A NUMERICAL MODELING OF MAGNETIC FIELD PERTURBATED BY THE PRESENCE OF SCHIP S HULL

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1 A NUMERCAL MODELNG OF MAGNETC FELD PERTURBATED BY THE PRESENCE OF SCHP S HULL M. Dennah* Z. Abd** * Laboratory Electromagnetc Sytem EMP BP b Ben-Aknoun 606 Alger Algera ** Electronc nttute USTHB Alger Abtract ---n th paper wreent a method for behavor computaton of the magneto tatc feld through thermeable thn layer. For the determnaton of nducton and magnetc feld n any pont n thace D, a formulaton n magnetc calar potental adopted, by applyng the fnte element method on the tructure wth thn layer and the boundary ntegral method to the urroundng medum. Wropoe two numercal approache of the fcttou magnetc charge ntroduced on the urface of the thn layer. The decompoton of the total magnetc feld n h r + h theoretcal. The determnaton of the magnetc feld of reacton h r make t poble to better hghlght the reult whch wll breented. Keyword : Magneto tatc feld; Thn Layer; Fnte element; Boundary ntegral ntroducton : The modelng of the tructure wth thn layer n order to know the behavor of the magnetc feld trough ther border, the everal ndutral applcaton obect (ytem of telecommuncaton, medcal magery, dentfcaton and gnature,...). Thee applcaton dffer n the mode of the magnetc feld ource, the nature of materal compong the thn layer and t tructure. Wropoe an electromagnetc ytem compoed of a tructure wth thn layer and t envronment. Th ytem preent an open problem that extend to nfnty. We conder a tructure wth thn layer of permeablty µ, embedded nto a magnetc ource feld H. t thckne ( ) very mall than the other dmenon n pace R. A cut of the ytem repreented by fgure. Some work on th ame model ha ummer made wth other numercal approache [] [2] []. Formulaton : The equaton that govern t nterpret the magnetotatc wthout current. n any pont of the ytem we have rot h 0. We uppoed that the materal doe not compre any ntal magnetzaton; one deduce from t that nducton and the magnetc feld are related to the magnetc polarzaton nduced by the relaton: b µ 0( h+ M ) and we have : hr+ h 0 ; h dvµ ( ) rot 0 Fg. cheme of a cut of the tructure wth thn Layer Subequently we ndcate the feld of reacton hr by a mple letter h. Then, the magnetc feld derve from a calar potental :. The unknown h grad quantty the reduced potental magnetc calar. To etablh a varatonal formulaton of throblem we ue the econd property whch verfer owe magnetc nducton : Ω dv µ ( h grad φ ) 0 gradφ. gradψ dv grad dv h. ψ φ ψ dγ n + 2 Ω φ ψ dγ n2 where Ψ an unpecfed functon tet defne n ame pace a that of tate functon. A volume Ω wth thn layer characterzed by a very mall thckne ep compared to t other dmenon geometrcal. Thu, one can confue the two border and 2, and ep become a parameter. The two border 0

2 and 2 wll be confued at a medan border. Ω become urface feld occuped by the tructure n R of permeablty magnetc µ µ 0 µ r. Ωe ndcate the medum external of magnetc permeablty µ 0. By holdng account that n the outgong normal wth the two ntegral relatng to the two border become: φ φ µ + Ψ Ψ h. n µ. h n n n2 2 φ µ + n n [ h. ] Ψ The ump of the normal component of the feld ource beng null, the formulaton of the nteror problem wrtten then : φ grad φ. grad ψ d γ + µ n h. grad ψ dγ () The econd term of the ntegral equaton called term of edge. Th tage, t hould be noted on the one hand that we cannot olve throblem wthout dentfyng the ump of the normal component of the magnetc feld on. On the other hand the oluton of the nteror problem could not be the oluton of the problem poed n open pace R. Only the term of edge whch enable u to take account of the behavor of the feld n the external medum. To treat th term of edge we frt of all wll formulate the external problem and wll expre t oluton accordng to the trace of the feld on the border. The ytem of equaton whch govern the behavor of the magnetc feld n the medum external of the tructure : dv h 0 0 We ndcate by ψ the urfacotental on aocated wth the magnetc feld of reacton. To know the functon only on the border would be enough wth the determnaton to n Ωe, and thu to the magnetc feld of reacton n the external medum. Throblem cont n fndng a potental uch a: 0 ur Ωe ψ ur Wth through, the ump of the normal component of magnetc nducton and the ump of the tangental component of the magnetc feld are null. The equvalent condton, mpoed on the calar potental, are thu: [ ] q ; [ ] 0 n t [.M ]. Wth : q n The magnetc potental can be to calculate tartng from the urface denty of Ψ magnetc load q. ndeed, by ung the technque of the mplotental layer [4], how that the oluton of exteror problem can be wrtten n the followng ntegral form: ( ) () ( 2 ) 4 q y x dy π x y The magnetc charge q, ntroduced on, are fcttou. t alo an ntermedate unknown ued to couple the exteror and the nteror problem. To keep thotental lke only unknown factor on, we wll have to expre q accordng to : [ n ] R. Our goal now to calculate R. Numercal mplementaton : Onroceed then n the way ndcated below. One adopt a trangular grd on and one aocate a value of for each node of the grd. On a trangle, one λ wrte:. n th work wropoe two form of approxmaton of the magnetc charge on. n a frt tme, we take the charge lke contant functon by trangle. We wrte then : q q t η. Wth q the value of q on trangle t and η 0 elewhere on t A varatonal formulaton of (2) make t poble to carry out the condton of couplng;.e. to calculate the operator R : η tk λ k tk qd y dy B q ' dx x y d xk q tk t x k y where q' defned lke tet functon : q' ( q ) η. t k K D k q n the matrx form we thu obtan : q (D - )B. n the econd tme, we take the charge lke lnear functon on each element of the grd. We wrte on each trangle: q q λ. n th cae the operator R take another form. n fact, the varatonal formulaton of (2) wrtten: q' λ q' λ tk tk S 2

3 qd y x y q' dx ( q ) q x λ λ tk tk tl tl Q q where q' defned lke tet functon : q ( q' ) '. n the matrx form we thu obtan : λ q (Q - )S, where and ndcate the total number of the node of the grd. Thee calculaton enable u to wrte the varatonal formulaton of throblem wth only unknown. The varatonal formulaton become: grad φ. grad ψ d γ + µ ( Q ) S h. grad ψ dγ From th formulaton, we calculate and n partcular h trace on, and conequently the denty of charge q. Once to calculate q, we calculate outde the tructure wth thn layer, the magnetc feld n any pont of the ytem and the nduced magnetzaton M of the tructure wth thn layer. Ψ y Table : Comparatve table between the analytcal and numercal value (Wth the approxmaton lnear charge). Computaton Reult : µr X Y Z CHXT HAX Table : Comparatve table between the analytcal and numercal value (Wth the approxmaton contant charge). The model a hollow phere of ray unt and Ep thckne. The ource feld followng X drecton. the value HAY and HAZ are null. We dd not preent the value HYT and HZT n order to mplfy the table. Thee value are almot null. µr X Y Z CHXT HAX

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5 edge, of the model wth contant charge, utlze le large matrce compared to thoe of the model wth lnear charge. However, the model, wth lnear charge, gve numercal reult n conformty and a fater convergence. We note that the approach wth lnear charge on gve a good repreentaton of the magnetc potental and n conequence of the magnetc feld on our phycal ytem. Th how a certan regularty of the feld, nde and outde the tructure wth thn layer, and check the theory concernng the conformty of the magnetc feld n thace of the acceptable feld. Th code can be wth throft of everal applcaton we can proect calculaton on other feld uch a the magnetc heldng and magnetc compatblty. Reference: The reult preented here are at dfferent alttude Z and the magnetc ource feld drected along the hp. Concluon : Each one of thee two approxmaton correpond to a computer code. A frt numercal mplementaton carred out wth charge taken contant by trangle on the border of the tructure wth thn layer. A econd numercal mplementaton carred out wth charge taken lke lnear functon on each trangle of the border. To tet the good functonng of thee computer code, we condered a phercal model n order to make a comparatve tudy between the numercal reult and analytcal calculaton (ee table and ). The numercal reult relatng to dfferent model are repreented by the equpotental ketched lne and the lne of the magnetc feld (ee fgure). We notce that the flux of the magnetc feld well channeled by the thn layer, the lne of magnetc feld are cloed agan n the area outde. n addton, the fgure how that the lne equpotental take form n conformty wth the phycal form of the tructure of the thn layer. []Chadebec O., Coulomb.L., Leconte V., Bongraud.P., Cauffet G «Modelng of tatc magnetc anomaly created by ron plate» EEE Tranacton on magnetc, vol,995,pp [2]Roux-Damdau F, Bandeler B., Penven P.«A fat and prece determnaton of the tatc magnetc feld n threence of thn ron hell» EEE Tranacton on Magnetc, vol,995,pp []Brunotte X., Meuner G., «Lne for effcent computaton of the magnetc feld created by thn ron plate, ept.990,eee Tran. Mag. [4]Nédélec.C. «Equaton ntégrale dan R», Analye mathématque et calcul numérquour le cence et le technque, Ed.R.Dautray and.l.lon Maon 988, vol6, chap XB,pp Thrncpal dffculty of our problem rede epecally on the calculaton of the term of edge. The approach that wropoed prove more flexble and le expenve from pont of vew capacty of the memory and tme CPU. The calculaton of the term of 5