grains so that each
surface is a triangle (triangulation of surfaces).
 Use ANSYSTM to convert this representation to the universal
IGES (Initial Graphics Exchange specification) format.
• Surface database : To ensure non-duplication of
surfaces, a database consisting of previously encountered
hyper-planes is searched. When a new surface is created,
if it is already in the database and if all the vertices of the
surface were not present in a previous grain, no new
surface is made.
 Domain smoothing: The regions of the microstructure
inside the region [0 1]3 is chosen. Edges are smoothed so that
the boundaries represent edges of a k-dimensional cube of unit
side.
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Meshing
Tetrahedral element
meshed. Grain
boundaries conform
with the mesh
shapes.
Frame 001  17 Nov 2005 
Z
X
Y
Pixel based meshing scheme. Boundary
is distorted since element shapes and
sizes are fixed.
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Mesh refinement
Tetrahedral mesh
Hexahedral mesh
Input to homogenization tool to obtain plastic
property and eventually property statistics
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(First order) homogenization scheme
How does macro loading affect the microstructure
(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)
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Homogenization of deformation gradient
How does macro loading affect the microstructure
Microstructure without cracks
Use BC:
= 0 on the boundary
Note w = 0 on the volume is the Taylor
assumption, which is the upper bound
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(a)
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Implementation
Largedef formulation for macro scale
Update macro displacements
Macro-deformation
gradient
Macro
Homogenized (macro) stress,
Consistent tangent
Boundary value problem for microstructure
Solve for deformation field
Consistent tangent formulation (macro)
Meso
(b)
meso deformation
gradient
Mesoscale stress,
consistent tangent
Integration of constitutive equations
Micro
Continuum slip theory
Consistent tangent formulation (meso)
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Homogenized properties
60
Equivalent stress (MPa)
50
40
Simple shear
30
Plane strain compression
20
10
Z
0
0.000
Y
)
0.010
0.020
0.030
0.040
0.050
0.060
Equivalent plastic strain
X
(a)Z
(b)
(b)
Z
Y
X
X
Y
Z
X
Y
Equivalent Strain: 0.04 0.08 0.12 0.16 0.2 0.24 0.28
0.4
0.6
X
0.8
Equivalent Stress (MPa): 20 30 40 50 60 70 80
(d) Equivalent Stress (MPa):
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(c)
19 27
36 45 53 62 70 79
Equivalent Stress (MPa): 20
30
40
50
60
70
80
(d)
Materials Process Design and Control Laboratory
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: Heyn’s intercept method. An equidistant network of parallel lines drawn on a
microstructure and intersections with grain boundaries are computed.
0.2
0.18
0.16
probability
0.14
Input constraints in the
form of first two moments.
The corresponding
MAXENT distribution is
shown on the right.
<Gsz>=10.97
<Gsz 2>=124.90
0.12
0.1
0.08
0.06
0.04
0.02
0
0
2
4
6
8
10
12
14
16
18
20
Grain Size( m)
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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)
Orientation distribution function
Orientation distribution function
0.14
Ensemble properties of ODF
from reconstructed distribution
0.14
0.12
0.12
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0
0.1
0.02
0
50
100
Orientation angle (in radians)
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U N I V E R S I T Y
150
0
-2
-1
0
1
Orientation angle (in radians)
2
Materials Process Design and Control Laboratory
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
Equivalent Strain (MPa)
70
60
Bounding plastic curves over a set
of microstructural samples
50
40
30
0
0.05
0.1
0.15
0.2
Equivalent Stress
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
Motivation
Uncertainties induced due to nonuniformities in grain growth
patterns
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
Input uncertainties
Problem inputs: Microstructures obtained using monte-carlo grain growth model
at different stages of the growth.
Sources of uncertainty: Anything that
changes the driving force for grain
growth (curvature driven, reduction in
surface energy) (e.g) ambient
conditions not exactly same in
microstructures near surface and in the
bulk.
Problem parameters:
1. 10 input microstructures used that
constraint the input information
2. Time lag of ~50 MC steps between
each sample.
3. Simulated on a 9261 point grid
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Materials Process Design and Control Laboratory
Maximum-entropic distribution of grain sizes
0.01
0.009
0.008
<Gsz>=383.4967
<std(Gsz)>=41.4490
Probability
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
0
100
200
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U N I V E R S I T Y
300
400
Grain size ( m3)
500
600
Materials Process Design and Control Laboratory
Sampling technique employed
0.2
0.18
0.16
Weakly
consistent
scheme
0.14
Probability
0.12
0.1
0.08
0.06
0.04
0.02
0
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U N I V E R S I T Y
0
5
10
15
Grain size
20
25
30
Materials Process Design and Control Laboratory
ODF reconstruction using MAXENT
Input ODF
Problem inputs/algorithm
parameters:
1. 145 degrees of freedom
2. MaxEnt algorithm using
Brent’s line search method
3. Eighty Gibbs iteration through
each grain of the
microstructure
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U N I V E R S I T Y
Some
representative
ODF samples
from the
MaxEnt
distribution
Materials Process Design and Control Laboratory
Ensemble properties
Input ODF
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U N I V E R S I T Y
Ensemble properties of
reconstructed samples of
microstructures
Materials Process Design and Control Laboratory
pdf
Final uncertainty representation
Microstructures sampled as points
from the joint pdf space
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
Microstructure models & meshes
Tetrahedral meshes
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
Obtaining statistics of non-linear properties
Different microstructural models of a polycrystal Aluminium
microstructure is obtained by sampling the resultant distribution. Each
of these specimens is subject to a pure axial tension along the x
direction. Plots of the resultant stress-contour and the resulting
homogenized stress-strain curves are plotted for different realizations
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
Homogenized stress fields on the microstructure
Hexahedral meshing
Pixel based meshing
Equivalent Stress (MPa)
84.8819
80.7536
76.6254
72.4971
68.3689
64.2406
60.1124
55.9841
51.8559
47.7276
43.5994
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U N I V E R S I T Y
Equivalent Stress (MPa)
125
115
105
95
85
75
65
55
45
35
25
Materials Process Design and Control Laboratory
Homogenized stress fields on the microstructure
Equivalent Stress (MPa)
84.4691
80.7536
77.0382
73.3228
69.6074
65.8919
62.1765
58.4611
54.7457
51.0302
47.3148
43.5994
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U N I V E R S I T Y
Equivalent Stress (MPa)
125
115
105
95
85
75
65
55
45
35
25
Materials Process Design and Control Laboratory
Comparison of pixel based versus hexahedral meshing schemes
The pixel based meshing
scheme distorts grain
boundaries and not only
increases their area but
also twists their shape
which leads to a higher
degree of stress
localization as viewed in
previous plot.
Equivalent stress (MPa)
50
40
Hexahedral mesh
Pixel based mesh
30
20
10
0
0.001
0.002
0.003
Equivalent strain
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Materials Process Design and Control Laboratory
Plots of homogenized stress-strain curves
45
40
Equivalent stress (MPa)
A plot showing three
different samples of
the stress-strain plots
obtained for different
statistical models of
the microstructure
generated using the
MaxEnt scheme.
35
30
25
20
15
10
0
0.0005
0.001
0.0015
0.002
Equivalent strain
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Materials Process Design and Control Laboratory
Stress contours across grain boundaries and triple junctions
Orientation
0.2071 -0.4142 0.0858
Orientation
0.4142 0.0858 -0.2071
Orientation
0.4142 -0.2071 -0.0858
Orientation
0.4142 0.0858 -0.2071
Orientation
-0.2929 -0.4142 0.2929
Extreme sharp
variation in texture
across the triple
junction. Hence, leads
to a large degree of
stress localization
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Materials Process Design and Control Laboratory
Applications (many …)
Z
X
Y
Equivalent Stress (MPa): 20
30
40
50
60
70
80
Statistics of
plastic
properties
Z
X
Y
Equivalent Stress (MPa): 20
30
40
50
60
70
80
Z
X
Y
Equivalent Stress (MPa): 20
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U N I V E R S I T Y
30
40
50
60
70
80
Materials Process Design and Control Laboratory
Discussion
• A statistical distributions of mictrostructure was obtained
incorporating variability in grain sizes and grain orientations.
• Stress field distributions show a significant difference between the
pixel based mesh and the hexahedral mesh. One possible reason
may be attributed to the fact that grain boundaries are distorted as a
result of which the localized stresses near the grain boundaries are
felt in some regions in the bulk of the grain. Also, for the hexahedral
grid 21960 elements were used while for the pixel based grid,
13824 elements were used. We are currently performing
convergence studies with respect to the mesh sizes but the number
of elements used were roughly equivalent. Also, sharp changes in
the field were noticed in the vicinity of the grain boundaries due to
steep variations in texture.
• Statistical samples of microstructure model were used to obtained
different samples of homogenized stress-strain curves.
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Materials Process Design and Control Laboratory
MODELING GRAIN BOUNDARY PHYSICS
Equivalent stress contours
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U N I V E R S I T Y
–Include failure mechanisms
–Grain boundary properties
–Local stress concentrations
develop to cause the emission
of a few partial dislocations
from grain boundaries, and
these high stresses drive the
partial dislocations across the
grain interiors
–MD studies indicate that this
is the major mechanism
of the limited inelastic
deformation in the grain
interiors of nanocrystalline
materials.
Materials Process Design and Control Laboratory