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doi:10.1016/j.ph

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Physica B 372 (2006) 373–377

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Nonlinear simulation of a nondestructive testing measurement system

Miklos Kuczmanna,�, Amalia Ivanyib

aSzechenyi Istvan University, Institute of Informatics and Electrical Engineering,

Department of Telecommunication, Egyetem ter 1, H-9026, Gy +or, HungarybBudapest University of Technology and Economics, Department of Broadband Infocommunications and Electromagnetic Theory, Hungary

Abstract

A nondestructive testing (NDT) equipment has been fabricated. The description of the measurement set-up with some measured crack

signals applying a Hall-type sensor, furthermore, the T;C–C potential formulation of the nonlinear eddy current field problem in the time

domain can be found in this paper. The hysteresis characteristic of the material has been simulated by the previously developed neural

network (NN) based isotropic vector hysteresis model. The nonlinear system of equations has been solved by the fixed-point iteration

scheme via the polarization method. Comparisons between the results of the 3D simulations and the measurements are also presented.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Fixed-point technique; Vector hysteresis; FEM; NDT

1. Introduction

A NDT equipment has been built to perform testmeasurements to check the developed 3D FEM software.The measuring arrangement allow to localize well definedartificial surface holes and slots. The nonlinear 3D eddycurrent field problem has been solved by the FEM usingtetrahedral elements. The T;C–C potential formulationhas been studied; the current vector potential T and themagnetic scalar potential C have been approximated byedge shape functions and nodal shape functions, respec-tively. The NN based isotropic vector hysteresis model hasbeen identified from first order reversal curves measured ona toroidal shape specimen [1,2]. After applying the weakform of the partial differential equations obtained fromMaxwell’s equations, the resulting nonlinear system ofequations has been solved by the fixed-point iterationscheme linearized by the polarization method. Compar-isons between measurements and simulations show goodagreements.

e front matter r 2005 Elsevier B.V. All rights reserved.

ysb.2005.10.115

ng author. Fax: +3696 429 137.

ss: kuczmann@sze.hu (M. Kuczmann).

2. The NDT measurement set-up

A photo about the installed NDT arrangement can beseen in Fig. 1. The U-shaped yoke can be magnetized bythe excitation coils (number of turns is 129). A specimenwith well defined artificial surface cracks can be insertedamong the legs of the yoke. The yoke and the specimen aremade of the same magnetic material (C19 structural steel)with the thickness of 5mm (the size of the wholearrangement is 320� 240� 5mm). In our experiments,we have applied the Hall-type sensor to measure theleakage magnetic field above the tested specimen. Theapplied Hall sensor can measure the three orthogonalcomponents of the magnetic field. In this case, the specimenis magnetized by a longitudinal magnetic flux.The schematic view of the built measurement set-up can be

seen in Fig. 2. This arrangement contains a positioning devicecontrolled by a pulse width modulation signalðf PWM ¼ 10 kHzÞ. The duty cycle of the PWM signal controlsthe position of the applied sensor above the tested specimenon the x–y plane. The measurement of the x, the y and the z

components of the magnetic field intensity vector can bepicked out by using a 3D sensor which contains threeorthogonal sensors with appropriate electronics. All taskshave been handled by the software package LabVIEW on a

ARTICLE IN PRESSM. Kuczmann, A. Ivanyi / Physica B 372 (2006) 373–377374

user-friendly graphical user interface. The excitation signalrepresented by its discrete set has been generated by asoftware based on the instrument control toolbox ofMATLAB and has been downloaded into the memory ofthe power supply KIKUSUI using the GPIB bus. KIKUSUIcan follow this signal continuously. The current generationand the measurements can be separated by this method. Atthe scanning points, measurements have been performedthrough the NI-DAQ card built in the PC. After measuringsome points, the effective value of the measured signal can becalculated. Acquired data are stored and saved to a fileautomatically during the scanning and measuring process.

A 3D GaAs Hall-type sensor (type OH10003) has beendeveloped [3]. In the NDT measurements, we used only thez directed sensor which is perpendicular to the surface ofthe specimen, because our experiment is that, the outputvoltage of the z directed sensor is the most effective forNDT measurements.

3. Measurement results

The core has been excited by sinusoidal signalðbI ¼ 8A; f ¼ 1HzÞ. In this case, the effect of hysteresis

Fig. 1. The installed NDT measuring arrangement.

Excitation c

Excitation waveform

KIKUSUI PBX

GPIB bus

Fig. 2. The schematic view of

characteristic can be sensed. Before measurements, theyoke and the specimen have been demagnetized by asinusoidal signal with decreasing amplitude.We have performed some preliminary measurements

along a scanning line through a crack to find thebest excitation. We have set the amplitudeðbI ¼ 2; . . . ; 10AÞ, and the frequency ðf ¼ 0:1; . . . ; 10HzÞof the exciting current to find the best set. Wehave experimented that, increasing the amplituderesults increasing amplitude of the measured cracksignal, but increasing the frequency in the givenrange results no significant modifications. If the amplitude,and the frequency are chosen as bI ¼ 8A and f ¼ 1Hz,respectively, the best results can be obtained. Wetried to use small frequency, because the hysteresis modelapplied in the simulations is a frequency independentmodel. The scanning area was 40� 40mm around thecrack and the measurements have been performed 41� 41points (the crack is in the origin). After measuring fiveperiods, the effective value of the measured voltage hasbeen calculated, then the crackless signal has beensubtracted from it.The output voltage of the sensor had zero value just

above the crack and had maximum value before and afterthe crack. These two peaks results accurate contour plots.In the first case, the effect of diameter of the surface holehas been measured (the depth of the cracks is d ¼ 2:5mm),then the effect of depth of the hole (the diameter is2mm) has been analyzed (Fig. 3(a)–(c)). Increasingthe diameter (or the depth) results increasing value of theoutput voltage. Finally, the orientation of two surface slotsð5mm� 1mmÞ has been detected. The first slot isperpendicular to the direction of the scanningline, the second one is transversal to it. The top viewof the measured voltages can be seen in Fig. 4(a) and (b).The perpendicular slot can be detected easily, becausethe peak value of the measured voltage is about0.15V, and the maximum value of the measured voltageaccording to the transversal slot is just about 0.06V, butthe contour plot of measurements show the orientation ofthe cracks. The orientation and the size of the slots isdeducible.

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M. Kuczmann, A. Ivanyi / Physica B 372 (2006) 373–377 375

4. Nonlinear simulation

In eddy current field problems, the electric and themagnetic fields are coupled as described by the well knownMaxwell’s equations [4].

The boundary conditions of the eddy current fieldproblem can be given as follows. At the artificial farboundaries of the nonconducting region (air), homoge-neous boundary conditions are assumed, i.e. the tangentialcomponent of the magnetic field intensity or the normalcomponent of the magnetic flux density is zero, H � n ¼ 0

or B � n ¼ 0, where n is the outer normal unit vector of theregion. Homogeneous boundary conditions can be sup-posed along the symmetry planes: H � n ¼ 0, i.e. J � n ¼ 0and E � n ¼ 0. On the interface between the conducting(yoke) and nonconducting (air) regions, the interfaceconditions must be satisfied: H � n and B � n must becontinuous and J � n ¼ 0.

Starting from r � J ¼ 0, the current density J can bedescribed by two current vector potentials [3,4]: one is theknown current vector potential T0 calculated by theBiot–Savart law, corresponding to the exciting current,the other is the unknown current vector potential T,defined only in the conducting material, taking intoaccount the effect of the eddy currents, i.e.J ¼ r� T0 þ r� T, that is J0 ¼ r� T0 and Je ¼ r� T.

The magnetic field intensity H can be approximated bythe magnetic scalar potential C and by the current vectorpotentials T and T0 as [4]:

H ¼T0 � rC in air containing the coil,

T0 þ T �rC in conducting material.

((1)

The problem can be divided into two parts: there is a timevarying magnetic field problem in air, and an eddy currentproblem in the conducting material. The two parts must becoupled by the interface conditions. This is the so calledT;C–C potential formulation.The constitutive relation between the magnetic field

intensity H and the magnetic flux density B can belinearized by the polarization method [3], i.e.

B ¼HfHg ¼ mFPH þ RFP, (2)

where mFP is an ideal permeability, selected by the relationmFP ¼ ðmmax þ mminÞ=2, where mmax and mmin are themaximum and the minimum slope of the major hysteresisloop of the hysteresis characteristic, and RFP is thenonlinear residual term. The value of mFP is constantduring the calculations. The residual term RFP is derivediteratively during the nonlinear iteration scheme at everytime step.

ARTICLE IN PRESSM. Kuczmann, A. Ivanyi / Physica B 372 (2006) 373–377376

Using the linearized constitutive relation (2) in nonlinearmedia, and the second potential formulation in (1) inMaxwell’s equations leads to the following partial differ-ential equations:

r �1

sr � T þ mFP _T � mFPr _C

¼ �r�1

sr � T0 � mFP _T0 � _RFP, ð3Þ

r � fmFPT0 þ mFPT � mFPrCþ RFPg ¼ 0. (4)

The partial differential equation r � fm0T0 � m0rCg ¼ 0valid in the air region must be coupled with the abovepartial differential equations (3) and (4). The boundaryconditions are the following: the magnetic scalar potentialalong the boundary and the symmetry plane as well as thetangential component of the unknown current vectorpotential along the symmetry plane and along the interfacebetween the conducting and the nonconducting materialare set to be zero.

The weak form of the above partial differentialequations leads to a system of linearized equations whichmust be solved by the fixed-point iteration scheme.

The size of a manufactured crack is very smallcomparing with the size of the whole arrangement: e.g.the diameter of a surface hole is maximum 2.5mm and itsdepth is maximum 5mm, the size of the U-shaped yoke is320� 240� 5mm, i.e. the order is two. The generatedmesh of the whole arrangement containing the crack is notaccurate to simulate the effect of a crack. If a very densemesh is used, then the number of unknowns is increasingvery much (about 80 000–120 000 elements or more). Onthe other hand, if a coarse mesh is applied, then the systemof equations is not very big, but the result around the crackis inaccurate, since the number of elements around thecrack is very few.

A global to local model (domain-decomposition) hasbeen applied in two steps. The domain-decompositionmethod has been generalized in the time domain based onthe static electric field calculations in [3,5]. First, a large-scale model without any crack is used to determine the

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boundary conditions of the local model, then the localdomain including the crack is investigated. The size of thelocal domain is 40� 40� 20mm, x 2 ½170; . . . ; 210�mm,y 2 ½�40; . . . ; 0�mm, z 2 ½�10; . . . ; 10�mm, and the place ofthe crack is x ¼ 190mm, y ¼ �20mm.

5. Numerical results

First, the magnetic field intensity map according to theholes has been analyzed. The output voltage of the sensoraccording to the holes with the diameter of 3mm and of2mm has been simulated, and the results are plotted inFig. 5(a) and (b). In this case, the Hall sensor is movingthrough the scanning line x ¼ 190mm, y ¼ �40; . . . ; 0mmwhich is along the center point of the cracks. A comparisonbetween the measured and the simulated signals accordingto the hole with the depth of 2:5mm can be seen in Fig.5(c). Simulated signals shown in Fig. 5(b) and (c) are thesame, because the two cracks are the same on anotherspecimen. The values of the measured signals are different,however the waveform is similar.

6. Conclusions

A more precise measurement arrangement would givemuch better results, however we could simulate accuratelythe test equipment. The comparison between the measuredand the simulated induced voltages has shown very goodagreement. This global quantity can be measured simplerthan crack signals, because the errors from measurementsare averaging.

Acknowledgements

The research work has been developed in the frame ofthe Hungarian–Romanian Bilateral IntergovernmentalS&T Cooperation Project RO-10/2002, and supported bythe Hungarian National Scientific Research Foundation,OTKA 2002, Pr. no. T 034 164 ELE Project.

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References

[1] M. Kuczmann, et al., Compel 22 (3) (2003) 730.

[2] P. Kis, J. Electr. Eng. 53 (10) (2002) 173.

[3] M. Kuczmann, et al., in: Proceedings of the 11th IGTE Symposium,

2004, p. 376.

[4] O. Bıro, Comput. Meth. Appl. Mech. Eng. 169 (1999) 391.

[5] I. Sebestyen, IEEE Trans. Magn. 38 (2) (2002) 1213.