Materials Process Design and Control Laborato Materials Process Design and Control Laborato C C O O R R N N E E L L L L U N I V E R S I T Y 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/ An information-theoretic approach for property prediction of random microstructures
63
Embed
Materials Process Design and Control Laboratory Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
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
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
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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
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
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
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.
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 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.
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
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
(closed form solution)(closed form solution)Gradient based methodsGradient based methods
A A comparisoncomparison
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
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
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
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
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
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
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
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
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.
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
Mesh refinementMesh refinement
Tetrahedral mesh Hexahedral mesh
Input to homogenization tool to obtain plastic property and eventually property statistics
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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
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
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
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
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
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
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
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 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
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
MotivationMotivation
Uncertainties induced due to non-uniformities in grain growth
patterns
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 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
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
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Homogenized stress fields on the microstructureHomogenized stress fields on the 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
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.
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
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.
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
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
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
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
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
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.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory