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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
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: zabaras@cornell.eduURL: http://mpdc.mae.cornell.edu/
An information-theoretic approach for property prediction of
random microstructures
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
NEED FOR UNCERTAINTY ANALYSIS
Variation in properties, constitutive relations
Imprecise knowledge of governing physics, surroundings
Simulation based uncertainties (irreducible)
Uncertainty is everywhere
Porous media
Silicon wafer
Aircraft engines
Material process
From DOEFrom Intel websiteFrom NIST From GE-AE website
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
UNCERTAINTY AND MULTISCALING
MacroMesoMicro
Uncertainties introduced across various length scales have a non-trivial interaction
Current sophistications – resolve macro uncertainties
Use micro averaged models for resolving physical scales
Imprecise boundary conditions Initial perturbations
Physical properties, structure follow a statistical description
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Initial preform shape
Material properties/models
Forging velocity
Texture, grain sizes
Die/workpiece friction
Die shapeSmall change in preform shape
could lead to underfill
Material ModelForging rate
Die/Billet shape
Friction
Cooling rate
Stroke length
Billet temperature
Stereology/Grain texture
Dynamic recrystallization
Phase transformation
Phase separation
Internal fracture
Other heterogeneities
Yield surface changes
Isotropic/Kinematic hardening
Softening laws
Rate sensitivity
Internal state variables
Dependance Nature and degree
of correlation
Process
UNCERTAINTY IN METAL FORMING PROCESSES
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
RANDOM VARIABLES = FUNCTIONS ? Math: Probability space (, F, P)
Sample space Sigma-algebra Probability measure
F
W : Random variableW
: ( )W
Random variable
A stochastic process is a random field with variations across space and time
: ( , , )X x t
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
SPECTRAL STOCHASTIC REPRESENTATION
: ( , , )X x t
A stochastic process = spatially, temporally varying random function
CHOOSE APPROPRIATE BASIS FOR THE
PROBABILITY SPACE
HYPERGEOMETRIC ASKEY POLYNOMIALS
PIECEWISE POLYNOMIALS (FE TYPE)
SPECTRAL DECOMPOSITION
COLLOCATION, MC (DELTA FUNCTIONS)
GENERALIZED POLYNOMIAL CHAOS EXPANSION
SUPPORT-SPACE REPRESENTATION
KARHUNEN-LOÈVE EXPANSION
SMOLYAK QUADRATURE, CUBATURE, LH
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
KARHUNEN-LOEVE EXPANSION
1
( , , ) ( , ) ( , ) ( )i ii
X x t X x t X x t
Stochastic process
Mean function
ON random variablesDeterministic functions
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
1( , , ) fn( , , , , )NX x t x t
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
GENERALIZED POLYNOMIAL CHAOS Generalized polynomial chaos expansion is used to represent the stochastic output in terms of the input
0
( , , ) ( , ) (ξ( ))i ii
Z x t Z x t
Stochastic output
Askey polynomials in inputDeterministic functions
Stochastic input
1( , , ) fn( , , , , )NX x t x t
Askey polynomials ~ type of input stochastic process
Usually, Hermite, Legendre, Jacobi etc.
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
SUPPORT-SPACE REPRESENTATION
Any function of the inputs, thus can be represented as a function defined over the support-space
1( , , ) : ( ) 0NA f ξ ξ
JOINT PDF OF A TWO RANDOM
VARIABLE INPUT
FINITE ELEMENT GRID REFINED IN HIGH-DENSITY REGIONS
2
2
1
ˆ
ˆ( ( ) ( )) ( )d
L
A
q
X X
X X f
Ch
ξ ξ ξ ξ
OUTPUT REPRESENTED ALONG SPECIAL COLLOCATION POINTS
– SMOLYAK QUADRATURE
– IMPORTANCE MONTE CARLO
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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.
0
n
fs
21 2( ,0, ,0) exp
r
b
p pR
2
01
( ) (1 ( ))i i ii
s s v
p p
V20.3398190.2390330.1382470.0374605
-0.0633257-0.164112-0.264898-0.365684-0.466471-0.567257
V10.4093960.3958130.382230.3686460.3550630.3414790.3278960.3143130.3007290.287146
Eigenvectors Initial and mean deformed config.
UNCERTAINTY DUE TO MATERIAL HETEROGENEITY
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Displacement (mm)
SD
Loa
d (N
)
Homogeneous materialHeterogeneous material
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
2
4
6
8
10
12
14
Displacement (mm)
Load
(N)
Mean
Load vs Displacement SD Load vs Displacement
Dominant effect of material heterogeneity on response statistics
UNCERTAINTY DUE TO MATERIAL HETEROGENEITY
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Parameters of interest in stochastic analysis are the moment information (mean, standard deviation, kurtosis etc.) and the PDF.
For a stochastic process
Definition of moments
NISG - Random space discretized using finite elements to
Output PDF computed using local least squares interpolation from function evaluations at integration points.
( , , ) ( , , ) , ,g x t g x t x X t T
( ( , , )) ( )ppM g x t f d
h
1 1
( ( , , )) ( ) ( ( , , )) ( )h
nel nh h p h h p h
p i e ie i iee i
M g x t f d w g x t f
1 1
( ( , ))nel nint
h h p hp i ei ei
e i
M w g x t f
ie
Deterministic evaluations at fixed points
NISG - FORMULATION
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Finite element representation of the support space.
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.
True PDF
Interpolant
FE Grid
Convergence rate identical to usual finite elements, depends on order of interpolation, mesh size (h , p versions).
NISG - DETAILS
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
SD Eq. strain0.2680860.2425070.2169280.1913490.165770.1401910.1146120.08903270.06345370.0378747
Mean
Initial Final
Using 6x6 uniform support space grid
SD-Void fraction0.01860.01720.01580.01430.01290.01150.0101
Void fraction0.04190.03880.03570.03250.02940.02630.0231
SD-Void fraction0.00980.00960.00940.00920.00910.00890.0087
Uniform 0.02
Material heterogeneity induced by random distribution of micro-voids modeled using KLE and an exponential kernel. Gurson type model for damage evolution
2
01
ˆ ˆ( ) (1 ( ))i n ii
f f v
p p
EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Load displacement curves
Displacement (mm)
Lo
ad
(N)
0.1 0.2 0.3 0.4
1
2
3
4
5
6
Mean
Mean +/- SD
Displacement (mm)
SD
Lo
ad
(N)
0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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 Void fraction: 0.002 0.004 0.007 0.009 0.011 0.013 0.016
Random ? Shape
Random ? friction
PROCESS UNCERTAINTY
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Force SD Force
PROCESS STATISTICS
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
0.00E+00
5.00E-05
1.00E-04
1.50E-04
2.00E-04
2.50E-04
3.00E-04
3.50E-04
0 2 4 6 8 10 12 14
Grid resolution (Number of elements per dimension)
Re
lativ
e e
rro
r
Parameter Monte Carlo (20000 LHS samples)
Support space 10x10
Mean 2.2859e3 2.2863e6
SD 297.912 299.59
m3 -8.156e6 --9.545e6
m4 1.850e10 1.979e10
Final force statistics
Convergence study
PROCESS STATISTICS
Relative Error
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
As the number of random variables increases, problem size rises exponentially.
1
10000
1E+08
1E+12
1E+16
1E+20
0 5 10 15 20 25
No. of variables
Fu
nct
ion
eva
luat
ion
s
(assume 10 evaluations per random dimension)
CURSE OF DIMENSIONALITY
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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 Importance spaced grid
PROPOSED SOLUTIONS
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
-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
-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 Sparse grid Scheme Dimension adaptive Scheme
Very popular in computational finance applications.
Has been used in as high as 256 dimensions.
PROPOSED SOLUTIONS
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Idea Behind Information Theoretic ApproachIdea Behind Information Theoretic Approach
Statistical Mechanics
InformationTheory
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
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.
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
MAXENT as a tool for microstructure reconstructionMAXENT 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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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
The MAXENT principleThe 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.
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
MAXENT : a statistical viewpointMAXENT : a statistical viewpoint
MAXENT solution to any problem with set of features is ( )ig I
Parameters of the distributioniInput 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.
Mean provided
( )ig I
1-parameter exponential family(similar to Poisson distribution)
Gaussian distribution
Mean, variance givenNo information provided(unconstrained optimiz.)The uniform distribution
Commonly seen distributions
-2 0 2 4 6 8 10 120
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
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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 sizesMicrostructural 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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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 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
CRYSTAL SYMMETRIES?Same axis of rotation => planes
Each symmetry reduces the space by a pair of planes
Particular crystal
orientation
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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 problemMAXENT as an optimization problem
Partition Function
Find
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Equivalent log-linear modelEquivalent log-linear model
Find that minimizes
Equivalent log-likelihood problem
Kuhn-Tucker theorem: The that minimizes the dual function L also maximizes the system entropy and satisfies the constraints posed by
the problem
Direct modelsDirect models Log-linear modelsLog-linear models
ConcaveConcave ConcaveConcave
Constrained (simplex)Constrained (simplex) UnconstrainedUnconstrained
““Count and normalize” Count and normalize”
(closed form solution)(closed form solution)Gradient based methodsGradient based methods
A A comparisoncomparison
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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)
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Improper pdf (function of lagrange multipliers)
Start from a random microstructure.
Go through each grain of the microstructure and
sample an ODF according to the conditional probability distribution (conditioned on
the other grains)
continue till the samples converge to the distribution
Processor 1 Processor r
…Each processor
goes through only a subset of the
grains.
Parallel Gibbs sampler algorithmParallel Gibbs sampler algorithm
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Optimization SchemesOptimization 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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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Voronoi structureVoronoi 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.
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Mathematical representationMathematical 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
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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CCOORRNNEELLLL U N I V E R S I T Y
Preprocessing: stage 1Preprocessing: 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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CCOORRNNEELLLL U N I V E R S I T Y
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.
Preprocessing: stage 2Preprocessing: stage 2
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
MeshingMeshing
X Y
ZFrame 001 17 Nov 2005
Pixel based meshing scheme. Boundary is distorted since element shapes and sizes are fixed.
Tetrahedral element meshed. Grain boundaries conform with the mesh shapes.
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Mesh refinementMesh refinement
Tetrahedral mesh Hexahedral mesh
Input to homogenization tool to obtain plastic property and eventually property statistics
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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)
How does macro loading affect the microstructure
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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CCOORRNNEELLLL U N I V E R S I T Y
Homogenization of deformation gradientHomogenization of deformation gradient
How does macro loading affect the microstructure
Microstructure without cracks
(a) (b)
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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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
ImplementationImplementation
Largedef formulation for macro scale
Update macro displacements
Boundary value problem for microstructure
Solve for deformation field
Consistent tangent formulation (macro)
Integration of constitutive equations
Continuum slip theory
Consistent tangent formulation (meso)
Macro-deformation gradient
Homogenized (macro) stress, Consistent tangent
Mesoscale stress, consistent tangent
meso deformation gradient
(a) (b)
Macro
Meso
Micro
Homogenized propertiesHomogenized properties
X
Y
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
Equivalent Strain: 0.04 0.08 0.12 0.16 0.2 0.24 0.28
(a)
(c)
(b)
XY
Z
Equivalent Stress (MPa): 19 27 36 45 53 62 70 79(d)
X Y
Z
Equivalent Stress (MPa): 20 30 40 50 60 70 80
XY
Z
Equivalent Stress (MPa): 20 30 40 50 60 70 80
XY
Z0
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
(a)
(c)
(b)
(d)
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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)
2D random microstructures: evaluation of property statistics2D random microstructures: evaluation of property statistics
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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 4 6 8 10 12 14 16 18 200
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Grain Size( m)
prob
abili
ty
<Gsz>=10.97
<Gsz2>=124.90
Input constraints in the form of first two moments. The corresponding MAXENT distribution is shown on the right.
MAXENT distribution of grain sizesMAXENT distribution of grain sizes
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Assigning orientation to grainsAssigning orientation to grains
Given: Expected value of the orientation distribution function.
To obtain: Samples of orientation distribution function that satisfies the given ensemble
properties
-2 -1 0 1 20
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Orientation angle (in radians)
Orie
nta
tion
dis
trib
utio
n fu
nctio
n
0 50 100 1500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Orientation angle (in radians)
Ori
en
tatio
n d
istr
ibu
tion
fun
ctio
n
Input ODF (corresponds to a pure shear deformation, Zabaras et al. 2004)
Ensemble properties of ODF from reconstructed distribution
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0 0.05 0.1 0.15 0.230
40
50
60
70
80
Equivalent Stress
Eq
uiv
ale
nt
Str
ain
(M
Pa
)
Bounding plastic curves over a setof microstructural samples
Evaluation of plastic property boundsEvaluation 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
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MotivationMotivation
Uncertainties induced due to non-uniformities in grain growth
patterns
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Input uncertaintiesInput uncertainties
Problem inputs: Microstructures obtained using monte-carlo grain growth model Problem inputs: Microstructures obtained using monte-carlo grain growth model at different stages of the growth.at different stages of the growth.
Sources of uncertainty: Anything that Sources of uncertainty: Anything that changes the driving force for grain changes the driving force for grain growth (curvature driven, reduction in growth (curvature driven, reduction in surface energy) (e.g) ambient surface energy) (e.g) ambient conditions not exactly same in conditions not exactly same in microstructures near surface and in the microstructures near surface and in the bulk.bulk.
Problem parameters:Problem parameters:1.1. 10 input microstructures used that 10 input microstructures used that
constraint the input informationconstraint the input information2.2. Time lag of ~50 MC steps between Time lag of ~50 MC steps between
each sample.each sample.3.3. Simulated on a 9261 point gridSimulated on a 9261 point grid
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Maximum-entropic distribution of grain sizesMaximum-entropic distribution of grain sizes
0 100 200 300 400 500 6000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
Grain size ( m3)
Pro
babi
lity
<Gsz>=383.4967<std(Gsz)>=41.4490
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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
Grain size
Pro
ba
bili
ty
Sampling technique employedSampling technique employed
Weakly consistent scheme
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Input ODF
Some representative ODF samples
from the MaxEnt
distribution
ODF reconstruction using MAXENTODF reconstruction using MAXENT
Problem inputs/algorithm parameters:
1. 145 degrees of freedom2. MaxEnt algorithm using
Brent’s line search method3. Eighty Gibbs iteration through
each grain of the microstructure
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Input ODF
Ensemble properties of reconstructed samples of
microstructures
Ensemble propertiesEnsemble properties
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Final uncertainty representationFinal uncertainty representation
Grain size
OD
F (a
func
tion
of 1
45 ra
ndom
par
amet
ers)
Microstructures sampled as points from the joint pdf space
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Microstructure models & meshesMicrostructure models & meshes
Tetrahedral meshes
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Obtaining statistics of non-linear propertiesObtaining statistics of non-linear properties
Different microstructural models of a Different microstructural models of a polycrystal Aluminiumpolycrystal Aluminium
microstructuremicrostructure is obtained by sampling the resultant distribution. Each is obtained by sampling the resultant distribution. Each
of these specimens is subject to a of these specimens is subject to a pure axial tensionpure axial tension along the x along the x
direction. Plots of the resultant stress-contour and the resulting direction. Plots of the resultant stress-contour and the resulting
homogenized stress-strain curves are plotted for different realizationshomogenized stress-strain curves are plotted for different realizations
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Homogenized stress fields on the microstructureHomogenized stress fields on the microstructure
Equivalent Stress (MPa)84.881980.753676.625472.497168.368964.240660.112455.984151.855947.727643.5994
Equivalent Stress (MPa)1251151059585756555453525
Pixel based meshing Hexahedral meshing
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Equivalent Stress (MPa)1251151059585756555453525
Equivalent Stress (MPa)84.469180.753677.038273.322869.607465.891962.176558.461154.745751.030247.314843.5994
Homogenized stress fields on the microstructureHomogenized stress fields on the microstructure
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Comparison of pixel based versus hexahedral meshing schemesComparison of pixel based versus hexahedral meshing schemes
Equivalent strain
Eq
uiv
alen
tstr
ess
(MP
a)
0 0.001 0.002 0.003
10
20
30
40
50
Hexahedral mesh
Pixel based mesh
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.
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Plots of homogenized stress-strain curvesPlots of homogenized stress-strain curves
Equivalent strain
Equiv
ale
ntst
ress
(MP
a)
0 0.0005 0.001 0.0015 0.002
10
15
20
25
30
35
40
45
A plot showing three different samples of
the stress-strain plots obtained for different statistical models of the microstructure
generated using the MaxEnt scheme.
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Stress contours across grain boundaries and triple junctionsStress contours across grain boundaries and triple junctions
Orientation 0.4142 -0.2071 -0.0858
Orientation-0.2929 -0.4142 0.2929
Orientation0.4142 0.0858 -0.2071
Orientation0.2071 -0.4142 0.0858
Orientation0.4142 0.0858 -0.2071
Extreme sharp variation in texture
across the triple junction. Hence, leads
to a large degree of stress localization
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Applications (many …)Applications (many …)
X Y
Z
Equivalent Stress (MPa): 20 30 40 50 60 70 80
X Y
Z
Equivalent Stress (MPa): 20 30 40 50 60 70 80
X Y
Z
Equivalent Stress (MPa): 20 30 40 50 60 70 80
Statistics of plastic
properties
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DiscussionDiscussion
• A statistical distributions of mictrostructure was obtained A statistical distributions of mictrostructure was obtained incorporating variability in grain sizes and grain orientations. incorporating variability in grain sizes and grain orientations.
• Stress field distributions show a significant difference between the Stress field distributions show a significant difference between the pixel based mesh and the hexahedral mesh. One possible reason pixel based mesh and the hexahedral mesh. One possible reason may be attributed to the fact that grain boundaries are distorted as a may be attributed to the fact that grain boundaries are distorted as a result of which the localized stresses near the grain boundaries are 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 felt in some regions in the bulk of the grain. Also, for the hexahedral grid grid 21960 elements21960 elements were used while for the pixel based grid, were used while for the pixel based grid, 13824 elements13824 elements were used. We are currently performing were used. We are currently performing convergence studiesconvergence studies with respect to the mesh sizes but the number with respect to the mesh sizes but the number of elements used were roughly equivalent. Also, sharp changes in of elements used were roughly equivalent. Also, sharp changes in the field were noticed in the vicinity of the grain boundaries due to the field were noticed in the vicinity of the grain boundaries due to steep variations in texture. steep variations in texture.
• Statistical samples of microstructure model were used to obtained Statistical samples of microstructure model were used to obtained different samples of homogenized stress-strain curves.different samples of homogenized stress-strain curves.
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MODELING GRAIN BOUNDARY PHYSICS
Equivalent stress contours –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.