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Uncertainty quantification in multiscale deformation processes
Babak Kouchmeshky
Nicholas Zabaras
Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering
101 Frank H. T. Rhodes HallCornell University
Ithaca, NY 14853-3801
URL: http://mpdc.mae.cornell.edu/
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Problem definition
-Obtain the effect of uncertainty in initial texture on macro-scale material properties
Uncertain initial microstructure
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Deterministic multi-scale deformation process
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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)J r t A r t d A r d
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
REORIENTATION & TEXTURINGREORIENTATION & TEXTURING
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
( , )( , ) ( , ) 0
A s tA s t v s t
t
( , ) ( , )
s t A s t dv
'
'( ) ( , )
fv A s t dv
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
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Convergence of the deterministic problem
MPa MPa
Bulk modulus
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
MPa MPa
Convergence of the deterministic problem
Young modulus
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
MPa MPa
Convergence of the deterministic problem
Shear modulus
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Convergence of the deterministic problem
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Convergence of ODF
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Convergence of ODF
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Stochastic multi-scale deformation process
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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?
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL 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 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
A s t A s t
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’
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Reduced order model for a stochastic microstructure
Current method
minˆ( , , ) log( ( , , ) )a x s A x s A
1
ˆ( , , ) ( ) ( , )i ii
a x s s x
( , ) : ( ) ( )i j i j ijs s ds
( )i s( , )i x
#( , )i j ij #( , ) : ,D
f g f g dx , : ( ) ( ) ( )f g f g P d
where are modes strongly orthogonal in Rodrigues space and are spatial modes weakly orthogonal in space
1- D. Venturi, X. Wan, G.E. Karniadakis, J. fluid Mech. 2008, vol 606, pp 339-367
(1)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Reconstructing a stochastic microstructure
Step1: Construct the autocorrelation 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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T 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 .
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Comparison between the original microstructure and the reduced order one
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Comparison between the original microstructure and the reduced order one
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Comparison between the original microstructure and the reduced order one
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Comparison between the original microstructure and the reduced order one
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Comparison between the original microstructure and the reduced order one
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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)
Comparison between the original microstructure and the reduced order one
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
The effect of uncertainty in the initial texture of the work- piece on the macro-scale properties
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Conclusion
•A reduced order model for quantifying the uncertainty in multi-scale deformation process has been provided
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