sethuraman sankaran and nicholas zabaras

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

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: [email protected], [email protected]: http://mpdc.mae.cornell.edu/

Maximum entropy approach for statistical modeling of three-dimensional polycrystal

microstructures

mailto:[email protected]





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




Why do we need a statistical model?

When a specimen is manufactured, the microstructures at a sample point will not be the same always. How do we compute the class of

microstructures based on some limited information?

Different statistical samples of the manufactured specimen




Development of a mathematical model

Compute a PDF of microstructures

Grain size features

Orientation Distribution

functions

Grain sizeO

DF

(a fu

nctio

n of

145

rand

om p

aram

eter

s)Assign

microstructures to the macro specimen after

sampling from the PDF

Random variable 1(scalar or vector)

Random variable 2:High dimensions




The main idea

Extract features of the microstructure Geometrical: grain size

Texture: ODFs

Phase field simulations

Experimental microstructures

Compute a PDF of microstructuresMAXENT

Compute bounds on macro properties




Generating input microstructures: The phase field Generating input microstructures: The phase field modelmodel

Define order parameters:

q {1,2,..., }q Q where Q is the total number of orientations possible

Define free energy function (Allen/Cahn 1979, Fan/Chen 1997) :

21 2

1

( ) [ ( ( , ), ( , ),..., ( , )) ( ) ]2

Qq

o Q qq

F t f r t r t r t dr

Non-zero only near grain boundaries

2 2 2 2 2

1 1 1

({ }) ( , ) ( ) ( )2 4 2

Q Q Q Q

o q q q q sq q q s q

f r t




Physics of phase field methodPhysics of phase field method

Driving force for grain growth:

Reduction in free energy: thermodynamic driving force to eliminate grain boundary area (Ginzburg-Landau equations)

3 2 2( , ) ( , ) 2 ( , ) ( , ) ( , )q s

Qq

q q q q qs q

L r t r t r t r t r tt

qL kinetic rate coefficients related to the mobility of grain boundaries

Assumption: Grain boundary mobilties are constant




Phase Field – Problem parametersPhase Field – Problem parameters

• Isotropic mobility (L=1)Isotropic mobility (L=1)

• Discretization :Discretization :

problem size : 75x75x75problem size : 75x75x75

Order parameters:Order parameters:

Q=20Q=20

•

• Timesteps = 1000Timesteps = 1000

• First nearest neighbor approx.First nearest neighbor approx.

1; 2

0.1t 2x




Input microstructural samplesInput microstructural samples

3D microstructural samples

2D microstructural samples




The main idea

Extract features of the microstructureGeometrical: grain sizeTexture: ODFs








Microstructural feature: Grain sizes

Grain size obtained by using a series of equidistant, parallel

lines on a given microstructure at different angles. In 3D, the size

of a grain is chosen as the number of voxels (proportional to volume) inside a particular grain.

2D microstructures

3D microstructures

Grain size is computed from the volumes of individual grains




Cubic crystal

Microstructural feature : ODF

RODRIGUES’ REPRESENTATIONRODRIGUES’ REPRESENTATIONFCC FUNDAMENTAL REGIONFCC FUNDAMENTAL REGION

Crystal/lattice

reference frame

e2^

Sample reference

frame

e1^ e’1

^

e’2^

crystalcrystal

e’3^

e3^

ORIENTATION SPACEEuler angles – symmetries

Neo Eulerian representation

n

Rodrigues’ Rodrigues’ parametrizationparametrization

Orientation Distribution Function

Volume fraction of crystals with a specific orientation

Particular crystal

orientation




The main idea




Compute a PDF of microstructuresMAXENTTool for

microstructure modeling





Review

Grain size

OD

F (a

func

tion

of 1

45

rand

om p

aram

eter

s)

Know microstructures at some points

Given: Microstructures at some pointsObtain: PDF of microstructures




The MAXENT principle

The principle of maximum entropy (MAXENT) states that amongst the probability distributions that satisfy our incomplete information about the system, the probability distribution that maximizes entropy is the least-biased estimate that can be made. It agrees with everything that is known but carefully avoids anything that is unknown.

E.T. Jaynes 1957

MAXENT is a guiding principle to construct PDFs based on limited information

There is no proof behind the MAXENT principle. The intuition for choosing distribution with

maximum entropy is derived from several diverse natural phenomenon and it works in practice.

The missing information in the input data is fit into a probabilistic model such that

randomness induced by the missing data is maximized. This step minimizes assumptions about

unknown information about the system.




Subject to

Lagrange Multiplier optimization

Lagrange Multiplier optimization

feature constraints

features of image I

MAXENT as an optimization problem

Partition Function

Find




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)




The main idea





Compute bounds on macroscopic properties

Tool for microstructure

modeling




Microstructure modeling : the Voronoi structure

Division of n-D space into non-overlapping regions such that the union of all regions fill the entire space.

Voronoi cell tessellation :

{p1,p2,…,pk} : generator points.

1, 2{ ,..., } kn nS p p p

{ : , ( , ) ( , )}ki i jC x j i d x p d x p

Division of into subdivisions so that for each point, pi

there is an associated convex cell,

kCell division of k-dimensional space :

Voronoi tessellation of 3d space. Each cell is a microstructural grain.




Stochastic modeling of microstructuresStochastic modeling of microstructures

Sampling using grain size distribution Sampling using mean grain size

Match the PDF of a microstructure with PDF of grain sizes computed from MaxEnt

Each microstructure is referred to by its mean value.

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Mean Grain size

Pro

ba

bili

ty

Weakly consistent scheme

0 5 10 15 20 25 3000.020.040.060.080.10.120.140.160.180.2

Grain size

Pro

ba

bili

tyStrongly consistent scheme




Heuristic algorithm for generating voronoi centers

Generate sample points on a uniform grid from Sobel

sequence

0 5 10 15 20 25 3000.020.040.060.080.10.120.140.160.180.2

Mean Grain size

Pro

ba

bili

ty

( , ) ( ) ( ) ( , ) if ( ) ( ) ( , )

( , ) 0 otherwise

F i j r i r j dis i j r i r j dis i j

F i j

Forcing function

Objective is to minimize norm (F). Update the voronoi centers

based on F

Construct a voronoi diagram based on these centers. Let the

grain size distribution be y.

Rcorr(y,d)>0.95?

No

Yesstop

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

Grain size ( m)

Pro

ba

bili

ty m

as

s f

un

cti

on

CorrCoef=0.8689KL=0.0015

Given: grain size distributionConstruct: a microstructure which matches the given distribution




The main idea





Compute bounds on macroscopic properties

Tool for microstructure

modeling




(First order) homogenization scheme(First order) homogenization scheme

(a) (b)

1. Microstructure is a representation of a material point at a smaller scale

2. Deformation at a macro-scale point can be represented by the motion of the exterior boundary of the microstructure. (Hill, R., 1972)

0

10

20

30

40

50

60

0.000 0.010 0.020 0.030 0.040 0.050 0.060

Equivalent plastic strain

Equ

ival

ent s

tres

s (M

Pa)

Simple shear

Plane strain compression





Numerical Example: Strong sampling




Input constraints: macro grain size observable. First four grain size moments ,

expected value of the ODF are given as constraints.

Output: Entire variability (PDF) of grain size and ODFs in the microstructure is obtained.

MAXENT tool

3D random microstructures – evaluation of property statistics

Problem definition: Given microstructures generated using phase field technique, compute grain size distributions using MaxEnt technique as well as compute bounds in properties.




0 2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

0.05

0.1

0.15

0.2

0.25

Grain volume (voxels)

Pro

babi

lity

mas

s fu

nctio

n

Grain volume distribution

using phase field simulations

pmf reconstructed using MaxEnt

K.L.Divergence=0.0672 nats

Grain size distribution computed using MaxEnt

Comparison of MaxEnt grain size distribution

with the distribution of a phase field

microstructure

K.L( ; ) log( )iii i

pp q p

q




0 5000 10000 15000 20000 250000

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Pro

ba

bili

ty m

ass

fu

nct

ion

Rcorr

=0.9644

KL=0.0383

Reconstructing strongly consistent microstructures

Computing microstructures using the Sobel sequence method




0 5000 10000 15000 20000 250000

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Pro

babi

lity

mas

s fu

nctio

n

Rcorr=0.9830

KL=0.05

Reconstructing strongly consistent microstructures (contd..)

Computing microstructures using the Sobel sequence method




Input ODF

Reconstructed samples using

MAXENT

ODF reconstruction using MAXENT

Representation in Frank-

Rodrigues space




Input ODF

Expected property of reconstructed samples of

microstructures

Ensemble properties




0 1 2 3

x 10-4

0

10

20

30

40

50

60

Equivalent strain

Equ

ival

ent s

tres

s (M

Pa)

Mean stress-strain curve

Mean std

Statistical variation of properties

Statistical variation of Statistical variation of homogenized stress-homogenized stress-

strain curves. strain curves.

Aluminium polycrystal Aluminium polycrystal with rate-independent with rate-independent strain hardening. Pure strain hardening. Pure tensile test.tensile test.




Numerical Example: Weak sampling




A grain boundary network of one microstructural sample

3D microstructures: Grain boundary topology network

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1


Pro

ba

bili

ty m

as

s f

un

cti

on

Two grain size moments

Three grain size momentsFour grain size moments

Distribution of microstructures computed using MaxEnt technique using mean grain size as a microstructural feature




0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1


Pro

ba

bili

ty m

as

s f

un

cti

on

Two grain size moments

Three grain size momentsFour grain size moments

Samples of microstructures computed at different points of the PDF

Microstructures computed using the mean grain sizes, which are sampled from the PDF




Randomness in texture

Each grain is attributed an orientation that is sampled from a MaxEnt distribution of ODFs. Some of the samples of textures that are constructed are shown in the figure above.

Expected ODF distribution that is given as a constraint to the MaxEnt

algorithm

Samples of the reconstructed ODF function




Meshing microstructure samples using hexahedral elements (Cubit TM)




Equivalent strain

Equ

ival

ent

stre

ss (

MP

a)

0 0.5 1 1.5 2 2.5 3 3.5

x 10- 3

0

10

20

30

40

50

60

Equivalent strain

Equ

ival

ent

stre

ss (

MP

a)

Bounds on plastic

properties

0 0.5 1 1.5 2 2.5 3 3.5

x 10- 3

0

10

20

30

40

50

60

Equivalent strain

Equ

ival

ent

stre

ss (

MP

a)

Bounds on plastic

properties

Statistical variation of Statistical variation of homogenized stress-homogenized stress-

strain curves. strain curves.

Aluminium polycrystal with Aluminium polycrystal with rate-independent strain rate-independent strain hardening. Pure tensile hardening. Pure tensile test.test.

Extremal bounds of homogenized stress-strain properties




Limited set of input microstructures computed using

phase field technique

Statistical samples of microstructure at certain collocation points computed using

maximum entropy technique

Diffusivity properties in a statistical class of microstructures

Future work: Diffusion in microstructures induced by topological uncertainty

Diffusion coefficient

Pro

ba

bili

ty

Variability of effective diffusion

coefficient of microstructure




InformationInformation

RELEVANT PUBLICATIONSRELEVANT PUBLICATIONS

S.Sankaran and N. Zabaras, Maximum entropy method for statistical modeling of S.Sankaran and N. Zabaras, Maximum entropy method for statistical modeling of microstructures, Acta Materialia, 2006microstructures, Acta Materialia, 2006

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801Email: [email protected]

URL: http://mpdc.mae.cornell.edu/

Prof. Nicholas Zabaras

CONTACT INFORMATIONCONTACT INFORMATION

sethuraman sankaran and nicholas zabaras

Documents

macroscale point

materials process design

smaller scaledeformation

exterior boundary