Corrélation d'images numériques: Stratégies de régularisation et ...

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14 Νοε 2013 (πριν από 3 χρόνια και 4 μήνες)

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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




Relative
displacement
field ?

Image 1

Image 2




Image 1

Image 2




Reference image

Deformed image

Relative
displacement
field ?

Image # 1
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Image # 11
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Reference image

Deformed image

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Image # 11
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Displacement

field U
y

Displacement fields are nice,


but …

Can we get

more ?

Image 1

Image 2

-0.05
0
0.05
0.1
0.15
0.2
0.25
U
y
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300
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450
500
550
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650
700
750
800
850
Stress
intensity


Factor,

Crack
geometry

Reference image

Deformed image

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Image # 11
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Damage

field

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800
-2
-1.5
-1
-0.5
)
1
(
log
10
D

Reference image

Deformed image

Image # 1
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Image # 11
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Constitutive

law

D
2
eq

0
1
2
3
4
5
x 10
-3
0
0.2
0.4
0.6
0.8
1
1.2
Outline


A brief introduction to “global DIC”



Mechanical identification



Regularization


DIC IN A
NUTSHELL

From texture to displacements

Digital Image Correlation


Images (gray
levels
)
indexed

by time
t





Texture conservation (passive
tracers
)




(
hypothesis

that

can

be

relaxed

if
needed
)

)
)
(
(
)
(
1
1
0
t
t
x
u
x
f
t
x
f
,
,
,


)
,
(
t
x
f
Problem to solve


Weak formulation: Minimize wrt
u




where the residual is

)
)
(
(
)
(
)
(
1
1
0
0
1
t
t
x
u
x
f
t
x
f
t
t
x
,
,
,
;
,







t
x
t
t
x
d
d
0
2
2
)
;
,
(

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





Y
n
:


FEM
shape

function
, X
-
FEM, …


Elastic

solutions,
Numerically

computed

fields
,
Beam

kinematics



Y

n
n
n
t
x
u
t
x
u
)
,
(
)
,
(
The benefit of C
0

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
m
m

Example: T3
-
DIC

Example: T3
-
DIC


0.46


0.28



0.11


-
0.06


-
0.23

U
x

(pixel)


[H. Leclerc]

Example: T3
-
DIC


0.54


0.35



0.15


-
0.04


-
0.24

U
y

(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



and
Y

are
arbitrary

holomorphic

functions



m

is

the
shear

modulus
,



k

is

a
dimensionless

elastic

constant
(
related

to
Poisson’s

ratio)

)
(
)
(
'
)
(
2
z
z
z
z
U
Y





k
m
Plane
elasticity


It
suffices

to
introduce

a basis of test
functions

for
(
z
)

and
Y(
z
)

and
consider

that

and are
independent


Direct
evaluation

of 1/
m

and
k
/
m



)
(
z

)
(
)
(
'
z
z
z
Y


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
U
U
t




F
KU

0
)
..
(
div




f
U
C
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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400
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1000
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200
300
400
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600
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800
900
1000
Image # 12
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800
1000
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300
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500
600
700
800
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1000
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(



U
C
0
U
.
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

?

2
2
0
1
)
,
(
)
,
(
KU
A
t
x
f
t
U
x
f



Spectral
analysis


For a test
displacement

field


2
2
0
1
)
,
(
)
,
(
V
t
x
f
t
U
x
f



)
.
exp(
x
ik
V
U

2
4
2
V
k
KU

4


A
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


2
2
U

Regularization

at

voxel
scale


An
example

in 3D for a
modest

size
24
3

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
(
l
i
kl
i
i
U
K
D


Validation

< 5.3 %

ε
E
ε
)
1
(
:
)
2
/
1
(
0
D


Y
ε
E
ε
σ
)
1
(
0
D



Y


ε
E
ε
0
:
2
1


Y



D
Y
State potential (isotropic damage)

State laws

Y
E
D
)
/
2
(
of
function
)
1
(
0

0


D
Y
d

Dissipated power

0
and
0


Y
D


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










1
n
n
n
E
Y
a
D
(
)





1
)
/
exp(
1
n
n
n
y
Y
a
D
0
1
2
3
4
5
x 10
-3
0
0.2
0.4
0.6
0.8
1
1.2
Y
D
truncation

Identified damage image 10

Identified log
10
(1-D) image 10
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400
600
800
1000
200
400
600
800
-2
-1.5
-1
-0.5
Identified

damage image 11

Identified log
10
(1-D) image 11
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400
600
800
1000
200
400
600
800
-2
-1.5
-1
-0.5
log
10
(1
-
D)

Identified damage image 11

Identified log
10
(1-D) image 11
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400
600
800
1000
200
400
600
800
-2
-1.5
-1
-0.5
Image # 12
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1000
Image # 11
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Measured U
x
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1000
200
400
600
800
-4
-2
0
2
4
Measured U
y
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1000
200
400
600
800
-4
-2
0
2
Identified U
x
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1000
200
400
600
800
-4
-2
0
2
4
Identified U
y
500
1000
200
400
600
800
-4
-2
0
2
Validation image 10

Validation image 11

Measured U
x
500
1000
200
400
600
800
-5
0
5
Measured U
y
500
1000
200
400
600
800
-4
-2
0
2
Identified U
x
500
1000
200
400
600
800
-5
0
5
Identified U
y
500
1000
200
400
600
800
-4
-2
0
2
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