Fast Finite Element Computation for 3D Eddy Current Testing Problems with T- Ω Formulation
M. Hamel
Faculty of Engineering Science M'Hamed Bouguerra University
Boumerdès, Algeria h_meziane@hotmail.com
H. Mohellebi, F. Hocini
Numerical Modeling of Electromagnetic Phenomena and Components Laboratory, Mouloud Mammeri University
Tizi-Ouzou, Algeria mohellebi@yahoo.fr
Abstract— The paper deals with T-Ω formulation modeling of eddy current problems. The method is applied to the analysis of 3D eddy current testing of metallic plate with cracks, and the usefulness of the method is investigated by comparing calculated results with measured ones.
Keywords— nondestructive testing; eddy current; cracks; T-Ω formulation; 3D finite element.
I. INTRODUCTION
Eddy current testing (ECT) is one of the most effective nondestructive testing (NDT) techniques for detecting defects in conducting materials. The cracks and defects are real threats for reliability of a structure, as they can rapidly grow to cause failures of structural integrity, to prevent these failures ECT probes are used as a predictive approach to maintain the safety of the structures [1].
In order to improve the accuracy of the ECT it is considered to be important to use numerical analysis and to optimize the method of testing and the shape of probe coils. Since the electromagnetic phenomena in the ECT are 3D in nature, 3D numerical analysis is required in order to know eddy current distribution in the conductor and to improve the ECT technique.
Therefore one of recent research interests of the ECT is the development of more effective and accurate 3D eddy current analysis [2].
Based on the finite element method, the T-Ω method is one of the most effective methods for the numerical analysis of eddy currents. In this method, the magnetic field intensity is expressed as the sum of two parts: the gradient of a magnetic scalar potential Ω and electric vector potential T. The major advantage of the formulation is the use of the scalar potential in non-conducting region, which enables considerable savings of the number of unknowns [3].
In this paper, we report a T-Ω formulation and a 3D numerical solution an eddy current system dealing with a pancake type coil, placed above a copper plate with a crack.
II. ECT SYSTEM
The ECT system deals with a pancake type coil, placed above a copper plate with a crack [4]. The probe coil has axis- symmetric shape and moves along the crack length direction, it’s characterized by the following parameters: inner radius 2.35 mm, outer radius 4.5 mm, number of turns 170. The tested sample is a cooper plate of 4 mm thickness, 60 mm width and 110 mm length. The crack having the width of 2 mm and 12 mm length at 1.3 mm depth from the sample surface.
Fig.1. Geometrical representation of the system.
III. NUMERICAL MODEL
Maxwell’s equations represent the physical model used to describe electromagnetic eddy current problems. In this work these problems are solved with the finite element method. The T-Ω formulation is used, describing the physics through the use of the electric vector potential T and the magnetic scalar potential Ω. This formulation has the advantage to permit a reduction of computing cost by decreasing the degrees of freedom from three to one in all the non-conducting region [5][6].
The displacement current is neglected assuming that the conduction currents dominate which in general is valid for eddy current problems. Neglecting the displacement current in
Ampere’s law shows that the current density must have zero divergence, enforcing both the induced and source currents to perform closed loops.
The interaction between magnetic fields and electrical phenomena is described by the following subset of Maxwell’s equations:
∇ ∧ =
∇ ∧ = −
∇. = 0
The field’s vectors are not independent since they are further correlated by the material constitutive relationships:
=
= .
Where and are the magnetic permeability and the electrical conductivity of the material. They may be field dependent and may vary in space.
For the solution of the problem we consider also the continuity condition:
∇. = 0
This states the current density solenoidality.
J is the current density.
Starting from eq. (6), we can express the current density in terms of the electric vector potential:
= ∇ ∧
We note that from eq. (1) as well, so and differ by the gradient of a scalar and have the same units:
= − ∇Ω Where Ω is a magnetic scalar potential.
In eddy current region:
Using eq. (2), eq. (4) and eq. (5) we obtain:
∇ ∧ 1
∇ ∧ + − ∇Ω = 0
And from eq. (3) we obtain:
∇. µ − ∇Ω = 0
In current-free regions the magnetic field can be found from the scalar potential:
= −∇Ω
Thus from eq. (3) we obtain in current-free regions
−∇.µ∇Ω= 0)
The divergence of is not yet defined and consequently and Ω remain ambiguous. Defining the divergence of in addition to its curl is referred to as a choice of gauge. One of the two common gauge condition used in electromagnetic is the Coulomb gauge:
∇. = 0
This condition allows to append the left-side of eq. (9) by a term ∇(!∇. ):
∇ ∧ !∇ ∧ − ∇(!∇. ) + − ∇Ω = 0
(In eddy current region) The inductance matrix for the impedance computation is given
by:
#$% =&)*+ $. %∗(,-.
/0
Where $ , %∗ , I are the magnetic flux density, the complex conjugate of the magnetic field of the conductors i, j and the current following in the coil respectively.
To find the resistance, the system computes the Ohmic losses:
R = 1
σI04 J. J∗dV Where I is the total current in the coil. 7
IV. EXPERIMENTAL SETUP
The coil impedance data were recorded using the precision LCR meter GW-INSTEK 8101G at three frequencies 50, 100 and 150 kHz. The scan starts from the middle length of the crack and the impedance values were acquired by averaging 16 measurements for each coil position and frequency.
Fig.2. Photo of the measuring system.
(1)
(2) (3)
(4) (5)
(6)
(7)
(8)
(9)
(10)
(16) (15) (14) (13) (12) (11)
0 2 4 6 8 10 12 14 16 18 0
0.02 0.04 0.06 0.08 0.1 0.12
x(mm)
DeltaZ/Zmax
exp results num results
0 2 4 6 8 10 12 14 16 18
0 0.02 0.04 0.06 0.08 0.1 0.12
x(mm)
Delta Z /Zmax
exp results num results
Fig.3. Experimental measurements showing variation of coil impedance with frequency as a function of the distance from the center of the crack.
V. RESULTS
For each frequency, the numerical results (impedance variations) are compared to the experimental data, according to the common NDT data presentation B-scan. The B-scan presentation covers a line of 18 mm from the middle length of the crack with a step of 2 mm.
The signals have been produced by subtracting the coil impedance values with the coil above an intact area of the plate from the impedance of the coil when this one is located above the crack. By subtracting, we focus on the coil impedance change only due to the presence of the crack.
Fig.4. Comparison between the experimental and numerically computed data at 50 kHz.
Fig.5. Comparison between the experimental and numerically computed data at 100 kHz.
Fig.6. Comparison between the experimental and numerically computed data at 150 kHz
Fig. 4, Fig. 5 and Fig. 6, show the comparison of the experimental and numerical data according to the coil position.
In all cases the agreement is excellent.
VI. CONCLUSION
In this study an ECT system using 3D finite element computation has been implemented for the evaluation of cracks in conducting plates.
The T-Ω formulation has been used and it has the advantage to permit a reduction of computing cost by decreasing the degrees of freedom from three to one in the entire non- conducting region.
The numerical simulations, carried out with T-Ω formulation numerical model, cross-validate the experiments data. The agreement is remarkable in all considered cases.
References
[1] H. Meziane, M. Zaidi, F. Hocini and H. Mohellebi, “LabVIEW-Based Data Acquisition System for Eddy Current Testing Probe”, IC-WNDT- MI, Annaba, November 2014.
[2] O. Biro, G. Koczka, and K. Preis,’’ Finite Element Solution of Nonlinear Eddy Current Problems with Periodic Excitation and its Industrial Appliacations’’ , Applied Numerical Mathematics , pp.3-17, May 2014 . [3] Z. Ren, “T-Ω Formulation for Eddy-Current Problems in Multiply Connected Regions”, IEEE Transaction on Magnetics, pp. 557-560, March 2002.
[4] P. Burascano, E. Cardelli, “Numerical analysis of eddy current non destructive testing JSAEM benchmark problem #6 cracks with different shapes”, Proc ENDE-V Springer, pp 333-340, June 2000.
[5] F. Bouillault, Z. Ren, A. Razek, “Calculation of 3D eddy current problems by an hybrid T-Ω method”, IEEE Trans. Magn. Vol.26, pp. 478-481, 1990.
[6] T. Kang, T. Chen, Y. Wang, K. I. Kim, “A T- ψ formulation with the penalty function term for the 3D eddy current problem in laminated structures”, App. Math & Computation, vol.271, pp.618-641, 2015.
0 2 4 6 8 10 12 14 16 18
0 2 4 6 8 10
x(mm)
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exp for 50kHz exp for 100kHz exp for 150kHz
0 2 4 6 8 10 12 14 16 18
0 0.02 0.04 0.06 0.08 0.1 0.12
x(mm)
Delta Z/Zmax
exp results num results
