Modeling Uncertainty Propagation in Deformation Processes

8/14/2019 Modeling Uncertainty Propagation in Deformation Processes

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Modeling uncertainty propagation indeformation processes

Babak Kouchmeshky

Nicholas Zabaras

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

101 Frank H. T. Rhodes HallCornell UniversityIthaca, NY 14853-3801

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

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOOR R N NEELLLL U N I V E R S I T YCCOOR R N NEELLLL U N I V E R S I T Y




CCOOR R N NEELLLL U N I V E R S I T Y

CCOOR R N NEELLLL U N I V E R S I T Y

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 sufficientinformation 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

β β β β

β β β β

= − + + + +

− −

− − + + + −

L

Since incompressibility is assumed only eight components of

L are independent.

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

β 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)




CCOOR R N NEELLLL CCOOR R N NEELLLL U N I V E R S I T Y

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

Uncertain initial microstructure

use Frank-

Rodrigues space for continuous

representation

Limited snap shots of a random field0( , ) A s ω

Use Karhunen-Loeveexpansion 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 tohypercube

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 beexpressed 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 t A s t v s t

t

∂+ ∇⋅ =

∂

( , ) ( , )ℜ

Χ = Χ∫ s t A s t dv

'

'( ) ( , )ℜ

ℜ = ∫ f v 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( )ω = Ω

1 L FF −= &

Polycrystal plasticityInitial configuration

B o B

F * F p

F

Deformedconfiguration

Stress free (relaxed)configuration

n 0

s 0

n 0

s 0

n s

(2) Ability to capture material properties

in terms of the crystal properties

(1) State evolves for each crystal





Karhunen-Loeve Expansion:

0 01

( , ) ( ) ( , ) ( )ω λ ω ∞

=

= +

∑i i i

i

A r A r f r t Y

and is a set of uncorrelated random variables whose distribution depends on

the type of stochastic process.

( )ω i

Y

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

E n e r g y c a p t u r e d

Then its KLE approximation is defined as

where and are eigenvalues and eigenvectors of λ i

%Ci f

1

1=

= ∑M

i

i

A AM

1

1( ) ( )

1 =

= − −−

∑%M

T

i i

i

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 representingrealizations of then the unbiased estimate of the covariance matrix is

0( , ) A r ω

0 A1,..., M A A

2 L





Karhunen-Loeve Expansion

2

1 , , 1:λ

= − = j

i j i l i

Y A A f j N

( )ω i

Y can be obtained byRealization of random variables

where denotes the scalar product in .2l

N R

The random variables have the following two properties( )ω iY

[ ]( ) 0

( ) ( )

ω

ω ω δ

=

=

i

i j ij

E Y

E Y Y

1Y

2Y

3Y





( ) =1

E ( )=

∫ D

p d Y Y

gY 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 availableinformation as set of constraints

1 1

2 2

( ) ( )

( ) ( )

( ) ( )N k l

g E v

g E v

g E v v

=

=

=

v

v

v

M0

( ) exp( , ) µ

= − D

p 1 cYλ g(Y)





Maximum Entropy Principle

Target

M0 1.0001 1

M1 -1.30E-04 0

M2 2.51E-06 0

M3 4.83E-05 0

M4 9.98E-01 1

M5 -1.89E-04 0

M6 3.54E-04 0

M7 1.009E+00 1

M8 5.93E-04 0

M9 9.95E-01 1

Constraints at the final iteration

1Y

1( ) p Y

2Y

3Y

3( ) p Y

2( ) p Y





Inverse Rosenblatt transformation

1

2

1

1 1 1

1

2 2|1 2

1

|1:( 1)

( ( ))

( ( ))

( ( )) N N N N N

Y P P

Y P P

Y P P

ξ

ξ

ξ

ξ

ξ

ξ

−

−

−−

=

=

=

M

(i) Inverse Rosenblatt transformation has been used to map

these random variables to 3 independent identicallydistributed 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

, 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

A s t A s t ω ξ ξ =

Since the Karhunen-Loeve approximation reduces the infinite size of stochastic

domain representing the initial texture to a small space one can reformulate the

SPDE in terms of these N ‘stochastic variables’

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

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http://www.ingentaconnect.com/content/klu/jomp/2006/00000027/F0030001/00009038;jsessionid=2dp0nh1nwptcf.alice

http://www.ingentaconnect.com/content/klu/jomp/2006/00000027/F0030001/00009038;jsessionid=2dp0nh1nwptcf.alice

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





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

andα α

Example 1 : The effect of uncertainty in process parameters on

macro-scale material properties for FCC copper

1 1

2 1 2

1 0 00 0.5 0 t<T

0 0 0.5

0 1 0

1 0 0 T <t<T

0 0 0

L

L

α

α

= −

−

=

Number of random variables: 2

(





( ) E MPa

1.28e05

2( ) (MPa)Var E

4.02e07

3.92e071.28e05

Adaptive Sparsegrid (level 8)

MC (10000 runs)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1ξ

2ξ

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 2 4 6 8 1 0

Interpolation lev

R e l a t i v e E r r o

rMean

Variance

Numerical Examples (Example 1)

N i l E l (





Example 2 : The effect of uncertainty in process parameter

(forging velocity ) on macro-scale material properties in a closed

die forming problem for FCC copper


L e v e

l

0 0.2 0.4 0.6 0.8 1

2

4

6

8

10

1ξ

Number of random variables: 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; ) ( , )

ω ω

ω ω

∂+ ∇ ⋅ =

∂=

A r t A r t v r t

t

A r A r


Example 3 : 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

E n e r g y

c a p t u r e

0 0

1

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

i

A r A r f r t ω λ ξ ω

∞

== + ∑

Step1. Reduce the random field to a set of random

variables (KL expansion)

N i l E l (






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

3Y 1Y

2Y

3Y

N i l E l (






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]^31 2 3, ,Y Y Y

1 2 3

, ,ξ ξ ξ

1

2

1

1 1 1

1

2 2|1 2

1

|1:( 1)

( ( ))

( ( ))

( ( )) N N N N N

Y P P

Y P P

Y P P

ξ

ξ

ξ

ξ

ξ

ξ

−

−

−−

=

=

=

M

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

N i l E l (




CCOORRNNEELLLLCCOORRNNEELLLLU N I V E R S I T Y


Step4. Use sparse grid collocation to obtain the stochastic characteristic of

macro scale properties

Mean of A at the endof deformation

process

Variance of A at theend 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

E f f e c t i v e s t r e s s ( M P a )

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

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Modeling Uncertainty Propagation in Deformation Processes

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