modeling uncertainty propagation in deformation processes
TRANSCRIPT
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 1/22
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
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 2/22
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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.
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 3/22
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
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
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 4/22
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
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)
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 5/22
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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.
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 6/22
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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
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
( , )( , ) ( , ) 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
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 7/22
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
Constitutive theoryConstitutive theory
D = Macroscopic stretch = Schmid tensor
= Lattice spin W = Macroscopic spin
= Lattice spin vector
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
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
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 8/22
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
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
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 9/22
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
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
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 10/22
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
( ) =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)
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 11/22
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
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
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 12/22
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
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.
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 13/22
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?
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
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’
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 14/22
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
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
(
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 15/22
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
( ) 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 (
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 16/22
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
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
Numerical Examples (Example 2)
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
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 17/22
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
Numerical Examples (Example 2)
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 18/22
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
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
Numerical Examples (Example 3)
Example 3 : The effect of uncertainty in initial texture on
macro-scale material properties for FCC copper
(
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 19/22
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
Numerical Examples (Example 3)
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 (
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 20/22
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
Numerical Examples (Example 3)
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 (
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 21/22
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
Numerical Examples (Example 3)
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 (
8/14/2019 Modeling Uncertainty Propagation in Deformation Processes
http://slidepdf.com/reader/full/modeling-uncertainty-propagation-in-deformation-processes 22/22
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLLCCOORRNNEELLLLU N I V E R S I T Y
Numerical Examples (Example 3)
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