Multiscale stochastic modeling in polycrystalline materials: Challenges, perspectives...

Materials Process Design and Control Laboratory 1

Multiscale stochastic modeling in polycrystalline materials: Challenges,

perspectives and open problems

Nicholas ZabarasMaterials Process Design and Control Laboratory

Sibley School of Mechanical and Aerospace Engineering101 Frank H. T. Rhodes Hall

Cornell UniversityIthaca, NY 14853-3801

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


Some problems of interest

Develop data-driven non-linear strategies to encode the experimental or simulation-based information on microstructure and represent it in a finite low-dimensional (stochastic support) framework: examine the effects of different metrics in real, Fourier and Radon spaces.

Propagation of microstructural and process uncertainty to the final microstructure and material properties: error-bars on macroscopic material properties, failure probabilities, quantify needed input information for desired output variability.

Develop a stochastic framework to obtain the convex hull of properties of a material subjected to uncertain process parameters and initial microstructure: materials-by-design applications, exploring process/ structure /property probabilistic relations.

Materials Process Design and Control Laboratory

Stochastic multiscale modeling in polycrystalline materials

Solve micro SBVP

Microstructure

SRVE

Macro FE analysis

Integration pointI

II

IIIMacro, FE

Input microstructures Preprocessing, define metric

Grain volume

His

togr

am

0 200 400 600 800 10000

4

8

12

16

20

Grain volume

His

togr

am

0 200 400 600 800 10000

4

8

12

16

20

Grain size distribution

Low dimensional points from the pair-wise distances

-20 -10 0 10 20-20

0

20-20

-15

-10

-5

0

5

10

15

Convex hull of the points: The low dimensional model

-20 -10 0 10 20-20

020

-20

-15

-10

-5

0

5

10

15

-1 -0.5 0 0.5 1

-1

0

1-1

-0.5

0

0.5

1

Transform hull to unit hypercube

Reconstruction for arbitrary points on the hypercube

Low-dimensional representation of microstructures

1) Geometry can be preserved if the distances between the points are preserved –Isometric mapping.

2) The geometry of the manifold is reflected in the geodesic distance between points3) First step towards reduced representation is to construct the geodesic distances

between all the sample points4) The geodesic distance is computed by Dijkstra’s algorithm

Non-linear dimension reduction


• The metric of grain size distribution:

k'i

i'i

HH , S is an integer , 1 2S+1

H : the histogramwhere

H : metric of grain size

i S

k i S i bin

+

= −= =

⎧⎨⎩

∑

Defining the metric


• Reconstruction a polycrystalline microstructure from the lower dimensional space.

• Local linear interpolation:

• Find the point in the manifold

1

1

tan ( , ) 1tan ( , )

: the neighbors in lower dimension space: : the neighbors in higher dimension space

: the point in lower dimension space

ni

i in

i i

i

i

Rdis ce L lr

dis ce L lL

where Rl

=

=

=

⎧⎪⎨⎪⎩

∑

∑

' min ' , ' is in the mai o

f ldr

r r r− r

' r

Microstructure reconstruction


• Reconstruction a polycrystalline microstructure from the lower dimensional space.

• Local linear interpolation:

Microstructure reconstruction


• Finite elements method:

• Effective thermal conductivity:

,

m

meff

TKmK where A AdxT

mΩ

∂∂= =∂

∂

∫

Application: Effective thermal conductivity


Physical Modeling in Random Microstructures


Problem definition

Schematic of the problem

We are interested in modeling the thermalevolution of macro-scale devices.

We develop a strategy to incorporate theeffect of the micro-scale heterogeneousrandom media on the macro-scaletemperature distribution.

The characteristic length scale, L , of thedevices of the order of meters. At the lowerscale, the underlying microstructure isusually random and heterogeneous.

We limit ourselves to the analysis ofdevices made from two-phase materials.


Modeling multiscale diffusion process in random heterogeneous media

Each integration point isassociated with a stochasticmic ros t ruc tu re , wh ich isobtained from the non-linearmodel reduction technique.

The stochastic heterogeneous microstructure in the RVE is interrogated to compute the stochastic thermal response to prescribed temperature boundary conditions. Using this response, the effective stochastic thermal conductivity is determined.


Stochastic multiscale up-scalingThe starting point is a set of experimental statistics that the underlying microstructure

(thermal conductivity) satisfies. Assume that the largest correlation length is . A flow chart ofthe complete stochastic multiscale framework is given below:

cl

Step 1: Construct a coarse discretization of the macro-scale domain. Ensure that eachelement has element size ch l

Step 2: For each integration point in each element, consider a stochastic RVE, with adiscretization

Step 3: In each RVE construct a stochastic reduced-order model for the fine-scale stochasticthermal conductivity via non-linear model reduction technique.mk

Step 4: Virtually interrogate each RVE. Solve the stochastic PDE in the RVE, using .Compute the effective stochastic thermal conductivity in the RVE. Compute the PDF of ,from the stochastic effective thermal conductivity.

mkeffk

Step 5: Perform Monte Carlo sampling to compute statistics of the meso-scale temperature.For , perform the following steps p times: Assign effective thermal conductivity valuesat each integration points in the macro-scale, by sampling from the respective PDF of theeffective thermal conductivity. Solve the macro-scale problem for the temperature distribution

p +∈


Numerical results

Microstructure

Correlation PDF of the effective thermal conductivity


Numerical results

Mean

Standard Deviation


Macroscopic property variability induced by microstructural uncertainty

Limited set of input microstructures computed using

phase field technique

Statistical samples of microstructure at certain collocation points computed using maximum

entropy techniqueCompute the statistical variation of the macro-

scale properties

Statistical variation of homogenized stress-strain curve. Aluminum polycrystal with rate-independent strain hardening. Pure tensile

test.


Stochastic framework

17

PREPROCESSING: Set input stochastic model, set coarse discretization, set

threshold

Compute the stochastic multiscale basis functions for each coarse element

Compute the stochastic coarse-scale fluxes

Solution Methodology

Solve localized stochastic sub-grid problem with adaptive spare grid collocation

POSTPROCESSING: Compute multi-scale stochastic flux (moments and other statistics)

on-the-fly

Solve stochastic coarse-scale problem with adaptive sparse grid collocation

1 1( , ) ( . , ) ( , )ms msf i f i fk k gφ− −− ∇ = −v N v v

( , . ) ( , . )msf i fw w g∇ = ∇N

1 1

0

( , ) ( , ( )) ( . , ) ( . , )

c c c c c c

c

k k pn p

− −+ Φ − ∇= −

v u v u vv

( , . ) ( , )c c cw w f∇ =u


Information theory

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

MAXENT to generate microstructure

samples RVE

Stochastic homogenization

Effective stiffness tensor ( )C x

Macro FE Analysis


Underlying Microstructure

Continuum representation of texture in Rodrigues space

Fundamental part of Rodrigues space

Variation of final micro-structure due to various sources of uncertainty

Sources of uncertainty (texture)


Variation of macro-scale properties due to multiple sources of uncertainty on different scales

Uncertain initial microstructure

use Frank-Rodrigues space for continuous representation

Limited snap shots of a random field 0( , )A s ω

Use Karhunen-Loeve expansion to reduce this

random filed to few random variables

0 1 2 3( , , , )A s Y Y Y

Considering the limited information Maximum Entropy principle should be used to obtain pdf for these random variables

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0.000 0.002 0.004 0.006 0.008 0.010

Effective strain

Effe

ctiv

e st

ress

(MPa

)

Use Rosenblatt transformation to map these random variables to hypercube

Use Stochastic collocation to obtain the effect of these random initial texture on final macro-scale properties.


Evolution of texture

Any macroscale property < χ > can be expressed as an expectation value if the corresponding single crystal property χ (s ,t) is known.

• Determines the volume fraction of crystals within a region R' of the fundamental region R

• Probability of finding a crystal orientation within a region R' of the fundamental region

• Characterizes texture evolution

ORIENTATION DISTRIBUTION FUNCTION – A(s,t)

ODF EVOLUTION EQUATION – LAGRANGIAN DESCRIPTION

( , ) ( , ) ( , ) 0A s t A s t v s tt

∂+ ∇ ⋅ =

∂

'

'( ) ( , )ℜ

ℜ = ∫fv A s t dv

∫

∫

ℜ

ℜ=dvtsA

dvtsAtsXX

),(

),(),(


Karhunen-Loeve Expansion:

and is a set of uncorrelated random variables whose distribution depends on the type of stochastic process.

( )ωiY

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10

Number of Eigenvalues

Ener

gy c

aptu

red

Then its KLE approximation is defined as

where and are eigenvalues and eigenvectors of λi Cif

1

1=

= ∑M

ii

A AM

1

1 ( ) ( )1 =

= − −− ∑

MT

i ii

A A A AM

C

Representing the uncertain micro-structure

Let be a second-order stochastic process defined on a closed spatial domain D and a closed time interval T. If are row vectors representing realizations of then the unbiased estimate of the covariance matrix is

0 ( , )A s ω

0A1,..., MA A2L

∑∞

=

+=1

00 )()()(),(i

iii YsfsAsA ωλω


Karhunen-Loeve Expansion

2

1 , , 1:λ

= − =ji j i l

i

Y A A f j N

( )ωiY can be obtained byRealization of random variables

where denotes the scalar product in .2l

NR

The random variables have the following two properties( )ωiY

[ ]( ) 0

( ) ( )

ω

ω ω δ

=

⎡ ⎤ =⎣ ⎦

i

i j ij

E Y

E Y Y

1Y2Y

3Y


( ) =1

E{ ( )}=

∫D

p dY Y

g Y f

Obtaining the probability distribution of the random variables using limited information

•In absence of enough information, Maximum Entropy principle is used to obtain the probability distribution of random variables.

( ) =- p( )log(p( ))d∫S p Y Y Y

•Maximize the entropy of information considering the available information as set of constraints

0( ) exp( , )μ

= −Dp 1 cY λ g(Y)

⎪⎪⎩

⎪⎪⎨

⎧

=

==

lkN YYg

YgYg

)(

)()(

22

11

Y

YY


A simple compression mode is assumed with an initial texture represented by a random field A

The random field is approximated by Karhunen-Loeve approximation and truncated after three terms.

The correlation matrix has been obtained from 500 samples. The samples are obtained from final texture of a point simulator subjected to a sequence of deformation modes with two random parameters uniformly distributed between 0.2 and 0.6 sec^-1 (example1)

0

( , ; ) ( , ; ) ( , ) 0

( ,0; ) ( , )

ω ω

ω ω

∂+ ∇⋅ =

∂=

A r t A r t v r tt

A r A r

Numerical ExampleExample: The effect of uncertainty in initial texture on macro-scale material properties for FCC copper


Numerical Example

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10

Number of Eigenvalues

Ener

gy c

aptu

red

Step1. Reduce the random field to a set of random variables (KL expansion)

∑∞

=

+=1

00 )()()(),(i

iii YsfsAsA ωλω


Numerical Example

Enforce positiveness of texture

Step2. In absence of sufficient information,use Maximum Entropy to obtain the joint probability of

these random variables

1( )p Y

1Y

2( )p Y

2Y

3( )p Y

3Y1Y2Y

3Y


Numerical ExampleStep4. Use sparse grid collocation to obtain the stochastic characteristic of

macro scale properties

Mean of A at the end of deformation

process

Variance of A at the end of deformation

process

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0.000 0.002 0.004 0.006 0.008 0.010

Effective strain

Effe

ctiv

e st

ress

(MPa

)

Variation of stress-strain response

FCC copper

( )E MPa

1.41e05

2

( ) (MPa)Var E

4.42e08 Adaptive Sparse grid (level 8)

MC 10,000 runs4.39e081.41e05

Kouchmeshky & Zabaras (2009a)

http://mpdc.mae.cornell.edu/Publications/PDFiles/TextureUncertaintyA.pdf


Simple Karhunen-Loeve Expansion

The realizations of random variables

One can use methods from previous slides to construct the probability distributions of these random variables at each integration point j. Now if the random variables at different integration points are correlated to each other then the aforementioned methodology has no means of figuring that out in another words it can not see the correlation between the set of random variables from different integration points.

Bi-orthogonal Karhunen-Loeve Expansion

Quantifying the effect of uncertain initial texture

Kouchmeshky & Zabaras (2009b)

http://mpdc.mae.cornell.edu/Publications/PDFiles/UncertaintyMultiscale.pdf

An eigenvalue problem in Rodrigues space


Step3: Construct the Covariance using the snapshots

Step4: Obtain the eigenvalues and eigenvectors: ;

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15

Mode number

Cap

ture

d En

ergy

Step5: Obtain the spatial modes

Step6: Decompose the spatial modes using the polynomial Chaos:

are in a one to one correspondent to the Hermite polynomials .

Construct the reduced order representation of the texture Use the reduced order model to reconstruct

the texture

0

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8Polynomial order

Rel

ativ

e er

ror % B

GE

Top left: Distribution of Bulk modulus, Top right: Distribution of Youngmodulus, Bottom left: Distribution of Shear modulus. For one point on macro-scale. The bars represent the distribution obtained using the realizations of the texture and the solid line is the distribution obtained using the reduced order model for the texture.

The relative error with respect to the order of polynomial chaos.

Step1: Start from realizations of the texture

Step2: Transform the realization using


The effect of uncertainty in the initial texture of the work-piece on the macro-scale properties


Mean(G)

Var(G)

Mean(B)

Var(B)

Mean(E)

Var(E)

The effect of uncertainty in the initial texture

After the interpolants in the stochastic space for the texture have been obtained one can use them to obtain the realizations of the texture. Using these realizations statistics of the macro-scale properties can be obtained.


The effect of uncertainty in the initial texture

Comparison of Mean and variance of the macro-scale properties with MC, Top left: Bulk modulus, Top right: Young modulus, Bottom: Shear modulus

Relative error for Mean(G)

Relative error for Var(G)

Relative error for Mean(B)

Relative error for Var(B)

Relative error for Mean(E)

Relative error for Var(E)


Open related problems

The dimensionality of the stochastic space is currently assumed in the bio-orthogonal decomposition. A computationally efficient data-driven methodology is needed to compute the optimal stochastic dimensionality.

Extend the biorthogonal decomposition to ``non-linear stochastic POD’’ for the evolution of the microstructure and macro-system response.

Development of statistical models to substitute the expensive fine-scale solvers. Build a Bayesian regression from the high-dimensional input space to the lower-dimensional output space. These models are trained “on-the-fly”.

Condense the information furnished by the fine-scale details but also identify hidden physical connections between microscale features and macroscale response.

Multiscale stochastic modeling in polycrystalline materials: Challenges, perspectives...

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