Corrélation d'images numériques: Stratégies de régularisation et enjeux d'identification

1. 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

2. Image 2 Image 1 Relative displacement field ?

3. Image 2 Image 1

4. Deformed image Reference image Relative displacement field ?

5. Displacement field Uy Deformed image Reference image

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

7. Stress intensity Factor, Crack geometry Image 2 Image 1

8. Damage field Deformed image Reference image

9. Constitutive law Deformed image Reference image

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

11. From texture to displacements DIC in a nutshell

12. Digital Image Correlation • Images (gray levels) indexed by time t • Texture conservation (passive tracers) (hypothesisthatcanberelaxed if needed)

13. Problem to solve • Weak formulation: Minimize wrt u where the residual is Provides a spatiallyresolvedqualityfield of the proposed solution

14. Solution • The problemisintrinsicallyill-posed and highly non-linear ! • A specificstrategy has to bedesigned for accurate and robust convergence • It impacts on the choice of the kinematic basis

15. Global DIC • Decompose the soughtdisplacementfield on a suited basis providing a naturalregularization • Yn: • FEM shapefunction, X-FEM, … • Elastic solutions, Numericallycomputedfields, Beamkinematics…

16. The benefit of C0 regularization ZOI size / Element size (pixels) Key parameter = (# pixels)/(# dof)

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

18. Example: T3-DIC

19. Example: T3-DIC Ux (pixel) 0.46 0.28 0.11 -0.06 -0.23 [H. Leclerc]

20. Example: T3-DIC Uy (pixel) 0.54 0.35 0.15 -0.04 -0.24

21. Example: T3-DIC

22. Example: T3-DIC Residual 28 21 14 7 0 Mean residual = 3 % dynamic range

23. Identification

24. The real challenge • For solidmechanics application, the actual challenge is • not to get the displacementfields, but rather • to identify the constitutive law (stress/strain relation) • The simplest case islinearelasticity

25. Plane elasticity • A potential formulation canbeadoptedshowingthat the displacementfieldcanbewrittengenerically in the complex plane as whereF and Y are arbitraryholomorphicfunctions • mis the shearmodulus, • kis a dimensionlesselastic constant (related to Poisson’s ratio)

26. Plane elasticity • It suffices to introduce a basis of test functions for F(z) and Y(z) and considerthat and are independent • Direct evaluation of 1/m and k/m

27. Validatedexamples • Brazilian compression test • Cracks

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

29. Integrated approach

30. Integrated approach Identifiedproperties for the polycarbonate m 880 MPa n 0.45 In good agreement withliterature data

31. Need for coupling to modelling • Elasticity (or incremental non-linearbehavior) • FEM

32. Dialog DIC/FEA modeling • Local elastic identification R. Gras, Comptest2011

33. T4-DVC

34. More generalframework • Inhomogeneouselasticsolid • Non-linear constitutive law • Plasticity • Damage • Non-linearelasticity

35. Regularization

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

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

38. Tikhonov type regularization • Minimization of • Regularizationisneutralwith respect to rigid body motion • How should one chooseA ?

39. Spectral analysis • For a test displacementfield Regularization log(||.||2) DIC Cross-over scale log(k)

40. Boundaries • The equilibrium gap functionalisoperativeonly for interiornodes or free boundaries • Atboundaries, information maybelacking • Introduce an additionalregularizationterm(e.g. ) • Extendelasticbehavioroutside the DIC analyzedregion

41. Regularizationat voxel scale • An example in 3D for a modest size 243 voxels

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

43. Non-linear identification

44. Identification • As a post-processingstep, a damage lawcanbeidentifiedfrom the minimization ofwhereU has been measured and Kisknown • Manyunknowns !

45. Validation < 5.3 %

46. Constitutive law State potential (isotropic damage) State laws Dissipated power Thermodynamic consistency Growth law ~ equivalentscalarstrain

47. Use of a homogeneous constitutive law • Postulating a homogeneouslaw, damage is no longer a twodimensionalfield of unknowns, but a (non-linear) function of the maximum strainexperienced by an element of volume.

48. Damage growth law • Identifiedform or truncation

49. Identified damage image 10

50. Identified damage image 11 log10(1-D)