Keywords

1 Introduction

The modeling of an actual configuration of CND-CF can not generally be obtained analytically and uses numerical methods. Among them, the finite element method (FEM) which allows to take into account complex geometries of probes and parts [1].

Today a wide range of digital tools is available. It is based on the implementation of theoretical models using different mathematical tools including finite element technique. Numerical simulation makes it possible to study the functioning and the properties of a modeled system as well as to predict its evolution. It is very interesting to have a simulation environment that includes the possibility of adding different physical phenomena to the studied model. It is in this philosophy that Comsol Multiphysics has been developed. It is a modular finite element numerical computation software allowing to model a large variety of physical phenomena characterizing a real problem. It will also be a design tool thanks to its ability to manage complex 3D geometries [2].

2 Use of COMSOL Multiphysics in ECT

The release of the software used for the study is the 5.3 a release.

2.1 Formulation

The system of equations in magnetic vector and electric scalar potential to solve is:

(1)

\( \vec{\varvec{A}} \) and \( \varvec{V} \) are respectively the magnetic vector potential and electric scalar potential, \( \varvec{v} \) is the magnetic reluctivity and σ the electrical conductivity of the conductive plate [3].

2.2 Geometry

Geometric and electromagnetic characteristics are shown in Table 1.

Table 1. Geometric and electromagnetic characteristics of the system
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2.3 Boundary Condition

In all the boundaries of the study domain, the magnetic induction B is supposed tangent and then the magnetic insulation condition (nxA = 0) is imposed (default condition in COMSOL Multiphysics).

2.4 Mesh

The mesh is generated with tetrahedral elements. The entire mesh consisting of 25214 domain elements, 4091 boundary elements, and 419 edge elements (Fig. 2).

2.5 Resolution

The impedance variation ∆Z is a complex number. The real part is computed with the Joule Losses (JL) in the conductive media and the imaginary part is computed with the magnetic energy (WM) in the whole meshed domain (Fig. 3).

$$ {\text{Z}} = {\text{R}} + {\text{Xi}} $$

3 Experimental Procedures (Team Workshop Problem 15)

The experimental arrangement is shown schematically in Fig. 1. Here, a circular air-cored coil is scanned, parallel to the x-axis, along the length of a rectangular slot in an aluminum alloy plate. Both the frequency and the coil lift-off are fixed and ∆Z is measured as a function of coilcenter position. The parameters for this test experiment are listed in Table 2 (Fig. 4).

Fig. 1.
figure 1

Sensor-crack system.

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Fig. 2.
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Mesh

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Fig. 3.
figure 3

Distribution of the vectors of magnetic induction B in 3D.

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Table 2. Parameters of the Problem Benchmark TEAM Workshop Pb N ° 15-1
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Fig. 4.
figure 4

Detail of TEAM Workshop Pb N ° 15-1

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4 Validation

In this section we compared the experimental results with the numerical results of 3D simulations (Figs. 5 and 6).

Fig. 5.
figure 5

Variation of the resistance according to the displacement of the sensor

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Fig. 6.
figure 6

Variation of inductance as a function of sensor displacement

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5 Conclusion

This work presents a simulation of a 3D ECT problem with the COMSOL Multiphysics 5.3a software. The use of the software to solve this type of problem is detailed and explained. The results of our COMSOL Multiphysics calculations are in very good agreement with the experimental data, this definitely permits us to examine numerous other cases in a perfect way.