Simulation and Measurement of Magnetic Based Nondestructive Tester

Transcription

1 G. Kovács et al. / Journal of Advanced Research in Physics 2(1), (2011) 1 Simulation and Measurement of Magnetic Based Nondestructive Tester Gergely Kovács * and Miklós Kuczmann Laboratory of Electromagnetic Fields, Department of Telecommunications, Széchenyi István University, Egyetem tér 1, Győr, H-9026, Hungary Abstract The paper presents the analysis of a nondestructive testing equipment under investigation. There are three main parts of the research as well as this paper. The first part shows the present state of the developed nondestructive tester based on the Magnetic Flux Leakage (MFL) method, the second part reviews the simulation and the results, which have been made with the principle of the Finite Element method (FEM). The last part presents the measurement results, which have been performed with the nondestructive tester. In this paper the simulation results with the measurement results have been compared. Keywords Magnetic flux leakage method, Vector finite element method, 3D static magnetic field, LabVIEW I. INTRODUCTION THE aim of the nondestructive testing methods is to obtain some information about the specimen under test without any physical impression of the material. Here, only ferromagnetic materials have been used in the measurements. One of these methods is the so-called Magnetic Flux Leakage method, which detects the gaps, cracks, and flaws by the help of the magnetic field supplied by a source current. It is well know that the ferromagnetic materials drive the magnetic flux, but the generated flux emerges the gaps, and this effect can be measured by the appropriate sensor, such as using a little coil or a Hall-type sensor. The equipment has been built up in our laboratory, which can be used as a magnetic flux leakage tester. It can be seen in Fig. 1. The measurement system is based on a National Instruments Data Acquisition Card (NI-DAQ card connected to the computer through an USB interface) and a National Instruments LabVIEW software package [1]. The specimen and the sensor can be positioned in the x-y plane by using LabVIEW commands. There are two kinds of cracks, which have been studied. The first one is called ID crack (inner defect), that is the crack and the sensor are in the same side of the specimen, the other is the OD crack (outer defect), i.e. Manuscript received July 19, 2010 * Corresponding author (kovacs.gergely1984@gmail.com) the flaw and the sensor are in the opposite side of the material under test [2]. A U-shaped yoke placed above the specimen is used to generate the magnetic field inside the material under test. The current of coil wounded around the yoke can be a direct current or an alternating current supplied by a computer controlled current amplifier. The function of current can be set by LabVIEW functions. The leakage magnetic flux is measured by a detecting coil or a sensor, and its output is connected to the NI-DAQ card, and the measured signal can be post-processed by LabVIEW procedures or functions. For example, measured signals can be plotted, filtered, saved, or they can be the input of any further mathematical procedures. During the measurement process, the positioning, the signal generation and measurement of signal of the sensors have to be realized simultaneously. The schematic view of the measurement arrangement can be seen in Fig. 2. Fig. 1. Photo of the Magnetic Flux Leakage system

2 2 G. Kovács et al. / Journal of Advanced Research in Physics 2(1), (2011) Fig. 2. Block diagram of the measurement arrangement II. SIMULATIONS BY FEM First of all, the measurement system has been modeled as a static magnetic field problem, where the following Maxwell s equations can be used [3-7] H = J in, (1) 0, 0 y s, in 0 y s, B = 0 (2) ν B, in 0 y, H = (3) ν fpb + I, in s. Here H is the magnetic field intensity, B is the magnetic flux density, J 0 is the source current density, and ν is the reluctivity. The domain of interest has been split into three regions, the air region is denoted by 0, where ν = ν 0 ( ν 0 is the reluctivity of vacuum), the yoke is supposed to be magnetically linear, ν = ν 0 ν r ( ν r = 1/ 4000 ), and it is denoted by y, finally s is filled with ferromagnetic material. The second relation of (3) represents the nonlinearity by polarization formulation [2, 8, 9]. Here ν fp is the optimal value of reluctivity, ν max + ν ν min fp =, (4) 2 where ν max and ν min are the maximum and the minimum slope of the nonlinear characteristics (see in Fig. 3), moreover I is a nonlinear residual term determined iteratively by the fixed point technique [2, 8, 9]. The nonlinear characteristics of the specimen can be seen in Fig. 3, which has been approximated by an inverse tangent function [10]. It can be seen in Fig. 4a that the flaws are far away from each other, so they have no any effect to each other. That is why only one gap can be simulated, and some symmetry planes can be found. a) b) Fig. 4. a) The sketched model, b) The boundaries of the model The boundary surfaces are the Γ H and the Γ B boundaries (see Fig. 4b), where the following boundary conditions can be satisfied: H n = 0, on Γ H, (5) B n = 0, on Γ B. (6) On the artificial far boundary, which closes the problem region (6) must be prescribed. III. THE THREE-DIMENSIONAL MODEL A. The Vector Potential by EdgeFEM The problem has been solved by the help of the FEM. The three-dimensional model of a slot has been simulated by magnetic vector potential, B = A, (7) which satisfies (2) exactly. In 3D case, the magnetic vector potential is approximated by vector finite elements, and the edge element based FEM has been used [3, 10, 11, 12, 13]. In this case the unknowns are associated to the edges of the FEM mesh. The problem is supposed to be a static magnetic field. The partial differential equation, which solution is not sensitive to Coulomb gauge can be written as [3, 4, 10, 11] ( ν A ) = T0, in 0 y, (8) and ν Α = T - I, in (9) ( ). fp 0 s Here T 0 is the so-called impressed current vector potential, which represents the effect of the source current density through the following equation [3, 10] J 0 = 0, (10) that is T 0 = J 0. (11) The impressed current vector potential has to be represented by edge FEM approximation [3, 10]. The boundary conditions belonging to the partial differential equations of a two-dimensional static magnetic field problem can be formulated as [3] ν A + I n = 0 on Γ (12) ( ), fp, H n A = 0, on Γ B. (13) The partial differential equation (8) and the Neumann type boundary condition (12) can be summarized in the following weighted residual formulation [3, 4, 11] Fig. 3. The characteristic of the specimen

3 G. Kovács et al. / Journal of Advanced Research in Physics 2(1), (2011) 3 ΓH ( ν fp A) W d+ ( ν fp ) + W Α+ I n dγ = ( 0 ) ( ) = W T d + W I d (14) where n W = 0 on Γ, (15), B and W is a weighting function as well as the approximation function of the unknown vector potential. The second order edge finite elements have been used in this simulation [3, 10, 11]. The value of ν fp is equal to ν 0, or ν 0ν r in the air region, and in the yoke, respectively, moreover I is zero there. After using some mathematical identity and using some formulations [3, 4], the following weak equation can be obtained ( fp A) ( W) ν d = ( ) d ( ) = W T W I, 0 d (16) which solution results in the approximation of the magnetic vector potential, and from there the magnetic flux density can be obtained. The magnetic field intensity can be calculated from the magnetic flux density by using the nonlinear model. The fixed-point method The nonlinear partial differential equation has been linearized by the polarization method (the last expression in (3)), and the obtained system of equations has been solved by a fixed-point algorithm [3, 4, 10, 14]. The step of iteration is as follows: The algorithm is started by arbitrary values of vector I. The second step is the system of assembled equations is built according to (16), then the magnetic flux density B is calculated (7), and the magnetic field intensity H is determined by the nonlinear model (Fig. 4) [3, 10] B H = H 0tg, H B. (17) B0 Here H 0 = 1013A/m, and B 0 = 1.27T have been used. The nonlinear residual I is updated by the last expression in (3) I = H B. (18) ν fp The sequence 2-5 must be repeated until the error is higher than the predefined limit ε. The error is defined by ( n) ( n-1 B B ) ε, (19) where n is the number of the actual step. be realized by the functions of Comsol, and the fixed-point technique can be developed in the frame of Matlab. Two different flaws manufactured artificially have been studied, which had been compared with the measurements. One of them a longitudinal flaw, which is 1 mm wide and 5 mm long, and the width of the transversal flaw is 5mm and it is 1 mm long. The electromagnet and the artificial cracks can be seen in Fig. 5. In Fig. 6 the simulated results without any flaw can be seen. It can be seen that some variations of the components of the magnetic flux are resulted according to the legs of the yoke. Fig. 7 presents the y and the z components of the simulated magnetic flux density according to the longitudinal gap. Fig. 8 presents the y and the z components of the simulated magnetic flux density according to the transversal gap. Fig. 5. The electromagnet and the artificial cracks. IV. THE 3D SIMULATION RESULTS The FEM simulations have been performed by the help of the Comsol Multiphysics software [14]. The Comsol Multiphysics combined with the functions of Matlab can be used efficiently in the research work, since the geometry, the physics, the finite element mesh and the post-processing can Fig. 6. The y and z component without any flaw

4 4 G. Kovács et al. / Journal of Advanced Research in Physics 2(1), (2011) V. THE 3D MEASUREMENT RESULTS The aim of the measurements is to detect with the potential proportional to the magnetic flux density. The cracks have been measured with the nondestructive tester which has been expositioned in Section 1. Two different flaws manufactured artificially have been measured which have been compared with the simulations. One of them is a longitudinal flaw, which is 1 mm wide and 5 mm long, and the width of the transversal flaw is 5mm and it is 1 mm long. In the measurement a Hall sensor has been used. Fig. 9. presents the y and z components of the measured potential which is proportional to the magnetic flux density according to the longitudinal crack. The Fig. 10. from top view shows the y and the z components of the measured potential which is proportional to the magnetic flux density according to the longitudinal crack. Fig. 11. presents the y and z components of the measured potential which is proportional to the magnetic flux according to the transversal crack. Fig. 12. from top view shows the y and the z components of the measured potential which is proportional to the magnetic flux density according to the transversal crack. Fig. 7. The y and z component of the longitudinal gap Fig. 8. The y and z component of the transversal gap Fig. 9. The measurement result of the y and z component of the longitudinal crack

5 G. Kovács et al. / Journal of Advanced Research in Physics 2(1), (2011) 5 Fig. 10. The measurement result of the y and z component of the longitudinal crack top view Fig. 12. The measurement result of the y and z component of the longitudinal crack top view VI. CONCLUSIONS Comparing the measurement results with the simulation results, it can be experienced that it is possible to statement the manufactured cracks with the nondestructive tester, but it is necessary to continue to improve. The current nondestructive tester needs to be changed to more precise equipment. The sensor is not accurate. It is necessary to exchange this for more accurate sensor, for example to use a sensor matrix ACKNOWLEDGMENT This paper was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences (BO/00064/06), by Széchenyi István University ( ), by the Hungarian Scientific Research Fund (OTKA PD 73242), and by the Hungarian-Romanian Bilateral partnership (RO-46/2007). Fig. 11. The measurement result of the y and z component of the transversal crack REFERENCES [1] LabVIEW, [2] Mihalache O., Preda G., Uchimoto T., Demachi K., Miya K., Crack reconstruction in ferromagnetic materials using nonlinear FEM-BEM scheme and neural networks, Electromagnetic Nondestructive Evaluation, Edited by J. Pávó, Vol. V, IOS Press, 2001, pp [3] Kuczmann M., Iványi A., The finite element method in magnetics, Akadémiai Kiadó, Budapest, 2008.

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