### Stratégies de régularisation et enjeux d`identification

```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
•
Image 2
Image 1
Relative
displacement
field ?
•
Image 1
Image 2
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Deformed image
Reference image
Relative
displacement
field ?
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Image # 11
Image # 1
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Displacement
field Uy
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Deformed image
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Displacement fields are nice,
but …
Can we get
more ?
Stress intensity
Factor,
Crack geometry
Uy
Image 1
0.25
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0.2
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0.1
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-0.05
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Image 2
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Damage
field
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Deformed image
Reference image
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-0.5
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-1.5
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log
-2
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10
(1  D )
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Image # 11
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Constitutive
law
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Reference image
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D
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x 10
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 eq
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Outline
• A brief introduction to “global DIC”
• Mechanical identification
• Regularization
From texture to displacements
DIC IN A NUTSHELL
Digital Image Correlation
• Images (gray levels) indexed by time t
f ( x, t)
• Texture conservation (passive tracers)
f ( x , t 0 )  f ( x  u ( x , t1 ) , t1 )
(hypothesis that can be relaxed if needed)
Problem to solve
• Weak formulation: Minimize wrt u
    ( x , t ; t0 ) d x d t
2
2
where the residual is
 ( x , t1 ; t 0 )  f ( x , t 0 )  f ( x  u ( x , t1 ) , t1 )
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
u( x, t) 
u
n
Yn ( x , t )
n
• Yn:
– FEM shape function, X-FEM, …
– Elastic solutions, Numerically computed fields,
Beam kinematics…
The benefit of C0 regularization
ZOI size / Element size (pixels)
Key parameter = (# pixels)/(# dof)
Example: T3-DIC*
Pixel size = 67 mm
*[Leclerc et al., 2009, LNCS 5496 pp. 161-171]
Example: T3-DIC
Example: T3-DIC
Ux (pixel)
0.46
0.28
0.11
-0.06
-0.23
[H. Leclerc]
Example: T3-DIC
Uy (pixel)
0.54
0.35
0.15
-0.04
-0.24
Example: T3-DIC
Example: T3-DIC
Residual
28
21
14
7
0
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
2 mU  k ( z )  z  ' ( z )  Y ( z )
where  and Y are arbitrary holomorphic
functions
• m is the shear modulus,
• k is a dimensionless elastic constant
(related to Poisson’s ratio)
Plane elasticity
• It suffices to introduce a basis of test
functions for (z) and Y(z) and consider
that  ( z ) and z  ' ( z )  Y ( z ) are
independent
• Direct evaluation of 1/m and k/m
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)
   (1 / 2 )(  U   U t )

  C .. 
   f  0

• FEM
KU  F
 div ( C ..  U )  f  0
Dialog DIC/FEA modeling
• Local elastic identification
R. Gras, Comptest 2011
T4-DVC
33
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
div( C :   U )  0
or in FEM language
K .U  0
for interior nodes.
This can be used to help DIC
Integrated DIC
• Reach smaller scale
H. Leclerc et al., Lect. Notes Comp. Sci. 5496, 161-171, (2009)
Tikhonov type regularization
• Minimization of
f ( x  U , t1 )  f ( x , t 0 )
2
 A KU
2
• Regularization is neutral with respect to
rigid body motion
• How should one choose A ?
Spectral analysis
• For a test displacement field
f ( x  U , t1 )  f ( x , t 0 )
KU
log(||.||2)
2
k V
4
2
U  V exp( ik . x )
V
2
A
2
Regularization
DIC
Cross-over scale
log(k)
4
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
2
2
(e.g.  U )
– Extend elastic behavior outside the DIC
analyzed region
Regularization at voxel scale
• An example
in 3D for a
modest size
243 voxels
Voxel scale DVC
1 voxel  5.1 µm
Displacement norm (voxels)
Vertical displacement (voxels)
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
2
 (1  D )K
i
i
kl
Ul
elements i
where U has been measured and K is known
• Many unknowns !
Validation
< 5.3 %
Constitutive law
State potential (isotropic damage)
Y  (1 / 2 ) ε : E 0 (1  D ) ε
σ
Y
ε
State laws
Y 
 E 0 (1  D ) ε
Y
D

1
2
ε : E 0ε
Dissipated power
Thermodynamic consistency
d  Y D  0
D  0 and Y  0
Growth law
(1  D ) function
of ( 2 / E 0 )Y
~ 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
D 

1
n
0.8
truncation
D
n 1
Y 
an  
E
1.2
or
D 
0.6
0.4

n 1
a n (1  exp(  Y / y n ) )
0.2
0
0
1
2
Y
3
5
4
x 10
-3
Identified damage image 10
Identified log10(1-D) image 10
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Identified damage image 11
Identified log10(1-D) image 11
log10(1-D)
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Identified damage image 11
Identified log10(1-D) image 11
Image # 11
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Validation image 10
Identified Ux
Measured Ux
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0
-2
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Identified Uy
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Validation image 11
Identified Ux
Measured Ux
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Identified Uy
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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
```