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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

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Relative displacement field ? Image 1 Image 2

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Image 1 Image 2

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Reference image Deformed image Relative displacement field ?

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Reference image Deformed image Displacement field U y

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

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Image 1 Image 2 Stress intensity Factor, Crack geometry

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Reference image Deformed image Damage field

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Reference image Deformed image Constitutive law

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Outline A brief introduction to “global DIC” Mechanical identification Regularization

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DIC IN A NUTSHELL From texture to displacements

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Digital Image Correlation Images (gray levels) indexed by time t Texture conservation (passive tracers) (hypothesis that can be relaxed if needed)

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Problem to solve Weak formulation: Minimize wrt u where the residual is Provides a spatially resolved quality field of the proposed solution

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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

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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…

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The benefit of C 0 regularization ZOI size / Element size (pixels) Key parameter = (# pixels)/(# dof)

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Example: T3-DIC* *[Leclerc et al., 2009, LNCS 5496 pp. 161-171] Pixel size = 67 m

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Example: T3-DIC

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0.46 0.28 0.11 -0.06 -0.23 U x (pixel) [H. Leclerc]

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Example: T3-DIC 0.54 0.35 0.15 -0.04 -0.24 U y (pixel)

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Example: T3-DIC

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28 21 14 7 0 Residual Mean residual = 3 % dynamic range

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IDENTIFICATION

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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

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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)

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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 /

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Validated examples Brazilian compression test Cracks

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Example 1: Brazilian compression test Integrated approach: decomposition of the displacement field over 4 fields (rigid body motion + analytical solution)

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Integrated approach

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

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

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Dialog DIC/FEA modeling Local elastic identification R. Gras, Comptest 2011

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33 T4-DVC

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More general framework Inhomogeneous elastic solid Non-linear constitutive law –Plasticity –Damage –Non-linear elasticity

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REGULARIZATION

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

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Integrated DIC Reach smaller scale H. Leclerc et al., Lect. Notes Comp. Sci. 5496, 161-171, (2009)

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

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Spectral analysis For a test displacement field log(k) log(||.|| 2 ) Regularization DIC Cross-over scale

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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

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Regularization at voxel scale An example in 3D for a modest size 24 3 voxels

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Voxel scale DVC Displacement norm (voxels) Vertical displacement (voxels) 1 voxel 5.1 µm H. Leclerc et al., Exp. Mech. (2011)

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NON-LINEAR IDENTIFICATION

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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 !

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Validation < 5.3 %

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State potential (isotropic damage) State laws Dissipated powerThermodynamic consistency Growth law Constitutive law ~ equivalent scalar strain

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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.

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Damage growth law Identified form or truncation

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Identified damage image 10

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Identified damage image 11 log 10 (1-D)

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Identified damage image 11

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Validation image 10

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Validation image 11

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CONCLUSIONS

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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

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