Implementation in the ANSYS Finite Element Code of the Electric Vector Potential T-Ω,Ω Formulation

Transcription

1 Implementaton n the ANSYS Fnte Element Code of the Electc Vecto Potental T-Ω,Ω Fomulaton Peto Teston Dpatmento d Ingegnea Elettca ed Elettonca, Unvestà d Cagla Pazza d Am, 0923 Cagla Pegogo Sonato Dpatmento d Ingegnea Elettca, Unvestà d Padova Abstact A novel fomulaton (the T-Ω,Ω) has been mplemented n the fnte element code ANSYS. Wth espect to the ANSYS A-V,A fomulaton the Τ Ω,Ω fomulaton allows a educton of the degees of feedom fom thee to one n the non-conductve egons allowng to save CPU tme and memoy space fo the soluton. The mplemented fomulaton has been appled to the FELIX bck bench-mak poblem fom the Intenatonal Wokshops fo the compason of eddy cuent codes. The soluton s also compaed wth the esults fom the A-V,A fomulaton n ANSYS. Results obtaned wth the new Τ Ω,Ω fomulaton ae n vey good ageement wth those fom othe codes. Intoducton A good knowledge of space and tme dstbuton of both electc cuent and magnetc flux s necessay f the optmzaton of the devces whee eddy cuents ae nved has to be obtaned. Thee dmensonal fnte element eddy cuents poblems can be mathematcally fomulated n vaous ways. The vaable to be solved may be a vecto potental, a scala potental o a combnaton of those. Seveal fnte element commecal codes ae avalable on the maket. The fomulatons mplemented n these codes ae not eve geneally applcable to all knd of poblems and not eve ae the most convenent n tems of avalablty and powe of computng esouces. One of them s ANSYS, whch s maybe the most poweful and wde used fnte element commecal package. Fo the eddy cuent study, ANSYS mplements the magnetc vecto potental fomulaton whch uses n the non-conductng egons thee degee of feedom, the magnetc vecto potental components, and adds an exta degee of feedom, the tme-ntegated electc tage, n the conductng egons o a mxed fomulaton whch stll uses fou degees of feedom n the conductng egon and one n the non-conductve egons, but needs nteface elements n the nteface suface between egons wth dffeent fomulaton. Ths wok deals wth the mplementaton of a new fomulaton (the T-Ω,Ω) [] n ANSYS by usng ts customzaton capabltes fo ceatng new element types and addng them n the ANSYS lbay. The goal of ths wok has been to mplement n ANSYS a smple and economcal method fo calculatng 3-D eddy cuents educng the numbe of degees of feedom, fom thee to one, n the non-conductng egons. On the othe hand, the mplementaton of the new fomulaton n ANSYS allows to take advantage of the excellent and seveal capabltes of the commecal code tself: mesh geneaton, post-pocessng, gaphcs wndow, optmzaton, couplng and so on. Poblem defnton The nteacton between magnetc felds and electcal phenomena s descbed by the followng subset of Maxwell s equaton: H = J [2.]

2 B E = [2.2] t B =0 [2.3] In eq. [2.] the dsplacement cuent tem D / t s neglected; usually at low fequency the dmenson of the eddy cuent egons ae small compaed wth the wavelength of the pescbed felds. The feld vectos ae not ndependent snce they ae futhe coelated by the mateal consttutve elatonshp: B = µ H [2.4] J = σ E [2.5] whee µ and σ ae the magnetc pemeablty and the conductvty of the mateal. They may be feld dependent and may vay n space, hee we assume to neglect hysteess and ansotopy. Fo the soluton of the poblem we consde also the contnuty condton: J [2.6] whch states the cuent densty solenodalty. The poblem conssts (Fgue ) of an eddy cuent egon τ, whee eddy cuents can be nduced, wth non zeo conductvty σ and magnetc pemeablty µ bounded by a suface S 2 and a suoundng egon τ 2 fee of eddy cuents, whch may contan souce cuents J s, bounded by a suface S o whch may be extended to nfnty. The S o suface can be dvded nto two pats n accodance of the two types of bounday condton of pactcal mpotance: on S BN the nomal component of the flux densty s pescbed, on S HT the tangental component of the magnetc feld ntensty s pescbed. S o S 2 τ σ µ j τ s 2 µ 2 J s S HT S BN Fgue. Electomagnetc feld egons The whole poblem doman s done by the sum of τ and τ 2 and wll be denoted by τ 0. The τ 0 bounday condtons ae elated to the feld values of the components nomal o tangental to the boundaes: B nˆ on S BN [2.7] H nˆ on S HT [2.8] Of couse also the nteface condtons between the conductve and non conductve meda must be satsfed: B n + B nˆ 0 on S 2 [2.9] ˆ 2 2 =

3 H ˆ ˆ n + H 2 n2 on S 2 [2.0] J nˆ on S 2 [2.] whee nˆ s the oute nomal on the coespondng suface. Patcula cae must be taken n poblems nvng multply connected egons. The T-Ω, Ω fomulaton Maxwell s equatons ae fst ode coupled dffeental equatons whch can be vey dffcult to solve n bounday values poblems. The way fo educng mathematcal complexty s theefoe to fomulate poblems n tems of potentals. They, n fact, allow the constucton of effcent and economcal numecal algothms. Statng fom [2.6 ] we can expess the cuent densty n tems of an auxlay vecto T [], whch s called the electc vecto potental: J = T. [3.] by usng the well known vecto dentty T, whee T s any suffcently dffeentable vectoal functon. We note that fom eq. [2.] H = J as well, so T and H dffe by the gadent of a scala and have the same unts: H = T Ω n τ [3.2] whee Ω s a magnetc scala potental. Usng [2.2 ], [2.4] and [2.5] we obtan: and fom [2.3] we obtan: T + µ σ t ( T Ω) n τ [3.3] µ ( T Ω) n τ [3.4] It should be noted that by takng the dvegence of both sdes of [3.3] the solenodalty of the magnetc flux densty [3.4] s satsfed and would be at ths pont supefluous. In cuent-fee egons the magnetc feld can be found fom the scala potental: H = Ω n τ 2 [3.5] whee Ω esults fom [3.4]: µ Ω n τ 2 [3.6] The dvegence of T s not yet defned and consequently T and Ω eman ambguous. Defnng the dvegence of T n addton to ts cul s efeed to as a choce of gauge. One of the two common gauge condton used n electomagnetcs s the Coulomb gauge: T [3.7] Ths condton allows to append [3] the left-sde of [3.3] by a tem T : σ T T + µ ( T Ω) n τ [3.8] σ σ t Takng the dvegence of [3.8]: 2 T T + ( µ ( T Ω ) n τ [3.9] σ σ t The fst tem of [3.9]s zeo fo any vecto, the thd s zeo as well consdeng [2.3], so the scala

4 σ T satsfes Laplace s equaton σ 2 T n τ [3.0] σ If we set the bounday condton T on the nteface between the conductng and non- conductng meda, then fom [3.0] we conclude that T n τ. Ths satsfes the gauge condton n the conducto. The bounday condton of the electc vecto potental s detemned by [2.] and can be expessed as: nˆ T [3.] By takng the dvegence of both sdes of [3.8] the solenodalty of the magnetc flux densty s no moe satsfed and t must be enfoced. The nteface condtons [2.9], [2.0] must be satsfed n the nteface suface between the conductng and non conductng meda. Substtutng equatons [2.4], [3.3], [3.5] n [2.9] and [2.0] gves: µ T Ω n + µ Ω nˆ on S 2 [3.2] σ ( ) ˆ ( ) 2 ( Ω ) n + ( Ω ) nˆ 0 ˆ 2 = T on S 2 [3.3] Also the bounday condtons [2.7] and [2.8] must be satsfed, by usng [3.3 ], [3.6 ]. The fst one can be wtten as: µ T Ω nˆ = on S BN [3.4] ( ) 0 ( Ω ) nˆ µ on S BN [3.5] fo conductng and non-conductng egons espectvely, the second one s: ( Ω) nˆ T on S HT [3.6] ( ) nˆ Ω on S HT [3.7] fo conductng and non-conductng egons espectvely. In the dscetzaton by the fnte element method [3.2] s satsfed mplctly, the contnuty of the scala potental togethe wth [3.] ensues that [3.3] s satsfed too. The T-Ω, Ω fnte element fomulaton The fnte element mplementaton of the T-Ω, Ω fomulaton has been caed out by usng fou-noded, fst-ode, tetahedal elements. A tetahedal element n the global x, y, z system s shown n Fgue 2. The numbes on the element ndcate the local numbeng of the nodes. Insde each element the magnetc scala potental Ω and the electc vecto potental T can be expessed by a lnea combnaton of the shape functons assocated wth the nodes.

5 3 4 2 y x z Fgue 2. Fnte tetahedal element Wthn an element the scala potental Ω s appoxmated as: Ω = m = Ω N whee N s the nodal shape functon coespondng to node. The ndex m s the numbe of the element nodes and m = 4 fo tetahedal elements. The coeffcent Ω s the degee of feedom and t s the value of the magnetc scala potental Ω on the node. The electc scala potental T s teated as thee scala components, T x, T y, T z n the Catesan coodnate system. Each node has thee degees of feedom nstead of one. In each element the electc vecto potental T can be appoxmated as: m m T = T N = ( Tx x + Ty y + Tz z ) N [4.2] = = whee the coeffcent T s the value of the electc vecto potental T on the node, Tx, Ty, Tz ae the components of T. Fo a tetahedal element the shapes functons ae defned as: a + b x + c y + d z N = [4.3] 6 whee all the paametes depend on the element node coodnates. The element ume multpled by sx [4] s gven by the detemnant of the coeffcent matx: x y z x2 y2 z2 6 = x3 y3 z3 x4 y4 z4 whle the a, b, c, d constants can be obtaned calculatng the cofactos of the coeffcent matx: [4.]

6 a = x2 x3 x4 y2 y3 y4 z2 z3 z4 b = y2 y3 y4 z2 z3 z4 c = x2 x3 x4 z2 z3 z4 d = x2 x3 x4 y2 y3 y4 the othe coeffcents ae deved by cyclng ntechange of the subscpts. Dscetzaton of the T-Ω, Ω fomulaton equatons The Galekn s fom of equatons [3.8], [3.4] and [3.6] s: T T + µ ( T Ω) d σ σ t [5.] N µ ( T Ω) d t [5.2] N µ ( Ω) d [5.3] t By usng the Geen Gauss theoem and some vectoal algeba eq. [5.], [5.2], [5.3] can be ewtten as: ( N ) T + ( N ) T + N µ ( T Ω) d σ σ t N ˆ + ˆ n T ds N T n2 ds σ S2 σ [5.3] N ( Ω) + ( Ω) ˆ µ T d µ T n t t [5.4] S2 N ( Ω) + ( Ω) ˆ µ d µ n [5.5] t t by mposng the nteface condtons on the nomal components of B and J and on the tangental H and the condton nˆ T on the nteface suface between the conductng and non-conductng meda all the suface ntegals dsappea. Implementaton n ANSYS of the T-Ω, Ω fomulaton The ANSYS Use Pogammable Featues (UPFs) [5] capablty has allowed to ceate a custom veson of the ANSYS pogam by wtng some outnes n FORTRAN 77. UPFs allow to ceate new elements, add them to the ANSYS element lbay and use them as egula elements. The customsaton of the new fomulaton has been done by modfyng the uel0, uel02, uec0 and uec02. The uec outnes allow to descbe the element chaactestcs such us: 2-D o 3-D geomety, numbes of nodes, degee of feedom set, and so on. The uel outnes allow to calculate the element stffness and dampng matces and the element load vecto. The element pntout s also geneated, and the vaables to be saved ae calculated and stoed n the esult fle. The ANSYS dstbuton medum has also nclude-decks whch can be ncluded n the suboutnes. Ths nclude-decks also called commons contan mpotant amounts of data: soluton optons, output contol nfomatons, element

7 chaactestcs and so on. The nclude-decks allow the communcaton between the optons defned by the use dung the model ceaton and analyss soluton and the costumzed outnes. The modfed outnes and also the anscust.bat, makefle, ansysex.def, ansysb.dll and mnflb.dll fles (whch ae on the dstbuton medum) have to be moved n a wokng dectoy whee the FORTRAN fles wll be compled and a custom ANSYS executable veson wll be ceated by unnng the anscust.bat fle. In ths way, two elements have been ceated: the USER0 fo non-conductng egons and the USER02 fo conductng egons. Valdaton of the mplementaton of the T-Ω,Ω fomulaton To valdate the mplementaton n ANSYS of the new T-Ω,Ω fomulaton, the FELIX bck benchmak has been consdeed. Ths s the benchmak poblem 4 defned n the Intenatonal Wokshops fo Eddy Cuent Code Compason and has been poposed by A. Kameay [2]. The poblem has been solved by dffeent compute codes. A ectangula alumnum bck wth a ectangula hole s mmesed n a unfom feld whch decays exponentally wth tme. Symmety condtons allow to buld the model of only one eghth of the bck. The model used fo the analyss s shown n fgue 3 and 4. Fgue 3. Whole FELIX Bck Model Fgue 4. FELIX Bck Model The bck s made of alumnum alloy 600, wth a esstvty ρ= Ohm m. Dmensons n the X, Y and Z dectons ae espectvely m, 0.06 m and m. A ectangula hole m x m penetates the bck though the cente of the lage faces. The model ncludes 6668 elements and 364 nodes. To ovecome the pesence of a multple connected egon, an atfcal low conductvty egon has been ntoduced n the a-space nsde the conducto. The poblem conssts n detemnng the tme euton of the magnetc fled, the aveage powe loss n the bck and the eddy cuents at dffeent locatons. The appled magnetc feld s oented n the Z decton and t eves n the tme wth the followng low: Bz=Bo fo t<0 Bz= Bo e -t/τ fo t>0 whee Bo. T and τ.09 s. The tansent analyss poblem soluton has been obtaned consdeng a 20 ms tme ange and by dvdng t n 20 load steps, so usng 20 teatons. The tme euton of the total powe loss n the bck as calculated by usng the T-Ω,Ω fomulaton s shown n Fgue 5: ths esult can be compaed wth those pesented n [2] and shown n Fgue 6.. The esult obtaned s n good ageement wth the mean value of the othe esults confmng the valdty of ths appoach.

8 50 Loss Powe [W] Tme [ms] Fgue 5. Total powe loss tme euton T-Ω,Ω fomulaton Fgue 6. Total powe loss tme euton as epoted n [2] The total magnetc flux densty at t = 4, 8, 2 and 20 ms as calculated by usng the T-Ω,Ω fomulaton ae pesented n Fgue 7 and can be compaed wth those obtaned n [2] and shown n Fgue 8. Also n ths case a vey good ageement has been obtaned 4 ms 8 ms 2 ms 6 ms 20 ms 0,0 0,09 0,08 0,07 Bz [T] 0,06 0,05 0,04 0,03 0,02 0,0 0,00 0 0,05 0, 0,5 0,2 0,25 Z [m] Fgue 7. Magnetc feld on Z axs T-Ω,Ω fomulaton Fgue 8. Magnetc feld on Z axs as epoted n [2]

9 In table the peak of the nduced flux densty at the cente of the hole and at dffeent locatons, the aveage powe loss n the bck as computed fom the T-Ω,Ω fomulatons ae computed and compaed wth the mean values fom the othe codes. Table. Results Compason T-Ω,Ω fomulaton Mean B z (Z=0.0), T 3.95E-02 (2) 3.88 E-02 B z (Z=0.027), T 3.72E-02 () 3.63 E-02 B z (Z=0.0254), T 2.97E-02 () 2.96 E-02 Powe loss, W 2.4 (0) 0.6 The mplemented T-Ω,Ω fomulaton has been compaed wth the A-V,A ANSYS fomulaton consdeng the same FE model whch s made as above mentoned of 6668 elements and 364 nodes. In table 2 the numbe of degees of feedom n a and n the conductve mateal s shown fo both the fomulatons: Table 2. Degees of Feedom DOFs n a DOFs n conducto Total numbe of DOFs T-Ω,Ω fomulaton A-V,A fomulaton The compute stoage and the CPU tme fo the fst teaton as computed by ANSYS to obtan the solutons fo both the T-Ω,Ω fomulaton and the A-V,A fomulaton ae epoted n Table 3: Table 3. Runtme statstcs and compute stoage Soluton fle sze (MB) CPU tme pocessng CPU tme soluton CPU tme esults CPU tme total T-Ω,Ω fomulaton A-V,A fomulaton The A-V,A fomulaton eques moe CPU tme than the T-Ω,Ω fomulaton because most pat of the FE model s n a. The magnetc feld and the eddy cuents dstbutons n the conductng bck obtaned at tms wth the mplemented T-Ω,Ω fomulaton and wth the A-V,A ANSYS fomulaton ae shown n the followng pctues. On the left the esults fom the T-Ω,Ω fomulaton, on the ght fom the A-V,A fomulaton. Fgue Iteaton 9. numbe B 0 ms T-Ω,Ω fomulaton Fgue 0. B 0 ms A-V,A fomulaton

10 Fgue. J 0 ms T-Ω,Ω fomulaton Fgue 2. J 0 ms A-V,A fomulaton As can be noted the esults obtaned wth the two dffeent fomulatons ae n faly coespondence. Conclusons The am of ths wok has been to mplement a new fomulaton (the T-Ω,Ω) n the ANSYS package by usng ts customzaton capabltes fo ceatng new element types and addng them n the ANSYS lbay. The goal has been to mplement n ANSYS a smple and economcal method fo calculatng 3-D eddy cuents and magnetc feld dstbutons. Fo the eddy cuent study, ANSYS mplements the magnetc vecto potental fomulaton whch uses n the non-conductng egons thee degee of feedom, the magnetc vecto potental components, and adds an exta degee of feedom, the tme-ntegated electc tage, n the conductng egons. The mplemented fomulaton has been appled to the FELIX bck bench-mak poblem fom the Intenatonal Wokshops fo the compason of eddy cuent codes [2]. Results obtaned wth the new T-Ω,Ω fomulaton ae n vey good ageement wth those fom othe codes. It has been shown that the poblem of egons havng a multple connecton can be ovecome by ntoducng an atfcal egon of vey low conductvty to tansfom a multply connected poblem to a smply connected one. Some poblems stll eman n case of complex geometes, n patcula n case of thn cuts n the conductve egons. The unqueness of the poblem soluton can be ensued by mposng pope nteface condtons on the electc vecto potental n the nteface suface between conductng and non-conductng egons. It has been shown that theτ Ω,Ω fomulaton has a bg advantage n tems of compute stoage and CPU tme because they can be consdeably educed especally when most pat of the analyzed egon s cuent fee. Refeence [] K.J. Bnns, P.J. Lawenson, C.W. Towbdge, The Analytcal and Numecal Soluton of Electc and Magnetc Felds. John Wley & Sons, Inc.992 [2] A. Kameay,COMPEL The Intenatonal Jounal fo Computaton and Mathematcs n

11 Electcaland Electonc Engneeng BOOLE PRESS LIMITED, Vol. 7, No. &2, p [3] M.V.K. Cha, S.J. Salon, Numecal methods n electomagnetsm. Academc Pess 2000 [4] M. N. O. Sadku, Numecal Technques n Electomagnetcs. CRC Pess, 200 [5] Gude to ANSYS Use Pogammable Featues. ANSYS elease 5.6. Febuay 2000