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Journal of Sensors
Volume 2012 (2012), Article ID 785271, 6 pages
http://dx.doi.org/10.1155/2012/785271
Research Article

3D Finite Volume Modeling of ENDE Using Electromagnetic T-Formulation

1Institut de Recherche en Electrotechnique et Electronique de Nantes Atlantique (IREENA), 37 Boulevard de l’Université, BP 406, 44602 Saint Nazaire, France
2Département d’Electrotechnique, Université Mohamed Kheider, Briskra 07000, Algeria

Received 16 January 2012; Revised 21 March 2012; Accepted 6 April 2012

Academic Editor: Xinyong Dong

Copyright © 2012 Yue Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An improved method which can analyze the eddy current density in conductor materials using finite volume method is proposed on the basis of Maxwell equations and T-formulation. The algorithm is applied to solve 3D electromagnetic nondestructive evaluation (E’NDE) benchmark problems. The computing code is applied to study an Inconel 600 work piece with holes or cracks. The impedance change due to the presence of the crack is evaluated and compared with the experimental data of benchmark problems No. 1 and No. 2. The results show a good agreement between both calculated and measured data.

1. Introduction

The finite volume method (FVM) is a discretization method which is well suited for the various kinds of the conservation laws of numerical simulations. Originally, the FVM was extensively used in solving the dynamic fluid problems [1, 2]. Since the beginning of the eighties, it has been introduced for the solution of Maxwell’s equations, precisely in wave propagation domain [3, 4]. In 2001, the FVM was presented as a new method to investigate the eddy current distribution problems from the teamwork problem 21 [5]. After that, FVM was used to build a 2D model to solve magnetic problems with a thin conductor compared with its skin depth [6]. Recently, the FVM has been introduced for the solution of 3D practical nonlinear magnetostatic problems [7]. Similarly, a model of 3D eddy current nondestructive testing (NDT) problems by FVM method is presented for benchmark problem JSAEM No. 6 and furthermore, the movement of the probe coil is taken into account thanks to the nonconforming mesh generation [4]. However, the problem domain should be remeshed when the position of the coil is changed.

The FVM is quite different from the finite difference method (FDM) or finite element method (FEM). Roughly speaking, the FDM is the oldest numeric method to approximate the differential equations through the use of Taylor’s expansions. The FDM becomes difficult to use when the coefficients involved in the equation are discontinuous (in the case of heterogeneous media) which is not a problem for FVM if discontinuities of coefficients occur corresponding with the boundary of “control volume” [2]. There is also a potential bottleneck of the FDM when handling the multidimension geometry [8]. These shortcomings drive to replace the basic of differential equation by integral formulation and subsequently the development of FVM and the FEM. From the industrial point of view, the FVM is known as a robust and cheap method for the discretization of conservation laws [2].(i)“Robust” means that the FVM behaves well for different formulations and focuses on the physical meaning; the obtained solution does not appear without physical meaning.(ii)“Cheap” is obtained thanks to the simple and reliable computational coding for the complex problem.

The FVM maintains the FEM’s advantages, such as the facility to deal with the complex geometry [9] and boundary conditions, but the mathematical expressions is easier than FEM. For simple geometry, the FVM uses less CPU time and memories than the FEM to solve the IEEJ problem [10] and to solve composite materials’ problems [11].

This paper proposes an investigation work on 𝑇-iterative technique based on 𝑇-Omega formulation associated with the FVM method. With the algorithm we have developed, only the electrical vector potential is calculated. Moreover, thanks to the application of Biot-Savart’s law [1113], the movements of the coil are taken into account without re-meshing the problem domain and it permits to calculate directly the magnetic field generated by the eddy current in the conductor. This paper discusses also how to deal with the crack or hole using the FVM.

To test the proposed technique, the simulated results are compared with the experienced ones which are obtained from the E’NDE benchmark problems. The aim is to evaluate the impedance change due to the presence of a crack in the center of the isotropic conductor, taking the probe movements into account. The test system is showed in Figure 1. The pancake coil is located in the center over a plate conductor. The conductor has a thickness-through hole in the center. The coil moves along the direction of the crack length (𝑥-axis).

785271.fig.001
Figure 1: The system to be solved.

2. Formulation and Algorithm

A-V formulation and 𝑇-Omega formulation are used a lot to solve electromagnetic problems. Compared with A-V formulation, 𝑇-Omega formulation has the advantage of diminishing the number of unknown elements being solved in the system [14]. In 𝑇-Omega formulation, the electric vector potential 𝑇 exists only in the conductor and the magnetic scalar potential Omega does not exist only in the conductor but also in the air. However, solely the conductor is interesting in our problem. Therefore, we want to use a simple method similar with 𝑇-Omega formulation, considering only the electric potential 𝑇. Some authors have already proposed 𝑇-iterative method where the Omega is not included [13]. To begin with the Maxwell equations and the supplementary equations𝜕𝐵curl𝐻=𝐽,(1)curl𝐸=𝐻𝜕𝑡,(2)𝐽=𝜎𝐸,(3)𝐵=𝜇(4) we can obtain the final formulation as𝜎curl1𝑇𝜕𝐻curl=𝜇,𝜕𝑡(5) where 𝑇 is the electric vector potential, 𝐻 is the magnetic field, 𝜎 and 𝜇 are electrical conductivity and magnetic permeability; respectively, 𝐸 is the electric field, and 𝐵 is the magnetic flux density.

To explain the formulation to calculate the eddy current in the conductor, the scheme is shown in Figure 2.

785271.fig.002
Figure 2: The illustration of proposed calculation.

At the beginning, the magnetic field created by the coil is applied directly to the conductor by using Biot-Savart’s law [15]:𝐻𝑠=14𝜋𝑉𝐽𝑠×𝑟𝑟3𝑑𝑣,(6) where 𝐻𝑠 is the applied source magnetic field, 𝐽𝑠 is the source current density, and 𝑟 is the distance between the source and the conductor. Obviously, the eddy current excited by this magnetic field in the conductor is obtained by using (5). According to the equation of electric current conservation, the eddy current density can be derived as 𝐽div=0𝐽=curl𝑇.(7) Meanwhile, the eddy currents in the conductor generate another magnetic field 𝐻𝑐 which is opposite to the source magnetic field in the conductor. This field can be calculated by reusing the Biot-Savart law:𝐻𝑐=14𝜋𝑉𝐽×𝑟𝑟3𝑑𝑣,(8) where 𝑟 represents the distance between the volumes in the conductor. Afterwards, it is added to the source magnetic field to replace the former source field. Thus, the cycle continues to form an iterative process, while the vector 𝑇 and its normal component on the surface are concerned to calculate 𝐻𝑐 by using computer code “ELMES” [13, 16].

To carry out such a modeling, the association of FVM to this iterative process is used. Additionally, it is necessary to control the change of the variables in the iterative process of solution. A relaxation is introduced in the following equation which is based on relaxation factor 𝛼. The iteration continues until the convergence is reached. The convergence 𝜀 is assigned to 106. It can accelerate the convergence and reduce the number of iterations, consequently the computation time. For each iteration, the correction added to new estimation of the solution vector is given by𝑇𝑖𝑇=𝛼𝑖𝑇+(1𝛼)𝑖1,(9) where 𝑇𝑖 and 𝑇𝑖1 present the 𝑖th and (𝑖1)th iteration of electric vector potential, respectively. The algorithm of this iterative process is shown in Figure 3.

785271.fig.003
Figure 3: The algorithm of the calculation for FVM 𝑇-iteration.

3. Finite Volume Method

As introduced in [1, 4], the principle of the FVM method is to divide the whole domain into discrete control volume. The surfaces of control volume are positioned midway between adjacent points. Thus, each point is surrounded by a control volume or cell. A general point is indentified by 𝑃 and its neighbors in three-dimensional geometry, the nodes to the west and east along axis 𝑥 are identified by 𝐸 and 𝑊, the nodes to the west and east along axis 𝑦 are 𝑁 and 𝑆, the nodes to the top and bottom along axis 𝑧 are, respectively, 𝑇 and 𝐵 (see Figure 4). The integration of differential forms of this formulation for each volume 𝐷𝑃 belonging to the conductor can be written as𝐷𝑝curl𝜎1curl𝑇𝑑𝑣𝑃=𝐷𝑃𝜕𝐻𝜇𝜕𝑡𝑑𝑣𝑃,(10) where 𝑣𝑃 represents the volume of element 𝐷𝑃. By developing (10), we find that the inverses of the surface conductivities are used on the left side. The conductivity is constant everywhere in the conductor for the materials without crack. But if there is a crack, the conductivity is considered as zero which is a different material connected to the conductor. It brings a problem when calculating the inverse of it. To deal with this issue, the conductivity is set at a value which is several thousand times smaller than the one of the conductor [17]. Moreover, the control volumes are divided in order to coincide with the boundary of two materials; thus the method with which the conductivity of the interface is treated. As shown in Figure 4, the physical interface represents a surface in the y-z plane. Inside, the elements 𝐷𝑃 and 𝐷𝐸 are two connected control volumes in two different materials with a common interface 𝑒(𝑤).

785271.fig.004
Figure 4: Two neighbor volumes.

In fact, they belong to different volumes but should have the same conductivity. A simple approximation for the conditions of passage between two media is to make the geometric mean of the two physical volumes at the interface [7, 18]. Thus, for the interface 𝑒(𝑤), the inverse of conductivity can be written as𝜌𝑒/𝑤=𝜌𝑃Δ𝑥𝑃+𝜌𝐸Δ𝑥𝐸Δ𝑥𝑃+Δ𝑥𝐸,(11) where 𝜌𝑃 and 𝜌𝐸 are the inverse of the conductivities of the volumes 𝐷𝑃 and 𝐷𝐸, respectively. Equally, the inverses of the conductivities on the other interfaces keep their values.

4. Validation and Application

There were a lot of discussions on the E’NDE benchmark problems. Bowler et al. [19] describe theoretical predictions of eddy current probe responses for NDT and compared with the benchmark problem. A boundary integral vector potential formulation has been developed to evaluate the eddy current interactions [20]; the formulas of the electric dyadic Green’s function for a limited thickness plate are derived [21]. Some commercial software is used to simulate the eddy current distribution in the plate of this benchmark problem [22, 23]. The proposed algorithm associated with FVM method is applied to the E’NDE benchmark problems No. 1 and No. 2 (as shown in Figure 1). These problems deal with a pancake coil, placed above a conductive square plate in the centre with a lift-off of 0.5 mm. The thickness of the plate is 1.25 mm. The coil has ax-symmetric shape and is made of 140 turns, supplied with an alternative current of 0.007 A and 150 kHz or 300 kHz. Its inner and outer diameters are 𝑅1=1.2 mm and 𝑅2=3.2 mm. The height is 0.8 mm. The parameters of the coil are shown in Figure 5. The plate has an electric conductivity 𝜎=106S/m and a relative magnetic permeability 𝜇𝑟=1.

785271.fig.005
Figure 5: Pancake coil parameters.

For problem No. 1, the dimension of the plate is 40 mm × 40 mm. The coil is fixed over the centre of the plate. In this problem, the solution requires the determination of the impedance change of the plate. When we use Biot-Savart’s law to calculate the magnetic field in the conductor created by eddy current, it risks running “out of memory” in Matlab as the matrix is full. To reduce the full matrix to a “sparse” one, we suppose that the eddy currents far away from the coil create a small magnetic field that can be neglected. It means that the maximum distance for the computation is limited to several times the outer radius of the coil. The impedance change of the plate and the difference between the measured and simulated resulted are shown in Figure 6 at the frequency of 150 kHz.

fig6
Figure 6: (a) Impedance change of the plate. (b) Difference of the impedance.

From these figures, we find that the impedance change approaches the measurement from 4×𝑅2 and it begins to remain stable. An approximation of this fact is illustrated in Figure 7.

785271.fig.007
Figure 7: The limitation of the matrix using Biot-Savart’s law.

The impedance changes which are going be evaluated by the proposed method are compared with those obtained from measurements at the frequencies of 150 kHz and 300 kHz (see Table 1). Δ𝑍,Δ𝑅, and Δ𝑋 represent the impedance change, real part, and imaginary part of impedance change, respectively.

tab1
Table 1: Impedance change of the plate.

From the table, there is, respectively, 4.72% and 6.96% differences for the two frequencies between the simulated and measured results. The fact that we use the same meshes in both cases explains why in case of 300 kHz the error is higher than that in case of 150 kHz. To solve the system, a preconditioning method developed in [24] is used, and the stabilized biconjugate gradient solver (BICGSTAB) is applied. The algorithm used 237 iterations, and BICGSTAB used 27 iterations to inverse the matrix on a 2.53 GHz Intel Core 2 PC, where the plate has been divided into 14400 volumes.

For problem No. 2, the size of the plate is 140mm×140mm; the rectangular thickness-through crack has the dimension of 10 mm length and 0.2 mm width. The coil moves parallel to the 𝑥 axis, along the crack length direction from the centre (𝑥=0mm) to 𝑥=10mm, with a step of 2mm. The solution of this problem requires calculating the impedance change due to the presence of the hole. It can be regarded as the difference between the impedances of the plate without hole and with hole. The impedance change (Δ𝑍) in the function of the movements of the coil is represented in Figure 8. These figures show a comparison between the calculated values obtained with the proposed method and the measured results.

fig8
Figure 8: The impedance changes in function of the movement of the coil for two frequencies.

We can see that the impedance change becomes smaller when the coil is far from the centre of the hole. When the coil is at about 10 mm from the centre of the hole, the impedance remains the same as if there was not a hole in the plate. For the frequency of 150 kHz, the skin depth is a little greater than the thickness of the plate while it is smaller than the thickness for the frequency of 300 kHz. Due to the issue of skin depth, BICGSTAB used 500 iterations to inverse the matrix and the algorithm used 1500 iterations for every position on the same PC, where the plate has been divided into 34000 volumes at the frequency of 150 kHz. However, 1200 iterations to inverse the matrix, 2100 iterations for every position, and 45000 volumes are needed for 300 kHz.

5. Conclusion

The E’NDE benchmark problems No. 1 and No. 2 have been solved by the 𝑇-iteration technique associated with the FVM method. Thanks to Biot-Savart’s law, the mesh of the plate remains the same when the coil moves and the magnetic fields created by the eddy current are evaluated. A limitation of reducing the full matrix is applied. The approximation of interface conductivity calculation for the FVM has shown its efficiency to deal with cracks or holes in the conductor. The simulation results are in agreement with the benchmark problems.

References

  1. H. K. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Prentice Hall, New York, NY, USA, 2nd edition, 2007.
  2. R. Eymard, T. Gallouët, and R. Herbin, Finite Volume Methodes, vol. 7 of Handbook of Numerical Analysis, 2006, update version of 1997.
  3. R. J. Leveque, “High resolution finite volume methods on arbitrary grids via wave propagation,” Journal of Computational Physics, vol. 78, no. 1, pp. 36–63, 1988. View at Scopus
  4. A. Cheriet, M. Feliachi, and S. M. Mimoune, “3D movement simulation technique in FVM method application to eddy current non destructive testing,” The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 28, no. 1, pp. 77–84, 2009. View at Publisher · View at Google Scholar · View at Scopus
  5. X. Yaoqin, L. Feng, and M. Xinshan, “Computation of eddy current problems by the finite volume method,” in Proceedings of the 5th International Conference on Electrical Machines and Systems, vol. 2, pp. 1117–1120, 2001.
  6. J. Zou, Y. Q. Xie, J. S. Yuan et al., “Analysis of the thin plate eddy-current problem by finite volume method,” IEEE Transactions on Magnetics, vol. 40, no. 2, pp. 1370–1373, 2004. View at Publisher · View at Google Scholar · View at Scopus
  7. A. Cheriet, A. Zaoui, M. Feliachi, and S. M. Mimoune, “3-D nonlinear magnetostatic field computation with finite volume method by means of M-B iteration,” in Proceedings of the 5ème Conférence Européenne sur les Méthodes Numériques en Électromagnétisme (NUMELEC '06), Lille France, 2006, Proc., Art. 25.
  8. J. Peiro and S. Sherwin, Finite Difference, Finite Element and Finite Volume Method for Partial Differential Equations, Article 8.2, Handbook of Materials Modeling, 2005.
  9. G. Dong, J. Zou, R. H. Bayford et al., “The comparison between FVM and FEM for EIT forward problem,” IEEE Transactions on Magnetics, vol. 41, no. 5, pp. 1468–1471, 2005. View at Publisher · View at Google Scholar · View at Scopus
  10. A. Cheriet, A. Zaoui, M. Feliachi, and S. M. Mimoune, “Computational performance comparison between FVM and FEM for 3-D magnetostatic problems,” in Proceedings of the 5eme Conférence Européenne sur les Méthodes Numériques en Electromagnétisme (NUMELEC '06), 2006, Proc., Art. 47.
  11. Y. Li, G. Berthiau, M. Feliachi, and B. Ramdane, “Association of electromagnetic T-omega formulation and finite volume method for CFCM material modeling,” in Proceedings of the 15th International Symposium on Applied Electromagnetics and Mechanics, 2011.
  12. H. Menana and M. Feliachi, “3-D Eddy current computation in carbon-fiber reinforced composites,” IEEE Transactions on Magnetics, vol. 45, no. 3, pp. 1008–1011, 2009. View at Publisher · View at Google Scholar · View at Scopus
  13. T. Takagi, T. Sugiura, K. Miyata, S. Norimatsu, K. Okamura, and K. Miya, “Iterative solution technique for 3-D eddy current analysis using T-method,” IEEE Transactions on Magnetics, vol. 24, no. 6, pp. 2682–2684, 1988. View at Publisher · View at Google Scholar · View at Scopus
  14. F. Bouillault, Z. Ren, and A. Razek, “Calculation of 3D eddy current problems by an hybrid T-Ω method,” IEEE Transactions on Magnetics, vol. 26, no. 2, pp. 478–481, 1990. View at Publisher · View at Google Scholar · View at Scopus
  15. M. Gyimesi, D. Lavers, T. Pawlak, and D. Ostergaard, “Biot-savart integration for bars and arcs,” IEEE Transactions on Magnetics, vol. 29, no. 6, pp. 2389–2391, 1993. View at Publisher · View at Google Scholar · View at Scopus
  16. T. Takagi, M. Hashimoto, S. Arita, S. Norimatsu, T. Sugiura, and K. Miya, “Experimental verification of 3D eddy current analysis code using T-method,” IEEE Transactions on Magnetics, vol. 26, no. 2, pp. 474–477, 1990. View at Publisher · View at Google Scholar · View at Scopus
  17. O. Biro, K. Preis, W. Renhart, K. R. Richter, and G. Vrisk, “Performance of different vector potential formulations in solving multiply connected 3-D eddy current problems,” IEEE Transactions on Magnetics, vol. 26, no. 2, pp. 438–441, 1990. View at Publisher · View at Google Scholar · View at Scopus
  18. E. T. Chung, Q. Du, and J. Zou, “Convergence analysis of a finite volume method for Maxwell's equations in nonhomogeneous media,” SIAM Journal on Numerical Analysis, vol. 41, no. 1, pp. 37–63, 2003. View at Publisher · View at Google Scholar · View at Scopus
  19. J. R. Bowler, S. A. Jenkins, L. D. Sabbagh, and H. A. Sabbagh, “Eddy-current probe impedance due to a volumetric flaw,” Journal of Applied Physics, vol. 70, no. 3, pp. 1107–1114, 1991. View at Publisher · View at Google Scholar · View at Scopus
  20. Y. Yoshida and J. R. Bowler, “Vector potential integral formulation for eddy-current probe response to cracks,” IEEE Transactions on Magnetics, vol. 36, no. 2, pp. 461–469, 2000. View at Publisher · View at Google Scholar · View at Scopus
  21. L. Santandrea and Y. Le Bihan, “Using comsol multiphysics in eddy current non destructive testing context,” in Proceedings of the Comsol Conference Paris, vol. 1, Paris, France, 2010, HAL: hal-00557541.
  22. C. Dezhi, K. R. Shao, S. Jianni, and Y. Weili, “Eddy current interaction with a thin-opening crack in a plate conductor,” IEEE Transactions on Magnetics, vol. 36, no. 4, pp. 1745–1749, 2000. View at Publisher · View at Google Scholar · View at Scopus
  23. J. Cheng, J. Qiu, T. Takagi, and T. Uchimoto, “Eddy current numerical evaluation of carbon-fibre reinforced polymer by finite element method,” in Proceedings of the 15th International Symposium on Applied Electromagnetics and Mechanics, Naples, Italy, 2011.
  24. F. Azzouz, Contribution à la modélisation de tôles ferromagnétiques par impédance de surface et élément finis, Thèse de doctorat, Ecole Doctorale Sciences et Technologies de l’Information et des Matériaux, Université de Nantes-France, 2000.