Uncertainty quantification in multiscale deformation processes Babak Kouchmeshky Nicholas Zabaras...

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





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)J r t A r t d A r d




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


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




Convergence of the deterministic problem

MPa MPa

Bulk modulus




MPa MPa


Young modulus




MPa MPa


Shear modulus








Convergence of ODF




Convergence of ODF




Stochastic multi-scale deformation process




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




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

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’




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




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)




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




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




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 properties




Conclusion

•A reduced order model for quantifying the uncertainty in multi-scale deformation process has been provided

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Uncertainty quantification in multiscale deformation processes Babak Kouchmeshky Nicholas Zabaras...

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