A Simple Technique to Handle Fields Singularities in the Finite Element Time Domain Method

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1 1 A Smple Technque to Handle Felds Sngulates n the Fnte Element Tme oman Method alkelly Schmdke, lson A. Atuz, J., Membe, IEEE Abstact The use of dvegence-confomng edge bass functons of fst ode s poposed as a smple and effcent altenatve to deal wth sngulates of electomagnetc felds n analyses employng the fnte-element tme-doman method (FET). Ths teatment conssts n applyng ths knd of functon only to the edges of the tetahedal elements layng on egons whee a sngula behavo of the electcal feld s expected. The emanng elements n the computatonal doman ae kept wth the conventonal htney edge elements, whch ae culconfomng bass functons. Tests of ths technque wee made though the evaluaton of the chaactestc mpedance and the esonant fequency of some tansmsson lnes. Impoved esults have been obtaned usng ths appoach when compaed to the conventonal FET fomulaton. Index Tems Fnte Element Tme oman (FET), electomagnetc felds sngulates, edge elements, culconfomng and dvegence-confomng bass functons. S I. INTROUTION ngulates of electomagnetc felds have been subect of specal nteest n the development of seveal scentfc woks egadng the numecal methods appled to electcal engneeng [1]-[6]. These concen the phenomena assocated wth the abupt feld vaatons, n ntensty and decton, occung n the neghbohood of shap edges and wes [7], [8]. lasscal examples of electc feld sngulates ae the shap pont and edge effects [9]. The sngula behavo of the feld makes t dffcult to be pecsely calculated and ts modelng has been a challenge. Seveal appoaches developed n ths ambt have been mplemented fo the fnte element method [10], [11]. Some nclude modfcatons of the bass functons that ae used to appoxmate the feld n ts mathematcal expanson. Geneally, these bass functons ae elated to the geometcal enttes of elements, that s, tetahedal vetces, edges and faces, and they can be constant, lnea o hghe ode functons. The sutable choce of these s of geat mpotance n detemnng flexblty and effcency of the method, especally when dealng wth sngulates. lson A. Atuz J., Ph., epatment of Electcal Engneeng, Unvesdade Fedeal do Paaná - UFPR, utba, PR; e-mal: atuz@eletca.ufp.b alkelly Schmdke, maste student n the Electcal Engneeng Pogam at UFPR, APES fellowshp; e-mal: walkelly@tesla.eletca.ufp.b htney foms ae the most wdely used bass functons n the fnte-element tme-doman method (FET) [12]-[14]. These ae cul-confomng vecto bass functons whch ae pefectly solenodal. In analyses nvolvng electodynamc phenomena they popely model the tangental contnuty and the nomal dscontnuty of felds, necessay to coectly manpulate the bounday condtons between dffeent mateal ntefaces, and they elmnate spuous esponses. The accuacy of esults acheved when employng these conventonal functons, howeve, s spoled by egons whee a sngula behavo of the feld s expected. A sutable explanaton to ths fact s because n small aeas aound the sngulaty, the feld has a quas-electostatc chaactestc whch, n tun, s essentally otatonal [15]. Many of the appoaches found n lteatue to mpove the accuacy n these cases, deal wth 2- poblems and employ scala bass functons [11], [16] o [17]. Othes apply adaptve technques nvolvng mesh efnements (h technques), the ncease of the bass functons ode (p technques) o hybd pocedues of both [18], [19]. Nevetheless, these ae complex fomulatons and ae dffcult to mplement. Futhe, these algothms demand addtonal computatonal capacty n tems of memoy and tme pocessng equements. Ths wok s dstngushed fo the teatment of sngulates n 3- analyses and fo applyng vecto bass functons. It s a smple altenatve to mplement whch has demonstated to be effcent and computatonally nexpensve when compaed to the above mentoned methods. The methodology conssts n eplacng the conventonal cul-confomng edge bass functons by lnea dvegenceconfomng edge functons assocated only wth the edges that lay on conductve wedges whee sngulates occu. Results obtaned wth ths technque ae compaed to those esultng fom the applcaton of the conventonal FET method, n tems of the mpedance and the esonant fequency of fve dffeent tansmsson lne confguatons n whch the poblem of feld sngulates s pesent. II. PROPOSE TEHNIQUE Used n the mathematcal appoxmaton of the electc feld nsde tetahedal elements, the bass functons ae

2 2 elated to the spatal dscetzaton of the computatonal doman and ae assocated wth the edges of the tetahedal. The unknown coeffcents v assemble the expanson of the electc feld E as E = v, (1) wth and equal to 1, 2, 3 o 4. It s known fom the Helmholtz's decomposton theoem that any vecto feld can be sepaated nto otatonal (culfee and dvegence-confomng) and solenodal (dvegencefee and cul-confomng) vecto feld components. [20]. In electodynamc phenomena, felds pesent a behavo that s pedomnantly solenodal, but n egons whee feld sngulates ae encounteed they appoach the chaactestcs of statc felds becomng otatonal. A. ul-confomng Bass Functons In FET fomulatons, the most commonly used bass functons employed n (1) ae the htney edge bass functons whch ae expessed as = λ λ λ λ, (2) λ beng the baycentc coodnate and λ ts gadent. The edge element s a set of sx cul-confomng bass functons, and has sx degees of feedom, one pe edge, because the pemutaton of ndces n (2) do not geneate a new functon, only changes ts sgn. oeffcents v elated to the cul-confomng functons of (2) have a eady physcal ntepetaton: they ae the voltages between the nodes of elements. ul-confomng vecto edge functons ae chaactezed by the null dvegence and they ae manly tangental to ts assocated edge. In ode to llustate the solenodal behavo of one of these functons n a tetahedon face, Fg. 1 shows the vectos. It can be seen that tangental components ae constant along the edge 1-2 consdeed, whle nomal components vay lnealy. B. vegence-confomng Bass Functons 12 vegence-confomng vecto edge functons ae chaactezed by pesentng null cul and ae manly nomal to the assocated edge. oeffcents v 12 elated to these functons ae not the voltages on the edges and do not have an explct physcal meanng. One edge bass functon, of fst ode, whch has an otatonal chaactestc, s = λ λ + λ λ. (3) Thee ae also sx possble functons as (2) n a tetahedon. In contast wth the cul-confomng bass functons of (2), dvegence-confomng bass functons of (3) pesent lnea vaaton n both tangental and nomal dectons. Fg. 2 shows the behavo of 12 Fg. 1 Vectos of a cul-confomng edge bass functon. Fg. 2 Vectos of a dvegence-confomng edge bass functon.. Fomulaton The FET fomulaton appled n ths wok s essentally the same of that found n [21]. The dstncton conssts n the eplacement of one of the conventonal bass functons of the cul-confomng knd n (2) by a dvegence-confomng one n (3), only fo those edges layng on shap wedges of conductve obects, egons whee a sngula behavo of the electc feld s pedctable. The emanng elements n the computatonal doman ae kept as the functons n (2). Thus, ths method does not mply an ncease n the numbe of degees of feedom, because sx bass functons pe element ae mantaned. As a consequence, the tme pocessng equed fo smulaton s not affected when ths technque s compaed to the conventonal FET method. In ode to llustate the poposed pocedue, t s shown n Fg. 3 a stplne and one of ts tetahedal elements of dscetzaton. The tetahedon has an edge layng on the left bode of the conductve stp. As well as n the conventonal.

3 3 FET method, ths element s also modeled wth sx degees of feedom, but n ths case, fve of them efe to the culconfomng functons and one to a dvegence-confomng functon. Fg. 3 A tansmsson lne scheme llustatng the applcaton of the technque nvolvng the eplacement of an edge bass functon. In pefectly conductve wedges, coeffcents assocated to the cul-confomng edge bass functons ae null, theefoe, these functons do not nfluence the esults of smulatons, and so these can be elmnated and eplaced by the dvegence-confomng functons. Fom the Foue tansfom of V(t), the esonant fequency of the tansmsson lne s found. The chaactestc mpedance s obtaned takng nto account the mean value v mean n pemanent egme. The examned stuctues ae: a stplne wth nfntesmal thckness, a pa of coupled stplnes (both TEM popagaton modes), a mcostp lne and a stplne of fnte thckness. Tests wee epeated usng thee meshes of dffeent szes. A. Stplne of Infntesmal Thckness The smulated stplne, wth nfntesmal thckness, has the dmensons epesented n Fg. 5. The substate between the pefect electc conductos (PE) s a. In ths smulaton an electcal cuent pulse havng a wdth of 0.1 ns was appled n ode to excte the lowest esonant fequency whch s 25.0 GHz. The chaactestc mpedance esultng fom the smulatons ae compaed to ts theoetcal value of 73.8 Ω [22]. III. NUMERIAL EXPERIMENTS The effectveness of the poposed technque was evaluated by analyzng the chaactestc mpedance Z c and the esonant fequency f o of tansmsson lnes n fve dffeent confguatons, commonly used n mcowave applcatons. These stuctues pesent geometc chaactestcs that ognate feld sngulates,.e. conductve bodes o wedges. Solutons obtaned n smulatons ae compaed to the theoetcal values of Z c and f o, calculated fom analytcal o empcal expessons avalable n lteatue. Results ae expessed n tems of the absolute value of the elatve eos of each physcal quantty. Tansmsson lnes wee excted wth an electc cuent souce, havng Gaussan shape n tme wth a peak of 1 A, and the esponse acqued s a voltage wave V(t), used to calculate Z c and f o. Typcally, ths voltage wavefom esembles that shown n Fg. 4. B. oupled Stplnes Fg. 5 Stplne of nfntesmal thckness. Two coupled stplnes havng the same chaactestcs of the above mentoned one and dstant 1.0 mm apat fom each othe ae epesented n Fg. 6. Both even and odd TEM popagaton modes wee smulated n ths lne confguaton. The theoetcal chaactestc mpedance s 66.8 Ω, fo the fst mode, and 80.0 Ω, fo the second [22]. The theoetcal esonant fequency s also 25.0 GHz. Fg. 6 oupled stplnes. The symmety plane s a pefect electcal conducto (PE) fo the odd TEM popagaton mode, and t s a pefect magnetc conducto (PM) fo the even mode. Fg. 4 Typcal voltage wavefom obtaned n smulatons.. Mcostp lne A sketch of the smulated mcostp lne s shown n Fg. 7. Alumna s the substate wth elatve delectc constant ε =10. The computatonal doman s extended above the

4 4 mcostp wth an a volume. The chaactestc mpedance of ths mcostp lne s 41.1 Ω and the esonant fequency, 8.7 GHz [23]. Fg. 7 Mcostp lne.. Stplne of Fnte Thckness Fg. 8 shows the sketch of the stplne of fnte thckness that was smulated. Its theoetcal mpedance s 30.0 Ω and the esonant fequency s 25.0 GHz [24]. The tansmsson lne s flled wth a. In ths stuaton, the condton nvolvng the eplacement of bass functons s appled to the wedges of the nne conducto (dotted lnes). those of esonant fequency. Ths behavo can be explaned by the fact that the chaactestc mpedance s moe dependent on otatonal felds athe than the esonant fequency. Moeove, t can be noted that the eos of the esonant fequency ae moe affected by the ncluson of dvegence-confomng bass functons. Nevetheless, the nvestgaton of two physcal quanttes of dstnct natue, one beng pedomnantly of otatonal chaactestc (Z c ) and othe of solenodal (f o ), has played an mpotant ole to assue the effcacy of the poposed appoach. The lack of unfomty on pefomances when usng dffeent meshes ae due to the andomness of nonstuctued meshes. TABLE I. TESTE TRANSMISSION LINES Numbeng n onfguatons of tested tansmsson lnes abscssas a 1 Stplne of nfntesmal thckness 2 oupled stplnes n odd TEM popagaton mode 3 oupled stplnes n even TEM popagaton mode 4 Mcostp lne 5 Stplne of fnte thckness a Fo efeence n abscssas of gaphs n Fg. 9 and 10. Fg. 8 Stplne of fnte thckness. IV. RESULTS The esults of smulatons ae pesented n ths secton. The absolute value of the elatve eos of each evaluated physcal quantty, obtaned wth the applcaton of the poposed technque and the conventonal FET method, ae dsposed n the fom of ba gaphs, fo evey smulated stuctue. Tests wee epeated employng thee dffeent meshes and esults ae pesented n deceasng ode wth espect to the mean values of edge lengths adopted n smulatons (1.0 mm, 0.75 mm e 0.5 mm). The tansmsson lnes ae numbeed fom 1 to 5 n the abscssas of gaphs n accodance wth Table 1. Fg. 9 and Fg. 10 show the pecentual elatve eos of the chaactestc mpedance and the esonant fequency, espectvely. Results clealy demonstate that the use of dvegenceconfomng bass functons mpoves the accuacy n both physcal quanttes. It can be obseved by compang the two gaphs, that the chaactestc mpedance eos ae consdeably geate than Fg. 9 Eos of chaactestc mpedances of tansmsson lnes. Fg. 10 Eos of esonant fequences of tansmsson lnes.

5 5 V. ONLUSIONS A modfcaton of the conventonal FET fomulaton has been poposed as an altenatve method fo handlng feld sngulates. Ths conssts on the eplacement of culconfomng by lnea dvegence-confomng edge bass functons assocated only wth the tetahedal edges layng on conductve shap wedges. th ths smple technque, the esults nvolvng fve dffeent confguatons of tansmsson lnes pesented an accuacy mpovement n the evaluaton of both chaactestc mpedance and esonant fequency quanttes. REFERENES [1] Kost, A. and Shen, J. X., Teatment of sngulates n the computaton of magnetc felds wth peodc bounday condtons by the bounday element method IEEE Tans. Magnetcs, vol. 26, no. 2, pp , Mach [2] Boubanne, P., ellncé, F., Genon, A., Legos,. and Ncolet, A., one modelzaton fo FEM BEM couplng IEEE Tans. Magnetcs, vol. 27, no. 5, pp , Septembe [3] Belenhoff, K., Hench,. and Baun, F., Teatment of feld sngulates n the fnte-dffeence appoxmaton - Insttut fu Hochfequenztechnk, Technsche Hochschule amstadt, Gemany and Insttut fu Hochstfequenztechnk Beln, Gemany, IEEE, [4] Altman, Z., Renaud,. and Baudand, H., On the use of dffeental equatons of nonente ode to geneate ente doman bass functons wth edge sngulaty, IEEE Mcowave Theoy Tech., vol. 42, no. 10, pp , Octobe [5] Bown, J.. and lton,. R., Sngula bass functons and cuvlnea tangles n the soluton of the electc feld ntegal equaton, IEEE Tans. Antennas Popagat., vol. 47, no. 2, pp , Febuay [6] ecchn, P., Badat, F. and Ravanell, R., A geneal appoach to edge sngulaty extacton nea composed wedges n boundayelement method, IEEE Mcowave Theoy Tech., vol. 49, no. 4, pp , Apl [7] Mexne, J., The behavo of electomagnetc felds at edges, IEEE Tans. Antennas Popagat., vol. AP-20, no. 4, pp , July [8] Buas, N., Mans, H. and Ncolas, A., Physcal and geometc sngulates modelng technques fo electostatc and electomagnetsm poblems, IEEE Tans. Magnetcs, vol. MAG-21, no. 6, pp , Novembe [9] Shen, L.. and Kong, J. A. Appled Electomagnetsm, 3d Ed., PS, [10] Gagla, R.. and Lombad, G., Sngula hghe ode complete vecto bases fo fnte methods, IEEE Tans. Antennas Popagat., vol. 52, no. 7, pp , July [11] Juntunen, J. S. and Tsbouks, T.., On the FEM teatment of wedge sngulates n wavegude poblems, IEEE Tans. Mcowave Theoy Tech., vol. 48, no. 6, pp , June [12] Alan Bossavt, htney Foms: a lass of Fnte Elements fo Theedmensonal omputatons n Electomagnetsm IEEE Poceedngs, vol. 135, Pt. A, n. 8, pp , Novembe [13]. K. Sun, Z. J. endes, Tangental vecto fnte elements fo electomagnetc feld computaton IEEE Tans. Magnetcs, vol. 27, no. 5, Septembe [14] J. P. ebb, Edge elements and what they can do fo you, IEEE Tans. Magnetcs, vol. 29, no. 2, pp , Mach [15] Gl, J. M. and ebb, J. P., A new edge element fo the modelng of feld sngulates n tansmsson lnes and wavegudes, IEEE Mcowave Theoy Tech., vol. 45, no. 12, pp , ecembe [16] Gl, J. M. and Zapata, J., Effcent sngula element fo fnte element analyss of quas-tem tansmsson lnes and wavegudes wth shap metal edges, IEEE Mcowave Theoy Tech., vol. 42, no. 1, pp , Januay [17] Gl, J. M. and Zapata, J., A new scala tanston fnte element fo accuate analyss of wavegudes wth feld sngulates, IEEE Mcowave Theoy Tech., vol. 43, no. 8, pp , August [18] ebb, J. P., Heachal vecto bass functons of abtay ode fo tangula and tetahedal fnte elements - IEEE Tans. Antennas and Popagaton, vol. 47, n. 8, August [19] Lee, S.., Lee, J.F. and Lee, R., Heachcal vecto fnte elements fo analyzng wavegude stuctues - IEEE Tans. Mcowave Theoy Tech., vol. 51, n. 8, pp , August [20] Valey P. mtyev, On vecto potental of the oulomb gauge, Lomonosov Unvesty, Russa, (25/nov/2005). [21] Atuz J.,., Impovng the Newmak tme ntegaton scheme n fnte element tme doman methods - IEEE Mcow. eless ompon. Lett., ecembe [22] Bahl, I. J. and Bata, P., Mcowave Sold State cut esgn, ley, New Yok, [23] Renmut, K. H., Handbook of Mcowave Integated cuts, Atech House, Nowood, MA., [24] Guston, M. A. R., Mcowave Tansmsson-Lne Impedance ata, Van Nostand Renhold, London, 1972.