babak kouchmeshky uncertainty propagation in deformation processes materials process design and...

Babak Kouchmeshky

Uncertainty propagation in deformation processes

Materials Process Design and Control Laboratory

Sibley School of Mechanical and Aerospace Engineering169 Frank H. T. Rhodes Hall

Cornell University

Ithaca, NY 14853-3801

Email: [email protected]

Presentation for the B exam

Modeling uncertainty propagation in deformation processes

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


Variation of macro-scale properties due to multi-scale sources of uncertainty

•Macro-scale properties depend on the underlying micro-structure•Represent the micro-structure in a continuous framework using Orientation distribution function (ODF)

2Y 3Y

1YMaxEnt

All possible macro-scale properties due to the effect of multiple sources of uncertainties on

multi-scales

Error bars in stress-strain response due to the effect of multiple sources of uncertainties on multi-scales.

Continuum representation of texture in Rodrigues

spaceUnderlying

Microstructure

Obtain the variability of macro-scale properties due to uncertainties on multi-scales.

Problem definition

•Uncertainty in process parameters

•Uncertainty in underlying micro-structure

Approach•Using Karhunen-Loeve (KL) expansion the uncertainty on micro-structure is represented by a set of random variables.

•In absence of sufficient information, Maximum Entropy (MaxEnt) should be used to obtain the joint probability of these random variables.

KL

Approach (cont.)

•The effect of various multi-scale uncertainties on macro-scale properties can be quantified by solving the governing stochastic partial differential equations (SPDE).

•Use sparse grid collocation to solve the SPDE’s.

•All possible macro-scale properties due to uncertainty can be represented by a convex hull.

•The convex hull can be used in calculating the risk associated with obtaining macro-scale properties less than specific critical values.




Problem definition

•Sources of uncertainty:

- Process parameters - Micro-structural texture

•Obtain the variability of macro-scale properties due to multiple sources of uncertainty in absence of sufficient information that can completely characterizes them.




Sources of uncertainty (process parameters)

1 2 3 4

5 6 7 8

1 0 0 0 0 0 0 1 0 0 0 1

0 0.5 0 0 1 0 1 0 0 0 0 0

0 0 0.5 0 0 1 0 0 0 1 0 0

0 0 0 0 1 0 0 0 1 0 0 0

0 0 1 1 0 0 0 0 0 0 0 1

0 1 0 0 0 0 1 0 0 0 1 0

L

Since incompressibility is assumed only eight components of L are independent.

The coefficients correspond to tension/compression,plain strain compression, shear and rotation.

i




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 (Micro-structural 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

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

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 (r ,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



( , )( , ) ( , ) 0

A s tA s t v s t

t

( , ) ( , )

s t A s t dv

'

'( ) ( , )

fv A s t dv


Constitutive theoryConstitutive theory

D = Macroscopic stretch = Schmid tensor = Lattice spin W = Macroscopic spin = Lattice spin vector



Reorientation velocity

Symmetric and spin components

Velocity gradient

Divergence of reorientation velocity

vect( )

1L FF

Polycrystal plasticityInitial configuration

Bo BF*Fp

F

Deformed configuration

Stress free (relaxed) configuration

n0

s0

n0

s0

ns

(2) Ability to capture material properties in terms of the crystal properties

(1) State evolves for each crystal




Karhunen-Loeve Expansion:

0 01

( , ) ( ) ( , ) ( )i i ii

A s A s f s t Y

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

( )iY

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

En

erg

y c

ap

ture

d

Then its KLE approximation is defined as

where and are eigenvalues and eigenvectors of iCif

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

0A1,..., MA A

2L




Karhunen-Loeve Expansion

2

1, , 1:

j

i j i li

Y A A f j N

( )iY can be obtained byRealization of random variables

where denotes the scalar product in .2l

NR

The random variables have the following two properties( )iY

( ) 0

( ) ( )

i

i j ij

E Y

E Y Y

1Y

2Y

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( ))dS p Y Y Y

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

1 1

2 2

( ) ( )

( ) ( )

( ) ( )N k l

g E v

g E v

g E vv

v

v

v

0( ) exp( , )

Dp 1 cY λ g(Y)




Maximum Entropy Principle

TargetM0 1.0001 1M1 -1.30E-04 0M2 2.51E-06 0M3 4.83E-05 0M4 9.98E-01 1M5 -1.89E-04 0M6 3.54E-04 0M7 1.009E+00 1M8 5.93E-04 0M9 9.95E-01 1

Constraints at the final iteration1Y

1( )p Y

2Y

3Y

3( )p Y

2( )p Y




Inverse Rosenblatt transformation

1

2

11 1 1

12 2|1 2

1|1:( 1)

( ( ))

( ( ))

( ( ))NN N N N

Y P P

Y P P

Y P P

(i) Inverse Rosenblatt transformation has been used to map these random variables to 3 independent identically distributed uniform random variables in a hypercube [0,1]^3.

(ii) Adaptive sparse collocation of this hypercube is used to propagate the uncertainty through material processing incorporating the polycrystal plasticity.

STOCHASTIC COLLOCATION STRATEGYSTOCHASTIC COLLOCATION STRATEGY

Use Adaptive Sparse Grid Collocation (ASGC) to construct the complete stochastic solution by sampling the stochastic space at M distinct points

Two issues with constructing accurate interpolating functions:

1) What is the choice of optimal points to sample at?

2) How can one construct multidimensional polynomial functions?




1. X. Ma, N. Zabaras, A stabilized stochastic finite element second order projection methodology for modeling natural convection in random porous media, JCP

2. D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comp. 24 (2002) 619-644

3. X. Wan and G.E. Karniadakis, Beyond Wiener-Askey expansions: Handling arbitrary PDFs, SIAM J Sci Comp 28(3) (2006) 455-464

1

( ) ( ,..., )N

ff

Assuming a finite dimension for the stochastic space, one can represent a function in random space as

f

http://mpdc.mae.cornell.edu/Publications/journalpapers.htm

http://mpdc.mae.cornell.edu/Publications/journalpapers.htm

From one-dimension to higher dimensions

Denote the one dimensional interpolation formula as

with the set of support nodes:

In higher dimension, a simple case is the tensor product formula

For instance, if M=10 dimensions and we use k points in each direction

Number of Number of points in each points in each direction, kdirection, k

Total number Total number of sampling of sampling pointspoints

22 10241024

33 5904959049

44 1.05x101.05x1066

55 9.76x109.76x1066

1010 1x101x101010

This quickly becomes impossible to use.

One idea is only to pick the most important points from the

tensor product grid.




1

1 1

1

1

11

( ( )) ( ).m

i i ij j

j

u ff a

1 1 1

1 1 1| 0,1 for 1,2,..., i i i

j jX j m

1

1 1 1

1 1

1

11 1

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

N N N

N N

N

m mi i i i i i

N j j j jj j

u u ff a a

FROM ONE-DIMENSION to MULTI-FROM ONE-DIMENSION to MULTI-DIMENSIONSDIMENSIONSExtension to multiple dimensions: Use

simple tensor product strategyNumber of Number of points in points in each each direction, direction, kk

Total Total number number of of sampling sampling pointspoints

22 10241024

33 5904959049

44 1.05x101.05x1066

55 9.76x109.76x1066

1010 1x101x101010

Results in combinatorial explosion

One idea is only to pick the more important points from the tensor product grid

1. S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl., 4 (1963) 240--243

Smolyak (1963) came up with a set of rules to construct such products1




1

1

1 1

1 1

1

1

1 1

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

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

N

N

N N

N N

N

i i

N

m mi i i i

j j j jj j

u u f

f a a

where , is the multi-index representation of the support nodes. N is the dimension of the function f and q is an integer (q>N). k = q-N is called the depth of the interpolation. As q is increased more and more points are sampled.

1

...d

i ii

ADAPTIVE SPARSE GRID COLLOCATIONADAPTIVE SPARSE GRID COLLOCATIONAnisotropic sampling for interpolating functions with steep gradients and other localized phenomena.

Have to detect it on-the-fly.

Utilize piecewise linear interpolating functions: local support

Utilize hierarchical form of basis function: provides natural stopping criterion

Define a threshold value. If magnitude of the hierarchical surplus is greater than this threshold, refine around this point.

Add 2N neighbor points. Scales linearly instead of O(2N)

Circumvent the curse-of-dimensionality?

1. B. Ganapathysubramanian and N. Zabaras, Sparse grid collocation schemes for stochastic natural convection problems, J. Comp. Phys 225 (2007) 652-685

2. X. Ma and N. Zabaras, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, submitted




ADAPTIVE SPARSE GRID COLLOCATIONADAPTIVE SPARSE GRID COLLOCATIONGiven a user-defined threshold, ε>0.

For points where w > ε, refine the grid to include 2N daughters. Compute the hierarchical surpluses at these new points.

Refine until all w< ε or maximum depth of interpolation is reached

Implementation:

Keep track of uniqueness of new points

Efficient searching and inserting

Parallelizability

Error estimate of the adaptive interpolant







Numerical Examples

A sequence of modes is considered in which a simple compression mode is followed by a shear mode hence the velocity gradient is considered as:

where are uniformly distributed random variables between 0.2 and 0.6 (1/sec).

1 2 and

Example 1 : The effect of uncertainty in process parameters on macro-scale material properties for FCC copper

Number of random variables: 2




( )E MPa

1.28e05

2( ) (MPa)Var E

4.02e07

3.92e071.28e05

Adaptive Sparse grid (level 8)

MC (10000 runs)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1

2

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 2 4 6 8 10

Interpolation level

Rel

ativ

e E

rror Mean

Variance

Numerical Examples (Example 1)




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; ) ( , )

A r tA r t v r t

tA r A r


Example 2 : The effect of uncertainty in initial texture on macro-scale material properties for FCC copper





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

En

erg

y c

ap

ture

d

0 01

( , ) ( ) ( , ) ( )i i ii

A s A s f s t

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





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

3Y1Y

2Y

3Y





Rosenblatt M, Remarks on multivariate transformation, Ann. Math. Statist.,1952;23:470-472

Rosenblatt transformation

Step3. Map the random variables to independent identically distributed uniform random variables on a hypercube [0 1]^3

1 2 3, ,Y Y Y

1 2 3, , 1

2

11 1 1

12 2|1 2

1|1:( 1)

( ( ))

( ( ))

( ( ))NN N N N

Y P P

Y P P

Y P P

1 1 2 1 2 3( ), ( , ), ( , , )p Y p Y Y p Y Y Y are needed. The last one is obtained from the MaxEnt problem and the first 2 can be obtained by MC for integrating in the convex hull D.

1( )p Y

1Y

2( )p Y

2Y

3( )p Y

3Y




Numerical Examples (Example 2)Step4. 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

Eff

ec

tiv

e s

tre

ss

(M

Pa

)

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

A model reduction of the multiscale input for uncertainty quantification in multiscale

deformation processes





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

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

Ca

ptu

red

En

erg

y

0.620.520.420.320.220.120.02

-0.08-0.18-0.28-0.38-0.48

0.480.380.280.180.08

-0.02-0.12-0.22

0.220.170.120.070.02

-0.03-0.08-0.13-0.18-0.23-0.28-0.33

0.20.10

-0.1-0.2-0.3-0.4-0.5-0.6-0.7

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 8

Polynomial order

Re

lati

ve

err

or

%

B

G

E

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




Use Rosenblatt transform to transfer the coordinates of the points inside a hypercube to the realizations of the random variables.

Propagation of uncertainty

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

In this method the stochastic space is sampled using collocation points. The coordinates of each of these collocation points correspond to the realizations of the random variables driving the problem. So, for each point we have the realizations of the random variables. These realizations can be used to solve a deterministic problem of deformation process and obtain the final texture.

Q: How to quantify the effect of uncertainty in initial texture on the final texture of a work-piece that has gone through a deformation process?

A: Collocation strategy is used for quantifying the effect of uncertainty.

Use the realizations of the random variables to construct the realizations of the initial texture

The results at the collocation points are used to construct interpolants in the stochastic space.

Q: What about using global polynomial chaos expansion method?

A: Needs significant changes in the formulation of the corresponding deterministic problem.

Q: But you said you are using global polynomial chaos expansion ?

A: Yes. It was used in the bi-orthogonal framework to approximate the initial random field. It could also be used in the stochastic PDEs along a stochastic Galerkin projection to reduce the SPDEs to a set of deterministic PDEs. But the latter step in not necessary and once you have a model to obtain the realization of the initial random field you can propagate uncertainty using any stochastic tools.




Problem definition

-Obtain the effect of uncertainty in initial texture on macro-scale material properties

Uncertain initial microstructure




Deterministic multi-scale deformation process




Implementation of the deterministic problem

Meso

Macro

formulation for macro scale

Update macro displacements

Texture evolution update

Polycrystal averaging for macro-quantities

Integration of single crystal slip and twinning laws

Macro-deformation gradient

microscale stressMacro-deformation gradient

Micro

( , )( , ) ( , ) 0

A s tA s t v s t

t




THE DIRECT CONTACT PROBLEM

r

n

Inadmissible region

Referenceconfiguration

Currentconfiguration

Admissible region

ImpenetrabilityImpenetrability ConstraintsConstraints

Augmented Lagrangian Augmented Lagrangian approach to enforce approach to enforce impenetrabilityimpenetrability

Polycrystal average of orientation

dependent property

Continuous representation of texture




REORIENTATION & TEXTURINGREORIENTATION & TEXTURING




Convergence of the deterministic problem

MPa MPa

Bulk modulus




MPa MPa


Young modulus




MPa MPa


Shear modulus




Convergence of ODF




Stochastic multi-scale deformation process




H

Curved surface parametrization – Cross section can at most be an ellipse

Model semi-major and semi-minor axes as 6 degree bezier curves

6

1

51

3 33

2 55

( ) ( )

(1.0 ) (1.0 5.0 )

20.0(1.0 )

6.0(1.0 )

i ii

R

4 2

2

2 44

66

( ) 0

15.0(1.0 )

15.0(1.0 )

R

/z H

Random parameters

2 3, N(1,0.2) 1 4 5 6 0.05 Deterministic

parameters

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

STOCHASTIC COLLOCATION STRATEGYSTOCHASTIC COLLOCATION STRATEGY

Use Adaptive Sparse Grid Collocation (ASGC) to construct the complete stochastic solution by sampling the stochastic space at M distinct points

Two issues with constructing accurate interpolating functions:

1) What is the choice of optimal points to sample at?

2) How can one construct multidimensional polynomial functions?




1. X. Ma, N. Zabaras, A stabilized stochastic finite element second order projection methodology for modeling natural convection in random porous media, JCP

2. D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comp. 24 (2002) 619-644

3. X. Wan and G.E. Karniadakis, Beyond Wiener-Askey expansions: Handling arbitrary PDFs, SIAM J Sci Comp 28(3) (2006) 455-464

Assuming a finite dimension for the stochastic space:

=




Mean(G)Mean(G)

Var(G)Var(G)

Mean(B)Mean(B)

Var(B)Var(B)

Mean(E)Mean(E)

Var(E)Var(E)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

The effect of uncertainty in the initial geometry




Error of Mean(B)Error of Mean(B)

Error of Var(B)

Error of Var(B)

Comparison with Monte-CarloComparison with Monte-Carlo

0.01050.00550.0005

0.0620.0320.002

0.00820.00420.0002

0.0420.0220.002

Error of Mean(E)Error of Mean(E)

Error of Var(E)

Error of Var(E)

0.02050.01050.0005

0.0840.0440.004

Error of Mean(G)Error of Mean(G)

Error of Var(G)

Error of Var(G)

m

m

X XError

X

: Macro-scale property

calculated using sparse grid

X

: Macro-scale property

calculated using MC mX




minˆ( , , ) ( , , ) ( , , ) log( ( , , ) )a x s a x s a x s A x s A

1

ˆ( , , ) ( ) ( , )i i ii

a x s s x

A MODEL REDUCTION of the MULTISCALE INPUT

Current method

( , ) : ( ) ( )i j i j ijs s ds

#( , )i j ij

#( , ) ,D

f g f g dx , ( ) ( ) ( )f g f g P d

1- D. Venturi, X. Wan, G.E. Karniadakis, J. Fluid Mech. 2008, vol 606, pp 339-367

where are modes strongly orthogonal in Rodriguesspace and are spatial modes weakly orthogonal in space

( )i s( , )i x

1




Reconstructing a stochastic microstructure

Step1: Construct the covariance using the snapshots

Step2: Obtain the eigenvalues and eigenvectors: ;0.620.520.420.320.220.120.02

-0.08-0.18-0.28-0.38-0.48

0.480.380.280.180.08

-0.02-0.12-0.22

0.220.170.120.070.02

-0.03-0.08-0.13-0.18-0.23-0.28-0.33

0.20.10

-0.1-0.2-0.3-0.4-0.5-0.6-0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15

Mode number

Ca

ptu

red

En

erg

y




Reconstructing a stochastic microstructure

Step3: Obtain the spatial modes

Step4: Decompose the spatial modes using the polynomial Chaos:

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

0

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8

Polynomial order

Re

lati

ve

err

or

%

BGE

BB

310 MPa

EE

310 MPa

310 MPa

GG

Comparison between the original microstructure and the reduced order one

Bars: obtained using the realizations of the texture that were used in constructing the covariance.Solid line: Obtained from the sampling the random variables and constructing the texture using the reduced order modeling.Bars: obtained using the realizations of the texture that were used in constructing the covariance.Solid line: Obtained from the sampling the random variables and constructing the texture using the reduced order modeling.




0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 2 4 6 8Polynomial order

Re

lati

ve

err

or

% EGB

BB

310 MPa

EE

310 MPa

310 MPa

GG





0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 2 4 6 8

Polynomial order

Re

lati

ve

err

or

%

EGB

BB

310 MPa

EE

310 MPa

310 MPa

GG





BB

310 MPa

EE

310 MPa

310 MPa

GG

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2 4 6 8

Polynomial order

Re

lati

ve

err

or

% E

G

B





0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 2 4 6 8

Polynomial Order

Re

lati

ve

err

or B

G

E

BB

310 MPa

EE

310 MPa

310 MPa

GG





Mean(G)Mpa

Mean(G)Mpa

Mean(B)Mpa

Mean(B)Mpa

Mean(E)Mpa

Mean(E)Mpa

OriginalOriginal

ReconstructedReconstructed

Mean(G)Mpa

Mean(G)Mpa

Mean(B)Mpa

Mean(B)Mpa

Mean(E)Mpa

Mean(E)Mpa





Var(G)Var(G)

Var(B)Var(B) Var(E)Var(E)

OriginalOriginal

ReconstructedReconstructed

Var(G)Var(G)

Var(B)Var(B)

Var(E)Var(E)





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




Mean(G)Mean(G)

Var(G)Var(G)

Mean(B)Mean(B)

Var(B)Var(B)

Mean(E)Mean(E)

Var(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.

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

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 Mean(G)

Relative error for Var(G)

Relative error for Var(G)

Relative error for Mean(B)

Relative error for Mean(B)

Relative error for Var(B)

Relative error for Var(B)

Relative error for Mean(E)

Relative error for Mean(E)

Relative error for Var(E)

Relative error for Var(E)




Conclusion

•The first half of the slides:

•The effect of uncertain process parameters and initial texture on the convex hull of the macro-scale properties is investigated.

•The second half of the slides:

•The previous method is extended to the multi-scale.

•A model reduction of the multi-scale input for the texture uncertainty in multi-scale deformation process has been provided.




Publications and presentations since Aug. 2006

• B. Kouchmeshky and N. Zabaras, A model reduction of the uncertain input for quantifying the effect of uncertainty in a multi-scale stochastic problem, In preparation.

• B. Kouchmeshky and N. Zabaras, The effect of multiple sources of uncertainty on the convex hull of material properties", Computational Materials Science, submitted.

• B. Kouchmeshky and N. Zabaras, Modeling the response of HCP polycrystals deforming by slip and twinning using a finite element representation of the orientation space, Computational Materials Science, Vol. 45, Issue 4, pp. 1043-1051, 2009

Publications:

Presentations:•B. Kouchmeshky and N. Zabaras, “Uncertainty quantification in multiscale deformation processes”, 2009, 10th U.S. National Congress on Computational Mechanics.

•B. Kouchmeshky and N. Zabaras, " Modeling uncertainty propagation in deformation processes", 2009 TMS Annual Meeting & Exhibition.

•B. Kouchmeshky and N. Zabaras, " Advances on multi-scale design of deformation processes for the control of material properties”, 2009 TMS Annual Meeting & Exhibition.

• B. Kouchmeshky and N. Zabaras, "A microstructure-sensitive design approach for controlling properties of HCP materials", 2008 TMS Annual Meeting & Exhibition.

•B. Kouchmeshky and N. Zabaras, "A simple non-hardening rate-independent constitutive model for HCP polycrystals deforming by slip and twinning”, 2008 TMS Annual Meeting & Exhibition.

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