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/
An Information-theoretic Tool forProperty Prediction Of Random
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
An overviewAn overview
Mathematical representation of random microstructures
Extraction of higher order features from limited microstructural
information : the MAXENT approach
MAXENT optimization schemes
Evaluation of homogenized elastic properties from microstructures
Effect of varying information content on property statistics
Numerical examples
Summary and future work
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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.
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
Representation of random microstructures
Indicator functions used to represent microstructure at different regions in the physical domain Indicator functions take values over a binary alphabet Statistical features of microstructure are mathematically tractable in terms of expected values over indicator functions
if
if
Two-phase material
if
if
n-phase material
Define IIii as the set comprising
(Ii(x1), Ii(x2), … Ii(xn)). IIi i
represents a random field of indicator functions over the domain. Microstructures are hierarchically characterized over a set of random variables of this field
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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
Defining correlation vectors using indicator functions
Two-point probability functions
Lineal Path Functions
n-point probability 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
Microstructure Reconstruction Schemes
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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
Reconstruction of microstructures
• Correlation features of desired microstructures is provided.• Aim to reconstruct microstructures that satisfy these ensemble statistical properties.• Ill-posed problem with many distributions that satisfy given ensemble properties.
Pb-Sn microstructuresHigh strength steel
microstructures obtainedby thermal processing
Media with short range interactions
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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
Current schemes for microstructure reconstructionCurrent schemes for microstructure reconstruction
D. Cule and S. Torquato ’99 Reconstruction of porous media using Stochastic Optimization
C. Manwart, S. Torquato and R. Hilfer ’00, Reconstruction of sandstone structures using stochastic optimization
N. Zabaras et.al. ’05 Reconstruction of microstructures using SVM’s
T.C.Baroni et al. ’02, Reconstruction of microstructures using contrast imaging techniques
A.P. Roberts ’97, Reconstruction of porous media using image mapping techniques from 2d planar images.
Input: Given statistical correlation or lineal path functions
Obtain: microstructures that satisfy the given properties Start from a random configuration over the specified problem domain such that the volume fraction information is satisfied.
Randomly choose two locations (pixels) and define a move by interchangingthe intensities of the two pixels.
If the error norm defined as the deviation of the correlation features fromtarget features reduces, accept the move, otherwise reject it.
Stochastic Optimization ProcedureStochastic Optimization Procedure
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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
A MAXENT viewpoint
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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
Information Theoretic Scheme: the MAXENT principleInformation Theoretic Scheme: the MAXENT principle
Input: Given statistical correlation or lineal path functions
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.
Additional statistical information is available using this scheme.
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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
A MAXENT viewpoint
Trivial case: no information is available about microstructure.
From MAXENT, the equiprobable case is the case with maximum
entropy for an unconstrained problem. This agrees with intuition
as to the most unbiased case
Information about volume fraction given.
The MAXENT distribution is one wherein we sample from the
volume fraction distribution itself at all material points
Correlation between material pointsto be taken into account. Result is
not trivial and needs to be numerically computed
Higher order information provided
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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 feature matching toolMAXENT as a feature matching tool
D. Pietra et al. ‘96, MAXENT principle for language processing.
Features of language extracted and MAXENT principle is used to develop a language translator
Zhu et al. ‘98, MAXENT principle for texture processing
Texture features from images in the form of histograms is extracted andMAXENT principle used to reconstruct texture images
Sobczyk ’03
MAXENT used for obtaining distributions of grain sizes from macro constraints in the form of expected grain size.
Koutsourelakis ‘05,
MAXENT for generation of random media. Correlation features of random media used as constraints to generate samples of random media.
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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 for microstructure reconstructionMAXENT for microstructure reconstruction
• MAXENT is essentially a way of MAXENT is essentially a way of generating a PDFgenerating a PDF on a hypothesis space which, on a hypothesis space which, given a measure of entropy, is guaranteed to incorporate only known given a measure of entropy, is guaranteed to incorporate only known constraints.constraints.
• MAXENT MAXENT cannotcannot be derived from Bayes theorem. It is fundamentally different, be derived from Bayes theorem. It is fundamentally different, as Bayes theorem concerns itself with inferring a-posteriori probability once the as Bayes theorem concerns itself with inferring a-posteriori probability once the likelihood and likelihood and a-priori probability are knowna-priori probability are known, while MAXENT is a guiding , while MAXENT is a guiding principle to principle to construct the a-priori PDFconstruct the a-priori PDF..
• We associate the PDF with a microstructure image and generate samples of the We associate the PDF with a microstructure image and generate samples of the image. image.
• MAXENT produces images with features (information) that are consistent with MAXENT produces images with features (information) that are consistent with the known constraints. Another way of stating this is that the known constraints. Another way of stating this is that MAXENT produces MAXENT produces the most uniform distribution consistent with the datathe most uniform distribution consistent with the data..
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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 optimization schemes
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Find
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
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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 maximizes
Equivalent log-likelihood problem
Kuhn-Tucker theorem: The that maximizes 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)Iterative methodsIterative 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
Optimization SchemesOptimization Schemes
• Generalized Iterative ScalingGeneralized Iterative Scaling
• Improved Iterative ScalingImproved Iterative Scaling
• Gradient AscentGradient Ascent
• Newton/Quasi-Newton MethodsNewton/Quasi-Newton Methods– Conjugate GradientConjugate Gradient– BFGSBFGS– ……
Start from a equal to 0. This is equivalent to uniform distribution Start from a equal to 0. This is equivalent to uniform distribution over sample space.over sample space.
Evaluate gradient at this point.Evaluate gradient at this point. Perform a line search on a direction based on the gradient Perform a line search on a direction based on the gradient
information.information. Evaluate the gradient information at the next point and continue the Evaluate the gradient information at the next point and continue the
procedure till it is within tolerance limit.procedure till it is within tolerance limit.
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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)
stochastic function
stochastic function
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CCOORRNNEELLLL U N I V E R S I T Y
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Sampling techniquesSampling techniques
• Sample from an exponential distribution using the Gibbs algorithmSample from an exponential distribution using the Gibbs algorithm
Choose a random point.Choose a random point. Evaluate the effective “energy” for various phases at that point Evaluate the effective “energy” for various phases at that point
using the updation algorithm to estimate “energy”.using the updation algorithm to estimate “energy”. Draw a sample from the given distribution and replace the pixel Draw a sample from the given distribution and replace the pixel
value at the material point.value at the material point.Continue the procedure till a sufficiently large number of Continue the procedure till a sufficiently large number of
samples are drawn. samples are drawn.
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Updation SchemeUpdation Scheme
r
zone of influence
Rozman,Utz ‘01Rozman,Utz ‘01
Two point Correlation FunctionTwo point Correlation Function Lineal Path FunctionLineal Path Function
A scheme to update correlation function of an image when the A scheme to update correlation function of an image when the phase of a single pixel is changedphase of a single pixel is changed
r
zone of influence (regionwhere correlation function is affected)
Material point whose intensity is changed
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CCOORRNNEELLLL U N I V E R S I T Y
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Line search and conjugate directionsLine search and conjugate directions
Brent’s parabolic interpolation used for line search.Brent’s parabolic interpolation used for line search.
Stabilization in conjugate gradient machinery (Schraudolph ’02)Stabilization in conjugate gradient machinery (Schraudolph ’02) Add a correction term so that as line search becomes increasingly Add a correction term so that as line search becomes increasingly
inaccurate, its effect on the conjugate direction is also subdued.inaccurate, its effect on the conjugate direction is also subdued.
Stabilization term
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CCOORRNNEELLLL U N I V E R S I T Y
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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0 20 40 60 80 100 120 140 160 180 2000
10
20
30
40
50
60
70
80
90
100
Iteration
Ent
ropy
(bits
)
Entropy variation during MAXENT algorithmic scheme
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Evaluation of effective elastic properties
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CCOORRNNEELLLL U N I V E R S I T Y
Effective elastic property of microstructures
Variational Principle: Subject to applied loads and other boundary conditions, minimize the energy stored in the microstructure.
Pixel based mesh with a single phase inside each pixel (E. Garboczi, NIST ’98). Each pixel attributed the property of that particular phase.
Homogenization: The effective homogenized property of the
microstructure is obtained by equating energy of microstructure with that of a
specimen with uniform properties
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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
Consolidated Algorithm
Experimental images
Analytical Correlation functions
Extract features and rephrase as mathematical constraints
Pose as a MAXENT problem and use gradient-based schemes for obtaining
solution
Use Gibbs sampling algorithm for sampling from
underlying distribution
Generate samples and interrogate using FEM
Obtain property statistics and use them for further analysis
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Numerical Examples
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CCOORRNNEELLLL U N I V E R S I T Y
Example 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
Reconstruction of 1d hard disks
Reconstruct one-dimensional hard disk microstructures based on two differentkinds of information: (a) two-point correlation functions (b) two point correlation and Lineal path function. Obtain elastic property statistics and compare for the two schemes.Input: Analytical two-point and lineal path functions (Torquato et.al. ’99)
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150 160 170 180 190 200 210 220 230 240 2500
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Effective young's modulus(GPa)
No.
of
sam
ples
Microstructures based on two-point correlation function
MAXENT distribution
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140 160 180 200 220 240 2600
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Effective young's modulus (GPa)
No.
of
Sam
ple
s
Microstructures based on two-point and lineal path function
MAXENT distribution
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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150 160 170 180 190 200 210 220 230 240 2500
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Effective young's modulus(GPa)
No.
of
sam
ples
140 160 180 200 220 240 2600
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Effective young's modulus (GPa)
No.
of
Sam
ple
s
Comparison of property statistics between two schemes
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Example 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
Porous Media with short range order
To generate microstructures of porous media which exhibit short range orders of given
specific structure. (S2 is the two point correlation function, k and ro depend depend on characteristic length scales chosen)
Problem Parameters
correlation length ro= 32
oscillation parameter
ao= 8
2
o
ka
Input: Analytical two-point correlation functions (Torquato et.al. ’99)
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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
Property statistics for media with short range order
MAXENT distribution
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CCOORRNNEELLLL U N I V E R S I T Y
Example 3
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Reconstruction using heterogeneous graded materials
Heterogeneous Graded Materials
Given a description of the gradation of phase-distribution in a graded material, reconstruct microstructures compatible with the given information, estimate statistics of
microstructure properties from this set.
Applications Tools with desirable properties at tips. Artificial joints for implants in humans
Input: Analytical volume fraction information throughout sample (Koutsourelakis ’04)
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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
Samples of bilinearly graded heterogeneous materialsSamples of bilinearly graded heterogeneous materials
at smooth resolution levels
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203 210 217 224 231 238 245 252 259 266 273 280 287 2940
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Effective Young's Modulus(GPa)
No.
of
sam
ples
Effective elastic properties for a tungsten-silver bilinear graded material at 25oC
Elastic properties of bilinear graded materialsElastic properties of bilinear graded materials
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Conclusions and future work
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CCOORRNNEELLLL U N I V E R S I T Y
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ConclusionsConclusions
Microstructures were characterized stochastically and scheme for
obtaining samples based on a MAXENT and time efficient update
scheme implemented.
Gradient based schemes and property of system entropy were
analyzed in detail.
Elastic properties were obtained using FEM and property statistics
developed
Schemes were discussed for numerical microstructures and effect of
incorporation of higher information on property statistics studied.
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CCOORRNNEELLLL U N I V E R S I T Y
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Future WorkFuture Work
Extend the method for polycrystal materials incorporating
information in the form of odf’s.
Couple the scheme with pixel based methods for obtaining plastic
properties.
Extend the method to physical deformation processes taking into
account the evolution of microstructure.
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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
ReferencesReferences
1. E.T. Jaynes, Information Theory and Statistical Mechanics I, Physical Review 106(4)(1957) 620—630.
2. D. Cule and S. Torquato, Generating random media from limited microstructural information via stochastic optimization, Journal of Applied Physics 86(6)(1999) 3428—3437
3.P.S. Koutsourelakis, A general framework for simulating random multi-phase media, NSF Workshop-Probability and Materials: From Nano to Macro scale (2005)
4. K. Sobczyk, Reconstruction of random material microstructures: patterns of Maximum Entropy, Probabilistic Engineering Mechanics 18(2003) 279—287
5. S.C.Zhu et al, Filters, Random Fields and Maximum Entropy (FRAME): Towards a Unified Theory for Texture Modeling, IJCV 27(1998) 107-126
6. A.Berger et.al., A maximum entropy approach to natural language modeling, (1996), Computational Linguistics 22 (1996),39-71
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