April 5, 2012 · 2012. 4. 16. · VALIDATION OF A RANDOM MATRIX MODEL FOR MESOSCALE ELASTIC...

VALIDATION OF A RANDOM MATRIX MODEL FOR MESOSCALE ELASTICDESCRIPTION OF MATERIALS WITH MICROSTRUCTURES

ROGER GHANEM, University of Southern California, Los Angeles, CA

ARASH NOSHADRAVAN, Massachusetts Institute of Technology, Cambridge, MA

JOHANN GUILLEMINOT, Université Paris-Est, MSME, Marne-la-Vallée, France

IKSHWAKU ATODARIA, PEDRO PERALTA, Arizona State University, Tempe, AZ

April 5, 2012SIAM-UQ 2012 - Raleigh, NC

R. Ghanem (USC) SIAM-UQ-2012, Raleigh, NC April 5, 2012 1 / 18

Introduction and motivation

Structural Health Monitoring (SHM) and Prognosis in Aerospace Systems

Future of SHM and Prognosis

Advances in experimentalcharacterization, sensing technology,computational facility.

human inspection, timed maintenance→ condition-based SHM, residual lifeestimation.

Anticipate damage from measureddata.

- number, type and location ofsensors

Sensing and Processing

Predictive modeling

Data Interrogation

Damage Prognosis

Structural Health Monitoring

Uncertainty Quantification & Model Validation


Challenges-Solutions

Challenge

Damage initiates at a very small scale.

Measured data is at a coarse scale.

The details of the microscale for the specimenbeing measured are not known.

Microscale Simulation ?microstructure unknown - can be characterized statistically in the lab.

to determine location of sensors we need to formulate an optimization problem.

mechanistic analysis of an ensemble of microstructures is very expensive.

SolutionDevelop a new stochastic mechanistic model with :

State of the model at same scale as experimental observables.

Model behavior sensitive to occurrences at the scale of damage initiation.

Scatter in predictions from model consistent with observed scatter.

Behavior of model honors known accepted conservation laws.


Physically-based Multiscale Modeling for SHM and Prognosis

• Microscopic images

• Ultrasonic techniques

Probabilistic physically-based multiscale model

• Damage detection

• Life time evaluation

• Prediction & Prognosis

Characterization of microstructural heterogeneity

Stochastic characterization and representation of

microstructureModel Verification

& Validation Stochastic upscaling


Outline

1 Statistical model for characterization and realization of microstructure

2 Probabilistic model for mesoscale bounded elasticity tensor of polycrystalls

3 Validation methodology

4 Summary and conclusions

Statistical model for characterization and

realization of microstructure

Probabilistic model for mesoscale description

of materials

Model Calibration

Model Validation

Microstructural measurements


Experimental characterization and database

Three EBSD maps of 12 mm×1.5 mm Al-2024 specimenInformation on geometry and crystallographic orientation of ≈12000 grainsThe microstructure is described by the equivalent aspect ratios and Euler anglesof all the grains

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Euler angle (rad)

pd

f

φ1

φφ

2

pdfs of Euler angles

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

Aspect ratio β

pdf

pdf of overall aspect ratio


Statistical model for characterization and realization of microstructures

I : Random geometry

Voronoi-G tessellation

Extension of classical Voronoi-tessellating by using the following distance

V(x(i)tes)def= {x ∈ Ω | dG(x(i)tes, x) ≤ dG(x

(i)tes, x

(j)tes)},

dG(x, y)def=√

(x, y)T [G] (x, y)

[G] =[

(1/s)2 00 1

]

s : Rate of growth of tessellation in e1 − direction1/s controls the aspect ratio of the grains


Statistical model for characterization and realization of microstructuresMinimum Relative Entropy estimates of parameter s

ŝ = arg mins∈R+

DKL (p̂β(β) ‖ pβ(β|s)) ,

DKL (p̂β(β) ‖ pβ(β|s)) =∫R+

p̂β(β) logp̂β(β)

pβ(β|s)dβ.

DKL (Kullback-Leibler divergence) : distance-like measure of the difference between two pdfs

1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Parameter s

Rel

ativ

e en

trop

y D

KL

Plot of minimum relative entropy functions

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6


Statistical model for characterization and realization of microstructures

II : Random crystallographic orientation

Objective

Euler angles are modeled as vector-valued random field.

x 7→ Φ(x) = (φ1(x), φ2(x), φ3(x))Approach

The vector-valued random field of Euler angles is modeled as a translationprocess with prescribing :

Marginal probability distribution functions estimated from measurementsFractile correlation functions estimated from measurements

Fractile correlation

um(x) = Fm(φm(x)), m = 1, 2, 3.

[R(x, x′)]mn =E{(um(x)− um(x))(un(x

′)− un(x′))}√

E{(um(x)− um(x))2}E{(un(x′)− un(x′))2}

= 12E{um(x)un(x′)} − 3.


Statistical model for characterization and realization of microstructuresEstimation of correlation functions from measurements :

0 30 60 90 120 150−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

η (micron)

[R] ii

[R]

11

[R]22

[R]33

0 30 60 90 120 150−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

η (micron)

[R] ij

[R]

12

[R]23

[R]13

Identification of the correlation structure : [R(η)]mn = amn exp(−η/lmn)

0 30 60 90 120 150−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

η (micron)

[R] 1

1

estimated valuesbest exponential fit (a

11=1.0, l

11=22.4553)

0 30 60 90 120 150−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

η (micron)

[R] 1

3

estimated valuesbest exponential fit (a

13=−0.9416, l

13=35.9678)


Probabilistic model for mesoscale elasticity tensor

Objective

Mesoscale material description that :(i) captures the effect of subscale heterogeneities (ii) could be used in a coarse scalemodeling.

Approach : MaxEnt principle & Random matrix theory

Constructing a probability distribution on the set of elasticity matrices.

Constrain random matrices to specified physics-based bounds.

Calibrate the random matrices from all the available information.

Prob. model for mesoscale elasticity

tensor

Calibration

Verification Validation

Prediction & Detection

....

Microstructure

Macroscale (RVE) Mesoscale (SVE)



Overview of model constructionLet :

N = (C− Cl )−1 − (Cu − Cl )−1 > 0,

Maximize : ∫M+n (R)

ln(p)pN(N) dN

subject to : ∫M+n (R)

pN(N) dN = 1,∫M+n (R)

N p[N](N) dN = N ∈ M+n (R),∫C

ln(det(N))pN(N)dN = cN , |cN | < +∞.

pN(N) = IM+n (R)(N)ĉ0 det(N)λ−1etr{−ΛN N}

ΛN and λ are Lagrange multipliers.


Probabilistic model for mesoscale elasticity tensorCalibration

Computing the realizations of the bounds for apparent elasticity matrix (Huet’s partitioningtechnique)Computing the realizations of the apparent elasticity matrix

Apparent properties

min ‖〈σ〉BC − [C]〈�〉BC‖subject to meaningful constraints

}⇒ [Capp]

BCs :

SUBC : t(x) = σ0n(x) ⇒ Lower bound [Cappσ ]

KUBC : u(x) = �0x ⇒ Upper bound [Capp� ]

MBC (Tension test, e.g.)⇒ Samples of apparent elasticity tensor [Capp]

Compute the statistical estimates of parameters for Nsim = 100 and Ω = 0.3× 0.3 [mm]

δ̃N =

1Nsim‖[Ñ]‖2FNsim∑k=1

‖[N(ωk )]− [Ñ]‖2F

1/2

= 0.66

[Ñ] =1

Nsim

Nsim∑k=1

[N(ωk )] = 10−3

0.2667 0.0879 −0.01890.0879 0.2214 0.0277−0.0189 0.0277 0.2366



VerificationWhether the model implementation accurately represent the intended conceptual descriptionof the model and the solution to the model

1.08 1.1 1.12 1.14 1.16

x 105

0

1

2

3

4

5

6x 10

−4

C11

PDF

1.08 1.1 1.12 1.14 1.16

x 105

0

1

2

3

4

5

6x 10

−4

C22

PDF

2.4 2.5 2.6 2.7 2.8 2.9

x 104

0

1

2

3

4

5

6

7

8x 10

−4

C33

PDF


Validation : Static

Credibility of the model in propagating the uncertainty in the fine scale into thecoarse scale response

x (mm)

y (

mm

)

0.0 1.0 2.0 3.0

0.0

1.0

2.0

3.0

0= 1000 N/mm2σ

Quantity of interest (QOI) :volume averaged strain energy density

ϕ = (1/2)〈�(x)Tσ(x)〉V

5.5 6 6.5 70

1

2

3

4

5

6

7

8

volume averaged strain energy density

pdf

coarse scalefine scale


Validation : Static

pdfs of QOI in subvolumes :

4 5 6 70

4

8

V11

4 5 6 70

4

8

V12

4 5 6 70

4

8

V13

4 5 6 70

4

8

V21

4 5 6 70

4

8

V22

4 5 6 70

4

8

V23

4 5 6 70

4

8

V31

4 5 6 70

4

8

V32

4 5 6 70

4

8

V33


Validation : Ultrasonic wave response

QOI : attenuation coefficient α

I2(xi ) = Io

2 e−2α‖xi−xo‖

0.0 1.5 3.0 6.0

0.0

1.5

3.0

6.0

y (

mm

)

x (mm)

4.5

4.5

0 0.05 0.1 0.15 0.2 0.25−0.5

−0.25

0

0.25

0.5

0.75

1Ricker pulse with fc=10 MHz

Time (µ sec)

Am

plit

ud

e

0 0.5 1 1.5 2 2.5 3−18

−16

−14

−12

−10

−8

−6

−4

−2

0

||xi−xo|| (mm)

log

(I2(

xi)

/ I2o)

computed valueslog(I

2(xi) / I

2o)=(−2α)||xi−xo||

2.8 2.9 30

10

20

30

40

50

60

α (1/mm)

pdf

fc=2 (MHz)

1.4 1.60

5

10

15

20

25

30

35

40

45

α (1/mm)

pdf

fc=5 (MHz)

0.6 0.7 0.80

5

10

15

20

25

30

35

40

45

50

α (1/mm)

pdf

fc=10 (MHz)


Summary

1 Developing a statistical tool for characterization and realization of randompolycrystal.

characterization and simulation of geometry with elongated grains.characterization and simulation of Euler angles random field.

2 Adopting a probabilistic model for mesoscale elasticity tensor of polycrystallinematerials.

3 Presenting a validation methodology for prediction of the probabilistic model

Validation in static case.Validation in dynamic case (ultrasonic wave response).


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April 5, 2012 · 2012. 4. 16. · VALIDATION OF A RANDOM MATRIX MODEL FOR MESOSCALE ELASTIC...

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