Modelling of Specimen Interaction with Ferrite Cored Coils by Coupling Semi-Analytical and Numerical Techniques

Transcription

1 Modelling of Specimen Interaction with Ferrite Cored Coils by Coupling Semi-Analytical and Numerical Techniques A. Skarlatos, E. Demaldent, A. Vigneron and C. Reboud 25 th 28 th of June, Bratislava, Slovak Republic Réunion de Labo LSME 12/09/2012 PAGE 1

2 OUTLINE Use of ferrite-cored probes in eddy-current testing (ECT) applications Problem splitting: motivation and coupling principle Axisymmetric probes: Finite integration technique (FIT) /semi-analytical coupling FIT discretisation scheme Internal problem solution using the FIT 2 ½ magnetostatic formulation Propagation of the solution Reflection treatment using the modal approach 3D probes: Surface integral equation (SIE) /semi-analytical coupling Overview of the SIE approach Demonstration example: tilted axisymmetric ferrite-cored probe over a conducting half-space Conclusions and perspectives ENDE, Bratislava 25-28/06/2013 PAGE 2

3 PROBLEM SCOPE: FERRITE-CORED PROBES IN ECT APPLICATIONS Ferrite-cored industrial probes for performance optimisation STT, STL (steam generator tube inspection, EDF France) +Point (steam generator tube inspection, US-nuclear park ) Rototest, probes with C and E cores (fastener inspection, aeronautical applications) Lift-off and tilt variations are a significant noise source to the measured eddycurrent signal Strong coupling between ferrite core and specimen ENDE, Bratislava 25-28/06/2013 PAGE 3

4 COUPLING PRINCIPLE Non-symetric (3D) arrangement of symmetrical objects rotational symmetry extrusion symmetry Problem decomposition into sub-problems of simpler geometry (Uniqueness theorem) (Equivalence theorem) ENDE, Bratislava 25-28/06/2013 PAGE 4

5 OVERVIEW OF THE FINITE INTEGRATION TECHNIQUE Grid Maxwell s equations: Discrete constitutive equations: Discrete grad operator: Yee-like dual system of orthogonal grids Applicable in rectangular cylindrical and spherical grid system Grid duality: Topological properties of the discrete operators: Weiland T (1977), A discretization method for the solution of Maxwell s equations for six-component fields, Electronics and Communications AEÜ 31, PAGE 5

6 INTERNAL PROBLEM: FIT 2½D MAGNETOSTATIC FORMULATION Magnetostatic formulation for loss-less medium Solution decomposition into a partial solution and the solution of the homogeneous problem: with Axisymetrical geometries: Modal decomposition along the direction of invariance and 2D meshing on the transversal plane Discrete equation for the scalar potential for with Boundary terms ENDE, Bratislava 25-28/06/2013 PAGE 6

7 COUPLING PRINCIPLE Non-symetric (3D) arrangement of symmetrical objects rotational symmetry extrusion symmetry Problem decomposition into sub-problems of simpler geometry ENDE, Bratislava 25-28/06/2013 PAGE 7

8 Second Green s theorem SOLUTION PROPAGATION TO THE EXTERNAL DOMAIN with the scalar Green s function of the free space Decomposition of the Green s function in cylindrical functions with Application to field integral representation with No need of special treatment for the Green s function singularity Memory efficient ENDE, Bratislava 25-28/06/2013 PAGE 8

9 COUPLING PRINCIPLE Non-symetric (3D) arrangement of symmetrical objects rotational symmetry extrusion symmetry Problem decomposition into sub-problems of simpler geometry ENDE, Bratislava 25-28/06/2013 PAGE 9

10 EXTERNAL PROBLEM: SEMI-ANALYTICAL SOLUTION 2D Fourier transform of the normal incident magnetic field Reflection coefficient in the Fourier Domain (half-space) Calculation of the reflected field via inverse Fourier transform with Straight-forward generalisation for multi-layered work-pieces Applicable to other canonical geometries (tubes, etc.) Theodoulidis T (2005), Analytical model for tilted coils in eddy-current nondestructive inspection, IEEE Trans. Magn. 41, PAGE 10

11 COUPLING PRINCIPLE Non-symetric (3D) arrangement of symmetrical objects extrusion symmetry Problem decomposition into sub-problems of simpler geometry ENDE, Bratislava 25-28/06/2013 PAGE 11

12 THE SURFACE INTEGRAL EQUATION APPROACH Linear isotropic piecewise homogeneous permeable media with boundary The magnetic field is given by where formulation is the unknown of the (reduced scalar potential) integral High-order polynomial expansion for both the geometry and the unknown The number of unknowns is reduced for a given accuracy It allows the use of a direct solver A single LU decomposition is performed for all positions and iterations ENDE, Bratislava 25-28/06/2013 PAGE 12

13 CASE STUDY Tilted axisymetrical ferrite-cored probe over conducting half-space ENDE, Bratislava 25-28/06/2013 PAGE 13

14 RESULTS: FIELD ON THE HALF-SPACE INTERFACE Normal magnetic field on the interface of the half-space FEM mesh SIE mesh FEM (COMSOL) FIT/Semi-Analytical SIE/Semi-Analytical FEM: 10 min (600,000 elements, PC 24 GB, Xeon 8 cores) FIT/Semi-Analytical: ~5 min SIE/Semi-Analytical: ~5 min (standard PC) PAGE 14

15 RESULTS: CONVERGENCE STUDY FIT/Semi-Analytical Coupling: Convergence criterion: Magnetic energy at i th iteration SIE/Semi-Analytical Coupling: Convergence criterion: Internal problem solution at i th iteration PAGE 15

16 COIL IMPEDANCE: NUMERICAL VALIDATION Coil impedance as a function of the tilt angle ENDE, Bratislava 25-28/06/2013 PAGE 16

17 COIL IMPEDANCE: EFFECT OF THE FERRITE CORE Ferrite-cored vs. air-cored coil impedance Results for the air-cored coil computed using (Theodoulidis 2005) ENDE, Bratislava 25-28/06/2013 PAGE 17

18 RESULTS: +POINT FIELD Normal component of the induced magnetic field on the half-space interface ENDE, Bratislava 25-28/06/2013 PAGE 18

19 CONCLUSIONS AND PERSPECTIVES Combination of semi-analytical and numerical approach for the modelling of ferrite cored probes over planar specimen Reduced computational time in respect to 3D meshing of the entire problem (~x3 speed-up) No limitations concerning the lift-off Reduced computational noise Automatic 2D grid, lift-off and tilt parameterisation Work under way Coupling with semi-analytical solutions for tubes and fastener structures Compatible with flaw response models of CIVA Coupling with 2 ½D FIT approach for taking into account pieces with irregular profile ENDE, Bratislava 25-28/06/2013 PAGE 19

20 PAGE 20 Réunion de Labo LSME 12/09/2012 Commissariat à l énergie atomique et aux énergies alternatives Institut Carnot CEA LIST Centre de Saclay Gif-sur-Yvette Cedex T. +33 (0) F. +33 (0) Département Imagerie Simulation pour le Contrôle Laboratoire de Simulation et Modélisation en Electromagnétisme Etablissement public à caractère industriel et commercial RCS Paris B