Finite Element Computational Model for Defect Simulation and Detection by Eddy Currents Non Destructive Testing

Transcription

1 Finite Element Computational Model fo Defect Simulation and Detection by Eddy Cuents Non Destuctive Testing M. RACHEK, M. ZAOUA, H. DENOUN, C. BROUCHE Faculté de Génie Electique et de l nfomatique. Dépatement d Electotechnique Univesité Mouloud Mammei de Tizi-ouzou BP 7 RP Tizi-ouzou Algeia Abstact: - This pape pesents numeical modelling of eddy cuents non-destuctive testing in bidimensionnal cylindical plane. The magnetostatic and magnetodynamic model in tem of magnetic vecto potential is solved using the finite element method. The displacement of the pobe opeand in diffeential mode is simulated with the geometical band technique based on the physical popeties assignment and the impedance change at each position of the pobe pemit to keep the pesence of defect and it influence on the eddy cuent distibution. A compaison of the esults given by the implemented model with expeimental ones is in good ageement. Key-Wods: Finite element method, eddy cuent non-destuctive testing, impedance calculating, geometical band of movement. ntoduction A vaiety of non-destuctive testing (NDT) based on physical pinciples ae available fo detection and chaacteization of geometical and physical anomalies in mateials. One widely NDT technique fo detection and evaluation of suface and subsuface defects in electically conducting mateials is the eddy cuents testing (ECT). This technique is employed in all types of engineeing industies, aeospace, nuclea, defense, tanspot and petochemical industies. The chaacteistics esponsible fo such populaity fo the technique incluse simplicity, higth sensitive, non-contact opeation, vesatility, high speed testing, and scope fo eal time analysis. Nevetheless (ECT) pesent some disadvantages such as, a geat sensitivity of the induced eddy cuents compaed to the lift-off, the mateials to be contolled must be conductive and the exact knowledge of a defect (position, dimension) equies use seveal fequencies [][]. The eddy cuents testing technique woks on the pinciples of electomagnetic induction, it s consisted on the detection of the magnetic field due to the eddy cuents induced on the tested specimen. The pesence of the defect modifies the eddy cuents patten and hence gives ise to field petubation closely elated to the position and shape of the defects. The distibution of the eddy cuent in the pobes depends on the vaious paametes such as, excitation fequency, conductivity and pemeability of the pobe, and also the pesence of mateial defect. The excitation field is caied out by using a coil fed by an altenating cuent and the changed impedance coil can be measued o computed to account the defect influence on the induced cuents. n the pesent wok, numeical model based on the finite element method is implemented to undestand inteactions between fields and defects mateials. Stating fom the Maxwell s equation eddy cuent testing phenomenon can be expessed in the fom patial deivative equation in tem of magnetic vecto potential and can be solved numeically fo obtained the fields and eddy cuents distibutions in ode to calculate the global quantities such as the impedance vaiation. The displacement of the pobe is simulated using the geometical band technique based on the physical popety assignment. Equations govening eddy cuent testing phenomenon Eddy cuants test phenomenon can be explained with the help of the Maxwell s equations with the absence of displacement cuents [3]: H = J s + J e () B = () B E = (3) t Whee the vaious quantities involved ae the magnetic field H, the applied souce cuent density J s, the induced eddy cuent density J e, the

2 electical field intensity E and the magnetic flux density B. The above field s equations ae supplemented by a constitutive elation that descibes the behaviou of electomagnetic mateials. B = µ. H (4) J e = σ E (5) Whee µ and σ is the pemeability and the conductivity of mateials espectively. The field s fomulations pesent the disadvantage of discontinuity at the intefaces; to mitigate these poblems the magnetic vecto potential defined by the equation () is intoduced. B = A (6) Substituting equation (6) in equation (), we get: A E = V (7) t Consideing a linea and isotopic homogeneous mateials and substituting (6), (7), (4), (5) in the equation () the esulting patial deivative equations in tem of the magnetic vecto potential fo such egions can be witten: ai υ A = J s pobe (8) σωa load ( ( ) With J s = σ V the cuent density souce elatively fo voltage excitation. n most pactical eddy cuent test situation the obect geomety and pobe ae otationally symmetic and can be simplified fo modelling puposes to D adial/axial plane of the cylindical coodinate system. n otationally symmetical geomety with (, ϕ, z) coodinates, the cuent in the excitation coil flows in the ϕ diection. n such a configuation, the magnetic vecto potential A has only one component A ϕ, with intoducing the modified magnetic potential A = Aϕ the equations (8) takes the following fom in the, z plane: ( ) υ A + σ A = J s z z ϕ σ ωa ai pobe load 3 Finite element fomulation and the calculating impedance vaiations (9) 3. Finite element fomulation The modified magnetic vecto potential is finite element appoached by intepolation functions α such as: A α (, z) = A A : Nodal potential values The space discetisations of the magnetodynamic equation using Galekin s finite element method and the substitution of natual bounday conditions lead to the following equation: the solution is obtained at discete nodes in the solution egion by applying the weighted esidual method. n υ i i = = n = αijs ddz = ϕ ddz n ( α ) ( α A ) + σω α ( α A ) ddz () When witing () fo all nodes in such egion, one obtains the following algebaic equations. [ Μ + L].[ Α] = [ Κ ] With: M υ ( α ) ( α ) i L i = e = i σω e ( α )( α ) i ddz Ki = α i J s ddz e ϕ ddz The magnetic vecto potential is obtained afte solving the algebaic equation and the othe physical quantities such as the magnetic flux, the induced eddy cuent density and the impedance pobe can be deduced.

3 3. Calculating of impedance pobe The detection of defect pesence and effect on load specimen passes by the measuement of impedance vaiation in tem of esistance and eactance [4 ]. 3.. Method based on the magnetic enegy and eddy cuent losses The pobe esistance R depending on eddy cuents losses P and the pobe eactance X depending on magnetic enegyw, ae calculated by the following expessions: P R = = J * s ϕ E d W ω * X = ω = H B d With is the cuent souce intensity and E * *, B the complex conugate electic field and magnetic induction density espectively 3.. Method based on the magnetic vecto potential Using the magnetic flux cosses elementay aea ds of pobe of coil and assuming the electomotive foce function of the magnetic flux, we obtained the eal and imaginay pat of the impedance pobe vaiation: N R = eal( Z ) = πf π J s S S N X = imag( Z ) = πf π J s S S ( A ) imag ( A ) eal With S, N the aea and numbe of tuns foming coils pobe, A the imaginay and eal pat A imag eal of magnetic vecto potential espectively and f the excited fequency. 4 Simulation of the pobe displacement The displacement of the pobe displacement along the load specimen is made using the geometical band technique [5]. This technique consists on two steps: - Ceate a geometical band, which is subdivided in elementay egions of height z. - Locate in the geometical band the finite element coesponding on the pobe and ai fo ds ds assignment thei physical popeties at each step displacement. z Geometical band Pobe z lad Afte one displacement, the pobe nodes and itsuounded ai ae localised fo assignment of thei popeties. This method is simply implemented fo conducting o non-conducting media with imposed fixed step displacement. 5 Applications The inspection of heat exchange tubes is usually caied out by using the eddy cuents testing though the analysis of the impedance vaiation of the axial pobe on diffeential mode. An altenative sinusoidal cuent with opposite diection excites the two coils placed on the load tube. The following Fig. epesents the studied device. ai Fig.. Geometical band of pobe displacement z, 9,84 A= A= 8,5 A= 7,75 A= Fig.. Dimensions of the device in [mm]

4 The coils ae composed of 7 tuns excited by sinusoidal cuent 5 3 [ A]. The conductivity and elative pemeability of tested load tube is 6 σ = [ S / m] and µ = espectively. The distance between the coils is,5mm. Diffeent types defect consideed such as ectangula extenal, intenal and middle, intenal stepwise, elliptical and slopes. The ectangula extenal defect have 4mm length and 4% of the load thick and the cicula defect have mm diamete. The othe defects have mm length and % of the load thick Given values [3] FE-model Fig. 5. Numeical esults of impedance-plane taectoy fo ectangula intenal defect, at Khz To check the validity of the finite element model, we have calculated the change impedance of the pobe fo othe defects at fequency Khz and 4 Khz. The pobe displacement is made using the geometical band with,5mm step Rectangula middle Fig. 3. Desciption of the defects geomety The geometical band is containing ai and the pobe egion. Fo each displacement the nodes and finite element pobe ae locate fo assignment thei popeties. Fo the validation and showing the effectiveness of the implemented finite element model, the impedance change fo the ectangula extenal and intenal defects calculated and showed in Fig.4 and Fig.5 ae compaed with expeiment esults elated in [6] x -3 Given values [3] FE-model Fig. 6. Computed Lissaous cuves at Rectangula middle Khz mpedance [Ohm] Displacement of the pobe [m] Fig. 4. Numeical esults of impedance-plane taectoy fo ectangula extenal defect, at Khz Fig. 7. Change impedance of the pobe at Khz

5 mpedance [Ohm] Rectangula middle Rectangula middle Fig. 8. Computed Lissaous cuves at Displacement of the pobe [m] Fig. 9. Change impedance of the pobe at 4Khz 4Khz numeical model is showed though the investigation of effect induced by slope, elliptical, stepwise and ectangula shape defects. Refeences [] G. Bethiau and B. De Bamon, Messine, and eddy cuent paametic model fo flaw chaacteisation, in eview of pogess in QNDE, edited by D.O Thompson and D.E Chimenti, Plenum, Vol. 8, pp 5-58, New Yok 999. [] S. Bakhtiai and D.S. Kuppeman, Modelling of Eddy Cuent pobe esponse fo steam geneato tubes, Nuclea Engineeing and Design Elsevie, Vol.94, 999, pp [3] G. Founet, Electomagnétisme à pati des équations locales, Masson Editeus Fance, 985 [4] J. Juillad, G. Pichenot and A. Masia, Semianalytical Method Fo Calculating of An Abitay Eddy-Cuent Pobe, EEE Tansactions on Magnetics, Vol.38, N 5,. [5] K. Saii, B. Bendima and M. Feliachi, Coupling models fo analysing dynamic behaviou of electomagnetic actuato, ELECTRMAC S, pp , Fance 996. [6] G. Pichenot and T. Sollie, Eddy Cuent Modelling fo Non-destuctive testing, Poceeding of 8 th Euopean Confeence On Non-destuctive testing, Bacelona,. The obtained impedance changes of the pobe fo the ectangula extenal and intenal defect ae compaed with thus given in [6] and show a good ageements. The impedance change value depends of the height of the defect and the phase depends of the thickness of the defect. 5 Conclusion n this pape, we have pesented the eddy cuent non-destuctive testing modelling tools implemented in Matlab softwae. The numeical appoach based on finite element method is used fo solving of the electomagnetic equations with magnetic vecto potential fomulation in D axisymmetical case. Fo testing the validity of the poposed model, the esults fo intenal and extenal ectangula shape defect is compaed with expeimental esults given and showed good ageement. Fo taking into account of the pobe displacement, the geometical band technique based on the physical assignment popeties is implemented. The effectiveness of the