Modeling Uncertainty Propagation In Deformation Processes: Babak Kouchmeshky Nicholas Zabaras

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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Materials Process Design and Control Laboratory

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) 1 L = β1 0  0 0 0 β5 0 0  0 1

0 − 0.5 0 0  1 β+ 6   0 

0  0 0 + β 2 0   −0.5   0 0 1− 0 1 0 0 β+  0 0 0

0 0 1 + 0β  0 −  1  0 0  0 0 7  1 0

 0  1 3  0 −1   0β+     0 

1 0 0+ β 0  0 0 0 8

0   0  4 1  0 0 0 0− 1  0 1 0

0 1 + 0 0  0 0

Since incompressibility is assumed only eight components of L are independent. The βi 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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Materials Process Design and Control Laboratory

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 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 to hypercube

A0(s, ω) Use Karhunen-Loeve expansion to reduce this random filed to few random variables

A0 ( s ,Y1 ,Y2 ,Y3 )

Use Stochastic collocation to obtain the effect of these random initial texture on final macro-scale properties.

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Materials Process Design and Control Laboratory

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 (ℜ') = ∫A(s ,t )dv ℜ'

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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Χ = ∫Χ ( s, t) A( s, t)dv ℜ

Materials Process Design and Control Laboratory

Constitutive theory Velocity gradient

& −1 L = FF Symmetric and spin components

Reorientation velocity

ω = vect( Ω)

Polycrystal plasticity Deformed

Initial configuration

F

s0 n0

Bo

configuration

F

p s0 n0

sn

F*

B

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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Materials Process Design and Control Laboratory

Representing the uncertain micro-structure Karhunen-Loeve Expansion: Let A0 ( r , ω ) 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 A0 then the unbiased estimate of the covariance matrix is 0.9 0.8

Energy captured

M 1 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

Then its KLE approximation is defined as ∞

A0 ( r, ω ) = A0 ( r ) + ∑ λi fi ( r, t )Yi (ω )

λi Yi (ω )

where

0 0

2

4

i =1

and

fi

6

8

10

Number of Eigenvalues

are eigenvalues and eigenvectors of

% C

and is a set of uncorrelated random variables whose distribution depends on the type of stochastic process.

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Materials Process Design and Control Laboratory

Karhunen-Loeve Expansion Realization of random variables

1 Yi = Aj − A, fi λi j

where

l2

l2

Yi (ω )

can be obtained by

, j = 1: N R. N

denotes the scalar product in

The random variables

Yi (ω ) have the following two properties

E [ Yi (ω ) ] = 0

Y3

E Yi (ω )Y j (ω )  = δ ij

Y2

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Y1

Materials Process Design and Control Laboratory

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 ))d Y Y ∫ )log(p(

∫ p( Y)d Y=1 D

E{ g( Y )}= f

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g1(v) = E (v1)  g2(v) = E (v2)  M g (v) = E (v v ) k l  N

p( Yλ) =g(Y) 1D c0 exp( −

,

µ

)

Materials Process Design and Control Laboratory

Maximum Entropy Principle p (Y1 )

p (Y2 )

Y2 Y1 Constraints at the final iteration

p (Y3 )

Target 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

Y3

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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 ( Pξ1 (ξ1 )) Y2 = P2|1−1 ( Pξ2 (ξ2 )) M YN = PN |1:( N −1) −1 ( Pξ N (ξ 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, ω) =A(s,t,ξ ,..., )N 1 ξ 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 m , 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: 1 L = α1 0  0 0 L = α 2 1   0

0 −0.5 0

0  0  t
Number of random variables: 2

1 0 0 0 T1
where α1 and α 2 are uniformly distributed random variables between 0.2 and 0.6 (1/sec).

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Numerical Examples (Example 1) 0.40

1

ξ2

0.35

Relative Error

0.30

0.8

Mean Variance

0.25 0.20

0.6

0.15 0.10 0.4

0.05 0.00 0

2

4

6

8

10

0.2

Interpolation level 0

E (MPa )

Var (E ) (MPa)

1.28e05

4.02e07

1.28e05

3.92e07

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0

0.2

0.4

ξ1

0.6

0.8

1

2

Adaptive Sparse grid (level 8) MC (10000 runs) Materials Process Design and Control Laboratory

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

1

0.6

0.8

1

ξ1

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; ω ) + A( r , t ; ω )∇ ⋅ v ( r , t ) = 0 ∂t A( r,0; ω ) = A0 ( r , ω )

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Numerical Examples (Example 3) Step1. Reduce the random field to a set of random variables (KL expansion)

∞

A0 ( r, ω ) = A0 ( r ) + ∑ λi f i ( r , t )ξ i (ω ) 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

Y2

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Y1

p(Y3 )

p(Y2 )

p(Y1 )

Y1

Y2

Y3

Materials Process Design and Control Laboratory

Numerical Examples (Example 3) Step3. Map the random variables Y1 , Y2 , Y3 to independent identically distributed uniform random variables ξ1 , ξ 2 , ξ 3 on a hypercube [0 1]^3 Y1 = P1−1 ( Pξ1 (ξ1 ))

Rosenblatt transformation

Y2 = P2|1−1 ( Pξ2 (ξ2 )) M YN = PN |1:( N −1) −1 ( Pξ N (ξ 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

1.41e05

4.42e08

Adaptive Sparse grid (level 8)

1.41e05

4.39e08

MC 10,000 runs

Variance of A at the end of deformation process

80.0

Effective stress (MPa)

70.0 60.0 50.0

FCC copper

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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