Evaluation of a 3-D bounded defect in the wall of a metal tube at eddy current frequencies: The direct problem

Effective eddy current nondestructive evaluation of metal tubes requires accurate modeling of the fields observed in known configurations in order to understand the behavior of such fields when a defect is present, to test inversion algorithms from synthetic data, and to appraise the output of these algorithms in real cases. Here we restrict ourselves to the interaction between a non-magnetic probe displaced along a straight part of the tube when a 3-D bounded defect is found nearby in the tube wall also assumed to be non-magnetic. Modeling of the anomalous fields at a given frequency and of the variations of impedance of the probe is carried out from an exact vector domain integral formulation of the eddy current phenomenon. This formulation results from the application of the Green's theorem to the wave equations and provides the sources induced in the defect volume, from which the variation of impedance follows by application of the reciprocity theorem. All depolarization effects in the flawed region are accounted for. This formulation is handled by a conjugate-gradient FFT-based Method of Moments. Difficulties met when carrying out the integrals involving the dyadic Green's function of the tubular structure are overcome by singularity extraction and the use of the nonlinear accelerating; Shanks' transform. Results of numerical simulations for an axisymmetric probe and typical defects illustrate the effectiveness of the approach, while consequences of neglecting depolarization effects are discussed. These results are compared to those obtained by a localized nonlinear approximation and by the Born approximation, which have the advantage to bypass in part or altogether the calculation of the field inside the defect volume, and to experimental data acquired from calibration tubes.