Transcript Document

Modeling uncertainty propagation in
deformation processes
Babak Kouchmeshky
Nicholas Zabaras
Materials Process Design and Control Laboratory
Sibley School of Mechanical and Aerospace Engineering
101 Frank H. T. Rhodes Hall
Cornell University
Ithaca, NY 14853-3801
URL: http://mpdc.mae.cornell.edu/
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Problem definition
•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
- Micro-structural texture
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Sources of uncertainty (process parameters)
0
0 
1
0 0 0 
0 1 0
0 0 1 
L  1 0 0.5
0    2  0 1 0    3 1 0 0    4  0 0 0  








0
0.5
0
0 0 1
0 0 0
1 0 0
0 0 0
0 1 0
0 0 1
0 0 0 
 5 0 0 1    6 1 0 0   7 0 0 0   8 0 0 1








0 1 0
0 0 0
1 0 0 
0 1 0 
Since incompressibility is assumed only eight components of
L are independent.
The  i coefficients correspond to tension/compression,plain
strain compression, shear and rotation.
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Sources of uncertainty (Micro-structural texture)
Continuum representation of
texture in Rodrigues space
Underlying Microstructure
Fundamental part of Rodrigues space
Variation of final micro-structure due
to various sources of uncertainty
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Variation of macro-scale properties due to multiple
sources of uncertainty on different scales
use FrankRodrigues space
for continuous
representation
Uncertain initial microstructure
Limited snap shots of a random field
Use Karhunen-Loeve
expansion to reduce this
random filed to few
random variables
Considering the limited information
Maximum Entropy principle should be used
to obtain pdf for these random variables
Use Stochastic collocation to obtain the
effect of these random initial texture on
final macro-scale properties.
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U
N
I
V
E
R
S
I
T
Y
A0 ( s, Y1 , Y2 , Y3 )
80.0
Effective stress (MPa)
Use Rosenblatt
transformation to map
these random variables
to hypercube
A 0 (s,  )
70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.0
0.000
0.002
0.004
0.006
0.008
0.010
Effective strain
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Evolution of texture
ORIENTATION DISTRIBUTION FUNCTION – A(s,t)
• 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
v f ( ) 
'
 A( s, t )dv
'
ODF EVOLUTION EQUATION – LAGRANGIAN DESCRIPTION
A( s, t )
 A( s, t )  v ( s, t )  0
t
Any macroscale property < χ > can be
expressed as an expectation value if the
corresponding single crystal property χ (r ,t)
is known.
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    ( s, t ) A( s, t )dv

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Constitutive theory
Velocity gradient
L  FF 1
Polycrystal plasticity Deformed
Initial configuration
configuration
F
s0
s n
n0
Symmetric and spin components
Bo
B
p
F
F*
s0
n0
Reorientation velocity
  vect()
Stress free (relaxed)
configuration
(1) State evolves for each crystal
(2) Ability to capture material properties
in terms of the crystal properties
Divergence of reorientation velocity
D = Macroscopic stretch
= Schmid tensor
= Lattice spin
W = Macroscopic spin
= Lattice spin vector
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Representing the uncertain micro-structure
Karhunen-Loeve Expansion:
Let A0 ( r,  ) be a second-order L2 stochastic process defined on a closed spatial
domain D and a closed time interval T. If A1 ,..., AM are row vectors representing
realizations of A then the unbiased estimate of the covariance matrix is
0
Then its KLE approximation is defined as

A0 ( r,  )  A0 ( r )   i f i ( r, t )Yi ( )
i 1
0.9
0.8
Energy captured
1 M
T
C
(
A

A
)
( Ai  A)

i
M  1 i 1
1 M
A
Ai

M i 1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
Number of Eigenvalues
i and f i are eigenvalues and eigenvectors of C
Yi ( ) is a set of uncorrelated random variables whose distribution depends on
where
and
the type of stochastic process.
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Karhunen-Loeve Expansion
Realization of random variables
Yi 
j
where
1
i
l2
Aj  A, f i
l2
Yi ( )
can be obtained by
, j  1: N
denotes the scalar product in
RN .
The random variables Yi ( ) have the following two properties
E Yi ( )  0
Y3
E Yi ( )Y j ( )    ij
Y1
Y2
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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.
•Maximize the entropy of information considering the
available information as set of constraints
S ( p ) =-  p(Y)log(p(Y))dY

p ( Y )dY =1
D
E{g (Y )}=f
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g1( v )  E (v1 )

g2( v )  E (v 2 )


g ( v )  E (v v )
k l
 N
p( Y)  1D c0 exp(  λ, g(Y)  )
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Maximum Entropy Principle
p(Y1 )
p(Y2 )
Y2
Y1
Constraints at the final iteration
p(Y3 )
Target
Y3
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M0
M1
M2
M3
M4
M5
M6
M7
M8
M9
1.0001
-1.30E-04
2.51E-06
4.83E-05
9.98E-01
-1.89E-04
3.54E-04
1.009E+00
5.93E-04
9.95E-01
1
0
0
0
1
0
0
1
0
1
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Inverse Rosenblatt transformation
(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.
Y1  P11 ( P1 (1 ))
Y2  P2|11 ( P2 ( 2 ))
YN  PN |1:( N 1) 1 ( P N ( N ))
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STOCHASTIC COLLOCATION STRATEGY
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’
A (s, t ,  )  A (s, t , 1, ..., N )
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.
2.
3.
X. Ma, N. Zabaras, A stabilized stochastic finite element second order projection methodology for modeling natural
convection in random porous media, JCP
D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci.
Comp. 24 (2002) 619-644
X. Wan and G.E. Karniadakis, Beyond Wiener-Askey expansions: Handling arbitrary PDFs, SIAM J Sci Comp 28(3)
(2006) 455-464
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Numerical Examples
Example 1 : The effect of uncertainty in process parameters
on macro-scale material properties for FCC copper
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:
Number of random variables: 2
where 1 and  2 are uniformly distributed random variables
between 0.2 and 0.6 (1/sec).
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Numerical Examples (Example 1)
1
0.40
2
0.35
0.8
Relative Error
0.30
Mean
Variance
0.25
0.20
0.6
0.4
0.15
0.10
0.2
0.05
0.00
0
0
2
4
6
8
10
Interpolation level
E ( MPa )
Var ( E ) (MPa)2
1.28e05
4.02e07
1.28e05
3.92e07
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0
0.2
0.4
1
0.6
0.8
1
Adaptive Sparse
grid (level 8)
MC (10000 runs)
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Numerical Examples (Example 2)
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
10
Level
8
6
4
2
0
0.2
0.4
0.6
0.8
1
1
1
Number of random variables: 1
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Numerical Examples (Example 2)
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Numerical Examples (Example 3)
Example 3 : The effect of uncertainty in initial texture on
macro-scale material properties for FCC copper
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)
A( r, t; )
 A( r, t; )  v( r, t )  0
t
A( r,0; )  A0 ( r,  )
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Numerical Examples (Example 3)
Step1. Reduce the random field to a set of random
variables (KL expansion)

A0 ( r,  )  A0 ( r )   i f i ( r, t )i ( )
i 1
0.9
Energy captured
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
Number of Eigenvalues
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Numerical Examples (Example 3)
Step2. In absence of sufficient information,use
Maximum Entropy to obtain the joint probability of
these random variables
Enforce positiveness of texture
Y3
Y1
Y2
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p(Y3 )
p(Y2 )
p(Y1 )
Y1
Y2
Y3
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Numerical Examples (Example 3)
Step3. Map the random variables Y1, Y2 , Y3 to independent identically
distributed uniform random variables 1,2 ,3 on a hypercube [0 1]^3
Y1  P11 ( P1 (1 ))
Rosenblatt transformation
Y2  P2|11 ( P2 ( 2 ))
YN  PN |1:( N 1) 1 ( P N ( N ))
p(Y1 ), p(Y1, Y2 ), p(Y1, Y2 , Y3 ) 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.
p(Y1 )
p(Y2 )
Y1
p(Y3 )
Y2
Y3
Rosenblatt M, Remarks on multivariate transformation, Ann. Math. Statist.,1952;23:470-472
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Numerical Examples (Example 3)
Step4. Use sparse grid collocation to obtain the stochastic characteristic of
macro scale properties
E ( MPa ) Var ( E )
(MPa) 2
Mean of A at the end
of deformation
process
Variance of A at the
end of deformation
process
1.41e05
4.42e08
Adaptive Sparse
grid (level 8)
1.41e05
4.39e08
MC 10,000 runs
80.0
Effective stress (MPa)
70.0
60.0
FCC copper
50.0
40.0
30.0
20.0
10.0
0.0
0.000
Variation of stress-strain
response
0.002
0.004
0.006
0.008
0.010
Effective strain
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