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 ?

Reference image Deformed image Displacement field U y

Displacement fields are nice, but … Can we get more ?

Image 1 Image 2 Stress intensity Factor, Crack geometry

Reference image Deformed image Damage field

Reference image Deformed image Constitutive law

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)

Problem to solve Weak formulation: Minimize wrt u where the residual is 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  n : –FEM shape function, X-FEM, … –Elastic solutions, Numerically computed fields, Beam kinematics…

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 ] Pixel size = 67  m

Example: T3-DIC

U x (pixel) [H. Leclerc]

Example: T3-DIC U y (pixel)

Example: T3-DIC

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  are arbitrary holomorphic functions  is the shear modulus,  is a dimensionless elastic constant (related to Poisson’s ratio)

Plane elasticity It suffices to introduce a basis of test functions for  z  and  z  and consider that and are independent Direct evaluation of 1/  and  / 

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

Identified properties for the polycarbonate   880 MPa  0.45 In good agreement with literature data

Need for coupling to modelling Elasticity (or incremental non-linear behavior) FEM

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

REGULARIZATION

Mechanical regularization The displacement field should be such that or in FEM language for interior nodes. This can be used to help DIC

Integrated DIC Reach smaller scale H. Leclerc et al., Lect. Notes Comp. Sci. 5496, , (2009)

Tikhonov type regularization Minimization of Regularization is neutral with respect to rigid body motion How should one choose A ?

Spectral analysis For a test displacement field 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

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 !

Validation < 5.3 %

State potential (isotropic damage) State laws Dissipated powerThermodynamic 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 truncation

Identified damage image 10

Identified damage image 11 log 10 (1-D)

Identified damage image 11

Validation image 10

Validation image 11

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