Refinement Strategy Using 3D FEM in Eddy Current Testing
In this paper, a method is used for evaluate quickly and accurately default for the non-destructive control. The established approach is based on two finite element formulations. Two error estimators coupled with adaptive mesh software are then proposed.
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bstract In this paper, a method is used for evaluate quickly and accurately default for the non-destructive control. The established approach is based on two finite element formulations. Two error estimators coupled with adaptive mesh software are then proposed. The efficiency of the procedure is showed on an example.
1 Introduction Eddy current testing (ECT) is now often employed. To reduce cost experiments related to the ECT and to the default characterization (shapes, sizes, . . .), analytical and numerical models are used. The finite element method (FEM) yields a precise description of the geometry problem but requires much attention to get a reliable result. In most case, the signal variations in ECT due to a default (rift in a steam pipe for example) are very low. In the FEM, the calculated signal due to the default is noised by the numerical error due to the discretisation. For calculating accurately the default perturbation, two calculations can be done considering or not default [1]. So, we show on Fig. 1 the two FE problems to be solved, the meshes used into the two cases are identical, only the material of the volume default is modified. In the cases with and without default, one calculates the linkage fluxes and this generally for several positions of the sensor. The FE movement techniques are not discussed in this paper but some are described for example in [2]. Time harmonic approach is well adapted to this kind of problem, where several series of calculations Krebs Guillaume and Clenet St´ephane Laboratoire d’Electrotechnique et d’Electronique de puissance de Lille, L2EP-LAMEL, ENSAM, 8, Bd Louis XIV, 59046 Lille, France [email protected] Abakar Ali EDF R&D, LAMEL, 1 avenue du G´en´eral de Gaulle, 92141 Clamart Cedex, France [email protected] K. Guillaume et al.: Refinement Strategy Using 3D FEM in Eddy Current Testing, Studies in Computational Intelligence (SCI) 119, 63–71 (2008) c Springer-Verlag Berlin Heidelberg 2008 www.springerlink.com
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K. Guillaume et al. Problem without defaut Air m0
Dc mr s
Problem with defaut Induced currents
js
G
Air m0
Dc mr`s
js
Inspected piece
Default
Sensor
Fig. 1 Schematic diagrams of the modelled problem in the eddy current testing
are required (due to the sensor displacements and to the high working frequency). By using this method of resolution, the sensor variations dues to the default are then expressed as (for the real part of the linkages fluxes for example): ∆Φr = (Φr nodef ± Φr mesh nodef ) − (Φr def ± Φr mesh def ).
(1)
With Φr nodef and Φr def that would be the results obtained with a “perfect” mesh by using a FEM formulation. The terms Φr mesh nodef and Φr mesh def are the perturbations introduced by the numerical errors. The weaker both last terms are, less ∆Φr will be perturbed. However to obtain such accurate variations, mesh adaptation is needed, this in order to reduce the terms Φr mesh nodef and Φr mesh def . In this paper, we present a mesh adaptation procedure where the error distribution is calculated from the variation o
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