Introduction

Eddy Current NonDestructive Testing is widely used to inspect conducting materials during manufacture or in service. In this context, modelling is a powerful tool for inspection improvements : it helps probe-coil designers to optimise sensors for each examination requirement, it gives better understanding of the involved physics, it helps operator training and it also increases defect analysis reliability. Moreover, the fast computation speed of these models facilitates greatly the process of the experimental data inversion. The models developed in CIVA/MESSINE, where MESSINE stands for « Models for Electromagnetic Simplified SImulation in Nondestructive Evaluation » are based on simplified (analytical or semi-analytical) fast-running 2D and 3D solutions for the Maxwell equations. With these models among which four are presented in this paper, various kinds of studies can be led in order to optimise the Eddy Current NonDestructive Testing.

The axisymmetrical model (figure 1) is particularly adapted to the study of the design of probes and to the evaluation of the thickness of materials.

Fig 1: Unflawed axisymmetrical configurations for testing

The multi-transformer model (figure 2) is dedicated to 2D simulation of flawed materials in axisymmetric geometry.

Fig 2: Flawed axisymmetrical configurations for testing

At last, two 3D defect-free models have been developed for incident field computation of arbitrary shape 3D coil. The first one is dedicated to planar configuration (figure 3) and the second one to cylindrical configuration (figure 4).

Fig 1: 3D model for arbitrary shape 3D coil in planar configuration

Fig 4: 3D model for arbitrary shape 3D coil in cylindrical configuration

Axisymmetrical Model

The CEA has developed an axisymmetrical model based on the separation of variables for the Helmoltz equation governing the Eddy Current (EC) distribution in a conductive medium. This model is derived from a previous work from Dodd & Deeds [1]. A specific analytical development was needed to handle correctly the modelling of ferromagnetic layers in the various axisymmetric configurations : coil over a slab, bobbin coil in a tube and coil encircling a rod, see figure 1. The scalar potential is split in two functions in cylindrical coordinates A(r,z) = R(r) x Z(z). These functions are composed of a weighted sum of sine, cosine, and modified Bessel functions of the first and of the second kind. The coefficients of these weighted sums are determined by the boundary conditions at the interfaces for the magnetic flux density B and the magnetic field intensity H. The accuracy of the model was tested against finite element method (FEM) and experimental impedance measurements. An excellent matching is observed between FEM and our method, the spreading of the absolute component of the impedance for various geometries listed in figure 1 is always below 1%. The comparison with experimental measurements exhibits slightly higher discrepancies as shown in figure 5 for a coil encircling an infinite rod of ferrite. This deviation is however in the range of the experimental uncertainties coming from the dubious value of the ferrite relative permeability µr (taken from data sheets given by the manufacturer) and of the small differences between the theoretical and the actual probe geometry.

Fig 5: Relative deviation between experimental measurements and computations

Fig 6: Plot of the flux lines distorted by a conductive and a ferromagnetic layers for a tubing inspection

Fig 7: Tubing inspection setup for a differential bobbin coil

Fig 8: EC signal at 100 kHz for a double flawed tube

An example of the flux lines distorted by a conductive and a ferromagnetic layers for a tubing inspection configuration is given in figure 6. The thickness of the first layer is 1 mm, its conductivity is 1 MS/m and its relative permeability is 1. The second layer is 2 mm thick, has a zero conductivity and a relative permeability of 100. The excitation frequency is 100 kHz. The impedance computation can be used to evaluate for example the thickness of the second layer.

Multi-Transformer Model

The multi-transformer model is designed for the simulation of EC tubing inspection with axisymmetric defects. This method is based on the decomposition of the conductive parts in a set of constant current loops [2]. For each loop, the resistance and all the inductances (with the other loops and with the probe-coil) are analytically derived from the conductor geometry and the probe characteristics. The actual complex impedance of a loop is thus expressed as a linear function of the currents flowing in each loop. This set of linear equations is known as the multi-transformer formulation and is solved accordingly. The current density is deduced for each loop as well as the coil impedance. An application of this model is given in figure 7.

A 40% through wall external groove and a 10% wall internal groove are introduced in a tube. The EC signals obtained at 100 kHz when the differential bobbin coil passes by the grooves are given in figure 8. This model runs at least 10 times faster than a FEM solver and does not require to build a specific mesh. Its accuracy, compared to FEM computations and experimental data, is better than1%. These features prove the multi-transformer model to be a reliable and efficient direct model for inversion purposes.

3D Models

A first 3D model has been developed to design coil of arbitrary shape above a conductive slab or a conductive half-space [3, 4]. This model is used to compute the incident fields, the impedance of the probe and the induced currents in the conductive medium. The PCS method is based on a decomposition of the driving coil in elementary Point Current Sources (PCS). The vector potential is calculated separately for the three spatial components of an elementary current source and then combined over all the PCS to obtain the general solution. An application of this model is given in figure 9 for a pancake coil tilted of 10° over a slab.

Fig 9: Distribution of the Eddy Current induced in a slab under a pancake coil for a tilt of 10°

Fig 10: Distribution of the Eddy Current induced on a cylinder by a loop for a tilt of 10°

A second 3D model has been developed to design coil of arbitrary shape and located outside a conductive cylinder (figure 4). This model is used to compute the incident fields and the induced currents in the conductive medium. The method used to solve the Maxwell equations is based explicitly on the Green dyads formalism, which is detailed in [5]. By expressing the dyadic Green function in a harmonic decomposition with respect to the z axis and the orthoradial direction, the huge advantage of this formalism is the introduction of generalised matrices of reflexion and transmission to take into account the interaction of the electromagnetic waves with the different media. In this context, it will be relatively easy to improve this model to simulate a stratified cylinder without changing the main structure of the developed algorithm. Moreover, as the source term in this formalism is apart from the terms describing the influence of the workpiece, coils of arbitrary shape can be studied. An application of this model is given in figure 10 for a current loop tilted of 10° over a cylinder.

With these models, values for the incident field obtained will serve as an input for the introduction of the defect interaction in 3D.

Conclusion

In this paper, the different models used in MESSINE have been presented. Depending on the configuration, an analytical or semi-analytical approach was considered. The CIVA/MESSINE models are helpful for the development of new NDT EC methods or the evaluation of existing inspection procedures. They are particularly useful to assess the impact of perturbation factors (e.g. lift-off, tilt).

References

1. Dodd C.V., Deeds W.E., "Analytical Solutions to Eddy-Current Probe-Coil Problems", Journal of Applied Physics, Vol. 39, Number 6, pp 2829-2838, May 1968.
2. Berthiau G. and de Barmon B., "MESSINE : Eddy Current Modeling in CIVA", 15 th WCNDT, Roma, 15-21 October 2001.
3. de Barmon B, Berthiau G., Juillard J., "MESSINE : Models for Electromagnetic Simplified Simulatioon in Nondestructive Evaluation", Rev. Progress Quantitative Nondestructive Evaluation, Des Moines, 1-3August 1999.
4. Juillard J., de Barmon B., Berthiau G., "Simple Analytical Three-Dimensional Eddy-Current Model", IEEE Trans. on Magnetics, Vol. 36, n° 1, pp 258-266, January 2000.
5. W.C. Chew, "Waves and Fields in Inhomogeneous Media", IEEE, Picataway, 1995.

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