Corrélation d'images numériques: Stratégies de régularisation

et enjeux d'identification

Stéphane Roux, François Hild

LMT, ENS-Cachan

Atelier « Problèmes Inverses », Nancy, 7 Juin 2011

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Relative displacement

field ?

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Relative displacement

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Displacement field Uy

Displacement fields are nice, but …

Can we get more ?

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Stress intensity Factor,

Crack geometry

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Damagefield

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Constitutivelaw

D

2

eq0 1 2 3 4 5

x 10-3

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Outline

• A brief introduction to “global DIC”

• Mechanical identification

• Regularization

DIC IN A NUTSHELLFrom texture to displacements

Digital Image Correlation

• Images (gray levels) indexed by time t

• Texture conservation (passive tracers)

(hypothesis that can be relaxed if needed)

))(()( 110 ttxuxftxf ,,,

),( txf

Problem to solve

• Weak formulation: Minimize wrt u

where the residual is

))(()()( 11001 ttxuxftxfttx ,,,;,

txttx dd022 );,(

Provides a spatially resolved quality field of the proposed solution

Solution

• The problem is intrinsically ill-posed and highly non-linear !

• A specific strategy has to be designed for accurate and robust convergence

• It impacts on the choice of the kinematic basis

Global DIC

• Decompose the sought displacement field on a suited basis providing a natural regularization

• Yn: – FEM shape function, X-FEM, … – Elastic solutions, Numerically computed fields,

Beam kinematics…

n

nn txutxu ),(),(

The benefit of C0 regularization

ZOI size / Element size (pixels)Key parameter = (# pixels)/(# dof)

Example: T3-DIC*

*[Leclerc et al., 2009, LNCS 5496 pp. 161-171]

Pixel size = 67 mm

Example: T3-DIC

Example: T3-DIC

0.46

0.28

0.11

-0.06

-0.23

Ux (pixel)

[H. Leclerc]

Example: T3-DIC

0.54

0.35

0.15

-0.04

-0.24

Uy (pixel)

Example: T3-DIC

Example: T3-DIC

28

21

14

7

0

Residual

Mean residual = 3 % dynamic range

IDENTIFICATION

The real challenge

• For solid mechanics application, the actual challenge is – not to get the displacement fields,

but rather – to identify the constitutive law (stress/strain

relation)

• The simplest case is linear elasticity

Plane elasticity

• A potential formulation can be adopted showing that the displacement field can be written generically in the complex plane as

where F and Y are arbitrary holomorphic functions

• m is the shear modulus,• k is a dimensionless elastic constant

(related to Poisson’s ratio)

)()(')(2 zzzzU

Plane elasticity

• It suffices to introduce a basis of test functions for (F z) and (Y z) and consider that and are independent

• Direct evaluation of 1/m and k/m

)(z )()(' zzz

Validated examples

• Brazilian compression test

• Cracks

Example 1:Brazilian compression test

• Integrated approach:

decomposition of the displacement field over 4 fields (rigid body motion + analytical solution)

Integrated approach

Integrated approach

Identified properties for the polycarbonate

m 880 MPa

n 0.45

In good agreement with literature data

Need for coupling to modelling

• Elasticity (or incremental non-linear behavior)

• FEM

0

..

))(2/1(

f

C

UU t

FKU 0)..(div fUC

Dialog DIC/FEA modeling

• Local elastic identification

R. Gras, Comptest 2011

33

T4-DVC

More general framework

• Inhomogeneous elastic solid

• Non-linear constitutive law– Plasticity– Damage– Non-linear elasticity Image # 11

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REGULARIZATION

Mechanical regularization

• The displacement field should be such that

or in FEM language

for interior nodes.

This can be used to help DIC

0):div( UC

0U.K

Integrated DIC

• Reach smaller scale

H. Leclerc et al., Lect. Notes Comp. Sci. 5496, 161-171, (2009)

Tikhonov type regularization

• Minimization of

• Regularization is neutral with respect to rigid body motion

• How should one choose A ?

22

01 ),(),( KUAtxftUxf

Spectral analysis

• For a test displacement field

22

01 ),(),( VtxftUxf

).exp( xikVU

242VkKU 4A

log(k)

log(||.||2) Regularization

DIC

Cross-over scale

Boundaries

• The equilibrium gap functional is operative only for interior nodes or free boundaries

• At boundaries, information may be lacking– Introduce an additional regularization term

(e.g. )– Extend elastic behavior outside the DIC

analyzed region

22U

Regularization at voxel scale

• An example in 3D for a modest size 243 voxels

Voxel scale DVC

Displacement norm (voxels) Vertical displacement (voxels)

1 voxel 5.1 µm

H. Leclerc et al., Exp. Mech. (2011)

NON-LINEAR IDENTIFICATION

Identification

• As a post-processing step, a damage law can be identified from the minimization of

where U has been measured and K is known

• Many unknowns !

2

elements

)1( likl

ii UKD

Validation

< 5.3 %

εEε )1(:)2/1( 0 D

εEε

σ )1(0 D

εEε 0:2

1

D

Y

State potential (isotropic damage)

State laws

YED )/2(offunction )1( 0

0 DYd Dissipated power

0and0 YD Thermodynamic consistency

Growth law

Constitutive law

~ equivalent scalar strain

Use of a homogeneous constitutive law

• Postulating a homogeneous law, damage is no longer a two dimensional field of unknowns, but a (non-linear) function of the maximum strain experienced by an element of volume.

Damage growth law

• Identified form

or

1n

n

n E

YaD

1

)/exp(1

n

nn yYaD

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Y

D

truncation

Identified damage image 10Identified log

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Identified damage image 11Identified log

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log10(1-D)

Identified damage image 11

Identified log10

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Measured Ux

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Validation image 10

Validation image 11Measured U

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CONCLUSIONS

Conclusions

• DIC and regularization can be coupled to make the best out of difficult measurements

• A small scale regularization is too poorly sensitive to elastic phase constrast to allow for identification

• Yet, post-treatment may provide the sought constitutive law description

• Fusion of DIC and non-linear identification is the most promising route