SIMULATION OF THE UT INSPECTION OF PLANAR DEFECTS USING A GENERIC GTD-KIRCHHOFF APPROACH

Transcription

1 ound M athematics L td. SIMULATION OF THE UT INSPECTION OF PLANAR DEFECTS USING A GENERIC GTD-KIRCHHOFF APPROACH V. DORVAL, M. DARMON, S. CHATILLON, L. FRADKIN presented by A. LHEMERY CEA, LIST, France Sound Mathematics, UK Review of Progress in QNDE Boise, July 04

2 OUTLINE OF THE PRESENTATION Context Crack scattering models Physical Theory of Diffraction Results

3 CONTEXT Modeling of the defect responses with CIVA UT Several types of defects and interaction models are available in CIVA Kirchhoff Voids flaws : cracks like or volumic shapes. Specular reflexion GTD (Geometrical Theory of Diffraction) Diffraction by edges SOV (Separation Of Variables) Spheres, SDH (Modified) Born Solid Inclusions For cracks: Kirchhoff is best suited for specular echoes, GTD is best suited for diffraction echoes Choosing beween the two models requires expertise In some cases, both specular and diffraction echoes are required CIVA : development of the Kirchhoff+GTD model, based on the Physical Theory of Diffraction Specular echo Edge diffraction

4 CRACK SCATTERING MODELS Modeling planar crack echoes in NDT experiments Reciprocity formalism for the expression of echoes: δγ ba = 4P v T v T Scattering coefficients can be derived from it: S F n ds B. A. Auld, "General electromechanical reciprocity relations applied to the calculation of elastic wave scattering coefficients, Wave Motion, 979. V R = KsV 0 V 0 A The Thompson-Gray measurement model, as described by L. Schmerr and Jung-Sin Song, "Ultrasonic Nondestructive Evaluation Systems, Springer Science, 007. For planar cracks, only the Crack Opening Displacement (COD) needs to be n calculated: δγ ba = v 4P T v T n ds = COD T S F 4P n ds S F + The Kirchhoff and GTD models are widely used to model crack echoes : field from the emitter, in the presence of defect : field from the receiver, in the absence of defect v: particular velocity T: stress n: normal to the surface P: emitter power K: proportionnality factor not detailed here s: reference signal : incident field : observed field V: amplitude of the plane wave that serves as an approximation to the field A: scattering coefficient, dependent on the directions and nature of the plane waves n S F + S F -

5 COD CRACK SCATTERING MODELS Kirchhoff approximation Also know as tangent plane approximation Field on the surface of the defect assumed to be locally equal to the field created by the reflection of waves on a tangent plane Tangent plane Incident L wave Reflected T wave Reflected L wave Defect surface Illustration of the Crack Opening Displacement in the Kirchhoff approximation in a canonical case: 3 x=- Incident unit plane wave x=.5.5 Advantages and drawbacks: Position Any defect geometry Incident field variations along the defect x No effect of the edge inaccuracies in edge diffraction

6 CRACK SCATTERING MODELS Geometrical Theory of Diffraction Aims at modeling crack edge diffraction Adds diffracted rays compared to geometric rays J. B. Keller, Geometrical Theory of Diffraction, Journal of the Optical Society of America,96. D: cylindrical wave 3D: conical wave Based on the asymptotic evaluation of an exact solution for a semi-infinite crack Edge diffraction x x x Field variations along the defect Complex geometries Regularity of coefficients: divergences

7 COD CRACK SCATTERING MODELS Geometrical Theory of Diffraction Divergences occur in the specular and transmission directions The Crack Opening Displacement corresponding to the GTD model is the exact solution for a semi-infinite crack 3 Penumbra zones Acoustic case inc Exacte GTD x=- Incident unit plane wave.5 0 Illustration of the exact solution for the COD of a semi-infinite crack. V. A. Borovikov, "Uniform Stationary Phase Method, IEE: London UK, Position

8 CRACK SCATTERING MODELS Overcoming the limitations of Kirchhoff and GTD Several methods have been developed with the aim and modeling tip diffraction without the coefficient divergence of GTD (particularly for electromagnetic waves) The Uniform Asymptotic Theory of Diffraction relies on fictitious rays to avoid the divergences encountered by GTD R. M. Lewis and J. Boersma, Uniform asymptotic theory of edge diffraction, Journal of Mathematical Physics,969. The Uniform Theory of Diffraction yields coefficients that avoid divergences thanks to a smoothing function that intervene in the calculations R. G. Kouyoumjian and P. H. Pathak, A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface, IEEE Trans. Antennas Propag.,974.. The Physical Theory of Diffraction can be interpreted as combination of Kirchhoff and GTD P. Y. Ufimtsev, Fundamentals of the Physical Theory of Diffraction, John Wiley & Sons., 007. Retains advantages of Kirchhoff (easily accounts for complex flaw geometries and for variations of the incident field over the surface) Relatively easy to implement if Kirchhoff and GTD are already available

9 PHYSICAL THEORY OF DIFFRACTION Physical Theory of Diffraction: principle Can be written as a combination of Kirchhoff, GTD, and a correction Kir D D PTD GTD Kirch U x U x x x e ikr kr V. Zernov, L. Fradkin, and M. Darmon, "A refinement of the Kirchhoff approximation to the scattered elastic fields," Ultrasonics, vol. 5, no. 7, pp , 0 Kirchhoff + corrected GTD The Kirchhoff contribution stems from an integration over the entire surface of the defect The GTD contribution stems from an integration over the edges Uniform scattered field (no divergence): Specular reflexion Diffraction U PTD x U Kir x U PTD x D GTD αβ x eikr kr UGTD x

10 COD COD COD COD COD PHYSICAL THEORY OF DIFFRACTION Physical Theory of Diffraction: interpretation Interpretation as a decomposition of coefficients for a D planar crack crack GTD Edge Kirchhoff Edge + Kirchhoff Surface + GTD Edge Kirchhoff Edge = PTD Interpretation as a decomposition of Crack Opening Displacements Position Position COD Position = Position Position Position Better behavior then Kirchhoff at the edges, no infinite quantities to integrate

11 PHYSICAL THEORY OF DIFFRACTION Physical Theory of Diffraction: results Comparison between PTD, GTD and an exact solution Kir D D PTD GTD Kirch U x U x x x e ikr kr Specular reflexion Diffraction U PTD x U Kir x U PTD x D GTD αβ x eikr kr UGTD x Exact UTD PTD GTD 330 PTD agrees with Kirchhoff and with GTD in their respective validity domains As expected, the PTD coefficients do not diverge, whereas the GTD coefficients do

12 Physical Theory of Diffraction: implementation. Meshing of the defects PHYSICAL THEORY OF DIFFRACTION. Computation of the field at each mesh point, for each probe and each mode (fields are computed using a paraxial ray method known as the pencil method and approximated by plane waves) 3. Application of the edge diffraction coefficient (GTD-KirchhoffEdge) at the border of the defect, application of the KirchhoffSurface coefficient on the surface 4. Sum of the contributions and convolution by the reference signals GTD Edge Kirchhoff Edge + Kirchhoff Surface = PTD

13 RESULTS Comparison between CIVA results for PTD and a FEM code P waves, direct echos, various angles of incidence FEM (Athena) Zoom Same maximum for Kirchhoff, PTD and FEM (specular reflexion) Agreement between PTD and FEM for diffraction: PTD corrects Kirchhoff

14 RESULTS Comparison between CIVA results for PTD, Kirchhoff and GTD P waves, direct echos, various angles of incidence PTD=Kirchhoff PTD=GTD Diffraction 45 Specular 90 PTD KA GTD Signal confirms that PTD behaves as expected: agrees with Kirchhoff for specular and with GTD for diffraction

15 RESULTS Unfavorable case: shear waves, small defect 5MHz specular direction () () Agreement for specular direction Interferences of head waves / diffracted waves near the critical angle (around 35 ) Increased performances for larger cracks (lesser interferences)

16 Time of flight Time of flight RESULTS Corner echoes of rectangular backwall breaking notches 45 P-waves, breakwall notches : Corner effect P Corner effect S Modes conversion Experiment Scanning Corner effect P Diffraction P PP : Diffraction P P-P-P : Corner effect echo, no conversion Mixed corner echo Corner effect S 5mm x mm 5mm x 5mm 5mm x 0mm 5mm x 0mm S-S-S : Corner effect echo, no conversion Simulation S-P-P : mixed corner effect echo R.Raillon et al., UT benchmark, Rev. Of Prog. QNDE 00 Corner echoes (specular) not modified compared to the 00 benchmarck

17 CONCLUSION Summary A combination of the Kirchhoff and GTD models based on the Physical Theory of Diffraction allows combining advantages: accurate prediction of edge diffraction (GTD) accurate prediction of corner and specular echoes (Kirchhoff) Implemented in CIVA extends the validity domain of the simulation Removes the need to choose between Kirchhoff and GTD Perspectives Improvement of the Kirchhoff+GTD model around the critical angle: accounting for head waves (collaboration with Sound Mathematics Ltd.) Improvement of the modeling of defect echos in CIVA: More detailed description of the field in defect interaction (avoiding the plane wave approximation) Coupling with numerical methods