Low-Frequency Full-Wave Finite Element Modeling Using the LU Recombination Method

Transcription

1 33 CES JOURNL, OL. 23, NO. 4, DECEMBER 28 Low-Fqucy Fu-Wav Fiit Elmt Modlig Usig th LU Rcombiatio Mthod H. K ad T. H. Hubig Dpatmt of Elctical ad Comput Egiig Clmso Uivsity Clmso, SC bstact I this pap, th low-fqucy istability of fu-wav fiit lmt mthods (FEM) is ivstigatd. Th cul pat of th FEM matix is show to b sigula. Th pap xplais how low-fqucy istabilitis a latd to this sigulaity. Basd o this aalysis, a LU combiatio mthod is implmtd i FEM to solv th low-fqucy poblm. This mthod, which has pviously b applid to th mthod of momts (MOM), ducs th os i th cul pat of th matix ad focs th coct gaug coditio. Moov, th mthod is stuctud to wok mo fficitly fo spas fiit lmt matics. I. INTRODUCTION Th fiit lmt mthod [] is w-suitd fo solvig poblms ivolvig ihomogous abitailyshapd objcts. May sachs hav obsvd that th cul-cul opatio that is fqutly mployd wh FEM is usd to solv th vcto Hlmholtz quatio ca sult i i-coditiod matics i som cicumstacs [2-4]. O situatio that gats i-coditiod matics ad ustabl solutios is modlig pfomd at low fqucis. Th xampls pstd i this pap iustat this bhavio. I [5], spcial palty tms w itoducd ad pottial fomulatios w usd to dal with this poblm. Most fu-wav sufac itgal tchiqus also suff fom low fqucy difficultis [6, 7]. Th low fqucy istabilitis ca b ascibd to th divgc opato applid to th ukow sufac cut dsity i th itgal quatio. Mathmaticay, ths istabilitis a latd to th sigula popty of th scala pottial pat of th impdac matix. mthod to cicumvt this poblm was ctly poposd [8, 9]. This appoach, cad th LU combiatio mthod, mploys lia tasfomatios of th momt matics i od to isolat ad limiat o-physical solutios. I this pap, th low-fqucy poblm with fiit lmt fomulatios is dscibd i tms of th sigula popty of th cul pat i th fiit lmt matix wh usig cul-cofomig Ndlc-typ basis fuctios. Th LU combiatio mthod is applid i od to isolat th sigulaity i th cul pat of th fiit lmt matix. It is ot cssay to itoduc ay palty tms, o cat w basis fuctios. Th appoach is futh fid so that th w matics aft LU combiatio a patiay spas. This ducs th computatio cost ad gatly impovs th pfomac. Fiay, a coupl of xampls a pstd. II. FORMULTION Fom Maxw s quatios, th vcto Hlmholtz quatio i tms of th E fild ca b witt as, ( ) E + jωε ε E it J ( ) it ( ) M ( ) wh J it ad M it a impssd lctic ad magtic soucs; ω is th agula fqucy; µ ad ε a th f spac pmability ad pmittivity; ad µ ad ε a th lativ pmability ad pmittivity. ft applyig a wightig fuctio w(), th FEM wak fom is [4, ], ( E( ) ) ( w( ) ) S ( ˆ H ( ) ) w( ) J it ds ( ) w ( ) it ( ) + M ( ) w( ) + jωε ε E wh S is th sufac closig volum. Th ukow E fild is xpadd usig culcofomig basis fuctios that a th sam as th wightig fuctios, d d () (2)

2 KE, HUBING: LF FULL-WE FINITE ELEMENT MODELING USING LU RECOMBINTION 34 ( ) E ( ) E w (3) wh E a ukow cofficits. Th sufac itgal o th ight had sid of quatio (2) is valuatd by usig sufac basis fuctios f (), which a latd to w () by, ( ) ˆ ( ) w f. (4) Equatio (2) is th disctizd ito a matix quatio, E B J+ S. (5) Th ight had sid psts th bouday coditio ad th souc tm. J is th quivalt cut dsity o th sufac. S is th souc tm. E is a vcto cotaiig th ukow cofficits i quatio (3). Th lmts of a, Lt m ( w ( ) ) ( w ( ) ) m + jωε ε w w m ( ) ( ) ( ) ( ) m ωε ε m d. (6) j w w d, (7a) which is th ight-most tm i th ight-had sid of quatio (6). Lt 2m ( w ( ) ) ( w ( ) ) m d, (7b) which is th lft-most tm i th ight-had sid of quatio (6). Evy lmt of appoachs zo at abitaily low fqucis. Th foowig asoig dmostats that, bcaus of th opato, 2 is a sigula matix wh usig th popula lowst od culcofomig basis fuctios. Th ak of th matix is dtmid by th total umb of ital ods i th fiit lmt msh []. Th basis fuctio fo a ttahdo ca b dfid o ach dg as [2], w 7 i f + g i th ttahda 7 i 7 i othwis i,2,...,6 (8a) wh ad f l7 i 7 i i i2 6, (8b) 6 g i 7 i. (8c) 7 i i H i ad i 2 a th od idics of dg i, dfid i Fig.. l is th lgth of th dg, is th uit vcto alog th dg, ad is th volum of th ttahdo. 4 Fig.. Th ttahdal lmt ad its dg-od latios. Cosid th local lmts, i.., th lmts valuatd withi o ttahdo. Th cul of th basis fuctio is a costat withi th ttahdo, Th local lmt i is, 2 li w 2g l. (9) i i 7 i g g 2ij i j i j l l i j 9 dg i i () wh th supscipt idicats local lmts. Thus withi o ttahdo, th local matix ca b witt as i quatio (), show o th top of th xt pag. Fo th ttahdo i Fig., dgs 4, 5, ad 6 fom a tiagl, which mas, l 4 +l 5 +l 6. (2) Cosqutly, th fist th ows i quatio () a lialy dpdt.

3 35 CES JOURNL, OL. 23, NO. 4, DECEMBER l6 l6 l6 l5 l6 l4 l6 l3 l6 l2 l6 l l5 l6 l5 l5 l5 l4 l5 l3 l5 l2 l5 l l3l 6 l4 l6 l4 l5 l4 l4 l4 l3 l4 l2 l4 l jωµ µ l3 l6 l3 l5 l3 l4 l3 l3 l3 l2 l3 l l2 l6 l2 l5 l2 l4 l2 l3 l2 l2 l2 l l l6 l l5 l l4 l l3 l l2 l l () O mo spcificay, cosid th omalizd local N whos lmts a giv by, matix ( 2 ) ( ) l l. (3) N 2ij i j 2ij 9 i j Th fist th ows of ( 2 ) ad a latd by itgs, ( ) ( j j) ( j) N a lialy dpdt N N N j,2,...,6. (4) This latioship is simila to quatio () i [9]. Th lialy dpdt ows cospod to th dgs i th ttahdo shaig a commo od. ft assmblig th local matics ito a global matix, th ows of th global matix cospodig to dgs shaig th sam ods a lialy dpdt, sultig i a sigula global 2. It is vidt fom quatio (7) that th FEM matix ( + 2 ) wi b ustabl at low fqucis, sic a matix ( ) with lmts appoachig zo wi b addd to a sigula matix ( 2 ). Du to limitd comput pcisio, th lmts of ca bcom buid i th oud-off o ad ot v affct th valus i th ova fiit lmt matix. Howv, th ifomatio i icopoats th gaug coditio fo th lctic fild [5]. Without this ifomatio, th matix quatio (5) is i-coditiod, sultig i sigificat os i th solutio. III. LU RECOMBINTION METHOD IN FEM Basd o th abov aalysis, th low fqucy poblm i FEM is aalogous to th low-fqucy poblm i th bouday lmt mthod [9]. Cosqutly, th LU combiatio mthod dvlopd fo th bouday lmt mthod ca b applid to th fiit lmt mthod. LU combiatio ca b usd to foc th sigula popty of 2 ad psv th coct gaug coditio i matix. Howv, th oigial 2 is a spas matix. pplyig LU combiatio i th ma dscibd i [9] would poduc a w, ds 2 matix. This would b highly udsiabl i a FEM fomulatio. Thfo, a icomplt LU combiatio tchiqu that is suitabl fo spas matics was dvlopd. I this appoach, oly pat of 2 is modifid ad th sultig w matix is sti spas. Th spasss of th w matix dpds o th umb of i ods i th msh. Th mthod bgis with th L-D-U dcompositio of 2, show i quatio (5) at th bottom of this pag, just as it dos wh applid to th bouday lmt mthod [9], L U U 2ii 2id ii ii id 2di 2dd L L U U T L ii D D 2ii 2id L ii L L D D 2di 2dd L L.(5) Th 2 matix is patitiod so that th lialy idpdt ows (pstd by th subscipt ii) a goupd togth ad th dpdt ows (subscipt dd) a movd to th d. Th LU combiatio mthod wi

4 KE, HUBING: LF FULL-WE FINITE ELEMENT MODELING USING LU RECOMBINTION 36 modify th sub-matics L di, D 2di, D 2id, ad D 2dd, whil L ii ad D 2ii a lft uchagd. Thfo, 2ii L ii D 2ii is uchagd aft costuctig a w 2. Th is o d to calculat 2ii aft th modificatios o L ad D 2. To accomplish that, th L matix is placd by, ii I L, (6) L I wh I is idtity matix. Th w dcompositio o 2 is th witt i quatio (7), as show at th bottom of this pag. Thus duig th LU combiatio, th 2ii pat mais th sam. No additioal lmts o os a itoducd. Th sam dcompositio i quatio (7) is applid to. ft LU combiatio, th w bcoms quatio (8) at th bottom of this pag. Not that L di is alady modifid, as dscibd i [9]. Th coct ifomatio i is psvd. But th w matix is sti i-coditiod at low fqucis sic is much sma tha 2. Th imbalac ca b aviatd by itoducig a scalig stp. Th sub-matics D di, D id, ad D dd a scald so that thy a compaabl to 2ii. This stp gatly impovs th coditio of th w matix. It is spciay bficial wh itativ mthods a usd to solv th matix quatios. I. NUMERICL RESULTS Two sampl stuctus w valuatd usig a fiit lmt modlig tchiqu with ad without LU combiatio. Th fist xampl is th ctagula pow bus stuctu show i Fig. 2. Th dimsios of th stuctu a 2 cm x 2 cm x cm. Th pow ad goud plas a modld as pfct lctic coductos (PECs). Th fou sid was of th boad a modld as pfct magtic coductos (PMCs). Th dilctic btw th plas has a lativ pmittivity of 4.5. Th boad is xcitd by a idal cut souc locatd i th dilctic, 6 cm fom o dg ad 7 cm fom a adjact dg. cm 2 cm 2 cm Fig. 2. pow bus xampl. Cut souc Th iput impdac of th pow bus was calculatd, ad th sults obtaid usig difft mthods a show i Fig. 3. Th solid dots show th sult obtaid usig th stadad FEM fomulatio. This tchiqu fails wh th fqucy is blow MHz. Th low fqucy limit is dtmid by th umb of sigificat figus usd wh maipulatig th lmts of th FEM matix. Th solid li idicats th sult wh th LU combiatio mthod is icopoatd ito th FEM. This sult is accuat v blow a fw Hz. ohms LU FEM cavity HFSS fqucy (MHz) Fig. 3. Th iput impdac of th pow bus. lso show i Fig. 3 a th sults calculatd usig a cavity modl ad soft HFSS [3]. Th cavity modl is a mod-xpasio mthod suitabl fo ctagula pow bus gomtis. It modls th pow bus as a TM z cavity ad dtmis th iput impdac by summig th cotibutios of a lvat soat mods [4]. This mthod has o poblm at low fqucis but it ca b difficult to apply to complicatd gomtis. HFSS, which is a FEM modlig cod, xtapolats fom th high fqucy sults to obtai a low fqucy appoximatio. I this cas, th xtapolatio was valid dow to a fw khz. I 2ii 2id ii ii id 2di 2dd L I U U T I ii D 2ii 2id I ii L I D D 2di 2dd L I I ii id ii L I T D ii id 2ii I ii + di dd D D L I. (7) (8)

5 37 CES JOURNL, OL. 23, NO. 4, DECEMBER 28 Th scod xampl is a stipli stuctu cosistig of a mtal tac imbddd i a dilctic btw two mtal plas as show i Fig. 4. Th dimsios of th plas a 2 mm x mm ad thy a 2 mm apat. Th dilctic costat is 4.5. Th tac has a width of mm ad a lgth of mm. Th tac is div by a.- cut souc at o d, ad is tmiatd by a 5-Ω sisto o th oth d. I this xampl th top ad bottom plas a modld as PECs, ad th quivalt cut o th dilctic bouday is st to zo. lso show i Fig. 4 is th top viw of th FEM msh. Figu 5 shows th magitud of th iput impdac calculatd at th souc pot. Th impdac should hav a al valu of 5 Ω at low fqucis. Th gula FEM sult xhibits sigificat os blow MHz. I fact, th a obsvabl istability poblms at fqucis abov MHz. With th hlp of th LU combiatio mthod, th o is coctd ad th sults a accuat dow to a fw Hz. Fig. 4. micostip xampl ad th msh. Iput Impdac (ohms) FEM LU fqucy (MHz) Fig. 5. Th iput impdac of th stipli. CONCLUSION Th sigula bhavio of th disctizd cul tm i th vcto Hlmholtz quatio causs low-fqucy istabilitis i fu-wav FEM fomulatios. Th LU combiatio mthod ca b applid to xistig FEM cods to solv this poblm. Th LU combiatio mthod uss lia tasfomatios to miimiz th ifluc of os i th cul pat of th matix. Poply applid, it is possibl to psv th spasss of th FEM matix. REFERENCES [] J. Ji, Th Fiit Elmt Mthod i Elctomagtics, Nw Yok: Joh Wily & Sos Ic., 993. [2] W. Boys, G. Mibo, K. Pauls, ad D. Lych, pplicatio of pottials to fiit lmt modlig of Maxw s quatios, IEEE Tas. Mag., vol. 29, o. 2, pp , Ma [3] J. L, D. Su, ad Z. Cds, Tagtial vcto fiit lmts fo lctomagtic fild computatio, IEEE Tas. Mag., vol. 27, pp , Sp. 99. [4]. F. Ptso, S. L. Ray, ad R. Mitta, Computatioal Mthods fo Elctomagtics, pp , Nw Yok: IEEE Pss ad Oxfod Uivsity Pss, 997. [5] R. Dyczij-Edlig, G. Pg, ad J. L, Efficit fiit lmt solvs fo th Maxw quatios i th fqucy domai, Comput Mthods i pplid Mchaics ad Egiig, vol. 69, o. 3, pp , Fb [6] M. Buto ad S. Kashyap, study of a ct, momt-mthod algoithm that is accuat to vy low fqucis, ppl. Computatioal Elctomag. Soc. J., vol., o. 3, pp , Nov [7] W. Wu,. W. Glisso, ad D. Kajfz, study of two umical solutio pocdus fo th lctic fild itgal quatio at low fqucy, ppl. Computatioal Elctomag. Soc. J., vol., o. 3, pp. 69-8, Nov [8] H. K ad T. Hubig, Usig a LU combiatio mthod to impov th pfomac of th bouday lmt mthod at vy low fqucis, Pocdigs of th 25 IEEE It. Symp. O EMC, Chicago, IL, ug. 25. [9] H. K ad T. Hubig, modifid LU combiatio tchiqu fo impovig th pfomac of bouday lmt mthods at low fqucis, ppl. Computatioal Elctomag. Soc. J., vol. 2, o. 3, pp , Nov. 25. [] Y. Ji, Dvlopmt ad applicatios of a hybid fiit-lmt-mthod/mthod-of-momts ( FEM / MOM) tool to modl lctomagtic compatibility

6 KE, HUBING: LF FULL-WE FINITE ELEMENT MODELING USING LU RECOMBINTION 38 ad sigal itgity poblms i pitd cicuit boads, Ph.D. disstatio, Uivsity of Missoui- Roa, 2. [] N. kataayalu, M. ouvakis, Y. Ga, ad J. L, Suppssig lia tim gowth i dg lmt basd fiit lmt tim domai solutio usig divgc f costait quatio, IEEE tas ad Popagatio Socity Itatioal Symposium, vol. 4b, pp , Jul. 25. [2] X. Yua, Th-dimsioal lctomagtic scattig fom ihomogous objcts by th hybid momt ad fiit lmt mthod, IEEE Tas. Micowav Thoy ad Tch., vol. 38, o. 8, pp , ug, 99. [3] HFSS vsio 9., soft Copoatio. [4] M. Xu ad T. Hubig, Estimatig th pow bus impdac of pitd cicuit boads with mbddd capacitac, IEEE Tas. dv. Packag., vol. 25, o. 3, pp , ug. 22. Haixi K civd his BSEE ad MSEE dgs fom Tsighua Uivsity i 998 ad 2, spctivly, ad his Ph.D. i Elctical Egiig fom Uivsity of Missoui-Roa i 26. H is cutly a Post-Doctoal sach at Clmso Uivsity. His sach itsts iclud computatioal lctomagtics, lctomagtic compatibility, ad vhicula lctoic systms. Todd Hubig civd his BSEE. dg fom th Massachustts Istitut of Tchology i 98, his MSEE dg fom Pudu Uivsity i 982, ad his Ph.D. i Elctical Egiig fom Noth Caolia Stat Uivsity i 988. Fom 982 to 989, h was mployd i th Elctomagtic Compatibility Laboatoy, IBM Commuicatios Poducts Divisio, i Rsach Tiagl Pak, NC. I 989, h joid faculty at th Uivsity of Missoui-Roa (UMR). t UMR, h wokd with faculty ad studts to aalyz ad dvlop solutios fo a wid ag of EMC poblms affctig th lctoics idusty. I 26, h joid Clmso Uivsity as th Michli Pofsso fo hicula Elctoics. Th h is cotiuig his wok i lctomagtic compatibility ad computatioal lctomagtic modlig, paticulaly as it is applid to automotiv ad aospac lctoic dsigs. Pof. Hubig has svd as a associat dito of th IEEE Tasactios o EMC, th IEEE EMC Socity Nwsltt, ad th Joual of th pplid Computatioal Elctomagtics Socity. H has svd o th boad of dictos fo both th pplid Computatioal Elctomagtics Socity ad th IEEE EMC Socity. H was th Psidt of th IEEE EMC Socity ad is a Fow of th IEEE.