Finite Element Method Modeling for Computational Electromagnetics Development of a Perfectly Matched Layer for Domain Termination

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1 1 Finit Elmnt Mthod Modling o Computational Elctomagntics Dvlopmnt o a Pctly Matchd Lay o Domain Tmination Fist Smst Rpot Fall Smst 214 -Full pot- By Aaon Smull Tam Mmbs: Ana Manic Sanja Manic Ppad to patially ulill th quimnts o ECE41 Dpatmnt o Elctical and Comput Engining Coloado Stat Univsity Fot Collins, CO 8523 Pojct adviso: D. Banislav Notaos Appovd By:

2 2 ABSTRACT In computational lctomagntics, th init lmnt mthod is an xtmly ctiv mthod o modling lctomagntics poblms with complicatd gomtis and mdiums. Sinc th init lmnt mthod involvs physical modling o computational domains, it quis a init domain o simulation. Fo computational poblms which qui ininit domain simulation (.g., scatting and antnna adiation poblms), w must us atiicial mthods to ctivly tminat th computational domain without intoducing lag amounts o numical o. Th pctly matchd lay is on o ths mthods. Th pupos o this pojct is to tak alady xisting cod o init lmnt mthod simulation, and optimiz and simpliy th cod, as wll as implmnting an ctiv pctly matchd lay. In od to ctivly tackl th poblm at hand, th tam has don xtnsiv sach into thoy bhind atiicial absoptiv boundais, including th squa and conomal pctly matchd lays, th ist and scond-od absobing conditions as wll as oth lss common oms o bounday tmination. W appoachd this poblm by doing litatu sachs among common jounals (IEEE Xplo) as wll as utilizing scholaly sach ngins (Googl Schola). In od to simpliy and optimiz th cod, w stippd it down to a ba-bons stuctu and valuatd which pats o th cod s main stuctu w ncssay, which pats could b valuatd, and which pats w unncssay to kp. So a, w hav dvlopd a ull thotical divation o th pctly matchd lay o scatting poblms, including a omulation o th lvant quations, and bgun implmnting sctions o cod ncssay to implmnt this omulation. As o optimization o th cod, w hav mad sval changs th stuctu o th cod, including patially witing cod o intgation and matix illing, spding up th ovall xcution. W w abl to mov a w sctions o cod, making it simpl o utu collaboatos to gt involvd. Additionally, documntation o th cod has bgun, in od to ctivly xplain how th cod woks and stamlin uth changs.

3 3 Tabl o Contnts Titl... 1 Abstact... 2 Tabl o Contnts... 3 List o Figus... 4 Intoduction... 5 a.) Computational Elctomagntics... 5 b.) Th Finit Elmnt Mthod... 5 c.) Pojct Dsciption... 5 d.) Pojct Spciications... 6.) Pojct Impact... 6 I. Finit Elmnt Mthod... 7 a.) Thoy... 7 b.) Choic o Basis Functions... 8 c.) Gomtical Elmnts... 9 d.) Anisotopic and Inhomognous Mdia... 9 II. Pctly Matchd Lay a.) Pupos b.) Thoy c.) Wav Equation Romulation III. Applications a.) Mdical b.) Militay c.) Communications IV. Tsting and Validation V. Conclusion and Futu Dvlopmnts Rncs Bibliogaphy Appndix A: Abbviations... 2 Appndix B: Budgt... 2 Appndix C: Pojct Plan Evolution Acknowldgmnts... 23

4 4 List o Figus Figu 1 Basis Functions o FEM 9 Figu 2 Gnalizd Hxahdal Elmnt 1 Figu 3 Pctly Matchd Lay 12 Figu 4a Modl o Human Bain 14 Figu 4b Nomalizd Elctic Fild insid Bain 14 Figu 5 Stalth Bomb Modl 15 Figu 6 Hon Antnna Modl 15 Figu 7 Sph Modl 16 Figu 8 Analytic Solution o Scatting om Sph 16

5 5 Intoduction a.) Computational Elctomagntics Computational lctomagntics is a ild that involvs inding numical solutions to Maxwll s quations in an abitay domain o intst. Its applications a numous Fom communications systms to mdical imaging, lctomagntics play a ky ol in all o ou livs vy day. Th ida bhind computational lctomagntics is to dvlop sotwa that icintly and accuatly simulats lctomagntic bhavio o vaious lctical systms; this is all in od to simpliy th dsign pocss, and potntially pvnt costly mistaks in pototyping (simulation o a modl can b don bo any initial poduction has bgun). Otn, dsign and tsting will b don ully in a CAD stting bo any physical componnts a v tstd. Chapt IV contains citical xampls o computational lctomagntics ol in th wold o ngining and scinc. b.) Th Finit Elmnt Mthod Th Finit Elmnt Mthod (FEM) is just on o many mthods usd in th ild o computational lctomagntics. It involvs disctizing th computational domain into small subdomains (lmnts), and thn appoximating th EM ilds within ach lmnt by a lina combination o basis unctions. On o FEM s gatst advantags is in its ability to handl unusual o complicatd domains. By choosing an ctiv st o lmnts and basis unctions, w can modl all sots o complicatd gomtis, as wll as anisotopic and inhomognous matials with as. Chapt I contains a thotical dsciption o th Finit Elmnt Mthod, as wll as dtails on basis unction and modling choics. Th Finit Elmnt Mthod is not th only commonly usd mthod o computational lctomagntics. Among oths a th init dinc tim domain mthod (FDTD) and th volum intgal quation (VIE) and suac intgal quation (SIE) omulations o th Mthod o Momnts (MoM). Each o ths mthods has its own plac, and ach o thm has stngths and waknsss. On o th unotunat dawbacks o th Finit Elmnt Mthod is that w must hav som mthod to tminat th computational domain. In poblms daling with cavitis o wavguids, w otn tminat th bounday o th domain with a pct lctic conducto (PEC), but whn daling with scatting o adiation poblms, th PEC causs spuious (nonphysical) lctions o wavs om th dg o th computational domain [1]. Ths nonphysical lctions lad to highly inaccuat solutions and mak computations lss than usul. Chapt II dscibs th pctly matchd lay, a computational tchniqu usd to accuatly and ctivly do this tmination.

6 6 c.) Pojct Dsciption Th main pupos o this pojct is to mak impovmnts to th alady xisting FEM lctomagntic dvlopd in pat by th lctomagntics lab. This cod has alady povn ctiv in applications to wavguids, cavitis, scatting, and adiation poblms (s publications on FEM by D. Banislav Notaos and D. Milan Ilic o uth dtails). Howv, th sotwa quid som amount o modnization and optimization, including implantation o nw tchniqus, and quick computation o old tchniqus. In paticula, on o th biggst goals o this pojct is to dvlop a pctly matchd lay (PML) schm o us in scatting and adiation poblms (as wll as oth utu dvlopmnts). Th nd goal o this long-unning pojct is to dvlop a ull-ldgd lctomagntic simulato o ctiv and ast simulation o all mann o computational lctomagntics poblms, om ull wav simulation o scatts and antnnas, to wavguid and cavity analysis. In od to hav an ctiv, multi-us solv, it must b abl to implmnt all mann o computational tools. Th pctly matchd lay is just on among a sco o ths tools. d.) Pojct Spciications As this is pimaily a sach ointd pojct, th pojct spciications bhind th dvlopmnt o th FEM cod a not xtmly stict. Howv, th a a w things to consid whn implmnting nw mthods such as th PML, and just updating th cod in gnal. Fo implmntation o th PML, w xpct to s impovmnt in accuacy om th cuntly usd absobing bounday conditions (ABC S) in th sults o scatting and adiation poblms. Fo optimization o th cod, w xpct a dcas in th computational tim quid to simulat a standad poblm. Additionally, w qui a dcas in th amount o cod. Sctions o cod can b combind o movd ntily, to dvlop a gnally mo obust and asy-toimpov pogam. As changs a mad to th cod, it is ssntial to th pojct to continually undgo dbugging and tsting. Tsting and validation a discussd in chapt IV. On impotant thing to consid whn daling with any sot o sotwa pojct is that o systm compatibility. Fotunatly, du to standads, w can wit ctiv cod that can b compild to un on any sot o hadwa systm. It is on o ou goals to vntually paallliz th cod, so it can b un on a high-pomanc computing systms such as th CSU Cay XT6m systm. Each platom has its own limitations, but w hav th potntial to vntually b abl to pot cod to an assotmnt o platoms..) Pojct Impact Th impact o this pojct is nomous. Licnss o lctomagntic simulatos on th makt can un om th low thousands to ov 1, dollas. As such, any pogam wishing to

7 7 suviv on this makt must b xtmly comptitiv. It must hav advancd atus compaabl with any oth lctomagntic solv. Fo instanc, ANSYS HFSS os a cubical PML implmntation o scatting analysis, but not a conomal PML (conomal PML can duc th numb o unknowns quid to solv o, and spd up th ovall analysis). In gnal, a comptitiv pic o sotwa should o all mann o computational options. To kp th lctomagntics lab at th vanguad o sach, it is ncssay to continually mak impovmnts upon xisting cod. Cuntly, only absobing bounday conditions hav bn implmntd, but ths hav bn known by th scintiic community [2] to b gnally lss accuat than th PML mthod. Th PML givs a mo accuat way to tminat th computational domain, and can actually b combind with th ABC to mak ou sults mo accuat than ith mthod alon. a.) Thoy I. Finit Elmnt Mthod [1] In th init lmnt mthod, an xtmly popula mthod o advancd lctomagntic modling, th computational domain is disctizd into multipl small domains, which w call lmnts. Th lctic ilds acoss ach lmnt a thn appoximatd using a combination o basis unctions. Ths basis unctions can b in gnal vctos o scala unctions. W opt to us vcto basis unctions (otn d to as dg-basd basis unctions). Fom Maxwll s quations in th quncy domain, w can asily div th doubl cul lctic ild wav quation: -1 E k 2 E Fom this quation, w apply th Galkin tsting pocdu. W multiply th quation by a tsting unction, and intgat ov all spac. V iˆ ˆjk ˆ -1 2 ( E) dv k E dv V ijk ˆˆ ˆ By applying th Gn s ist idntity o vcto unctions, w can wit this quation as: V ( iˆ ˆjk ˆ -1 ) ( E) dv k 2 V ijk ˆˆ ˆ E dv S ijk ˆˆ ˆ -1 E ds Th suac intgal on th ight hand sid dnot th suac nclosing th nti computational domain. In gnal, th ight hand sid o this quation is usd to apply vaious bounday conditions. Th intgals on th lt hand sid can b valuatd o ach basis and tsting unction to gnat a matix quation.

8 8 2 A k B G Th lctic ilds in ach lmnt a thn xpandd as a sis o basis unctions: E Nu 1 i Nv j2 N w k 2 uijk uijk Nu i2 Nv 1 j Nw k 2 vijk vijk Nu i2 Nv j2 Nw 1 k wijk wijk Wh uijk a th dg-basd basis unctions, and uijk a th cosponding coicints, which a unknowns to b solvd o. By choosing a sis o dint tsting unctions, th abov tsting pocdu givs us a st o lina quations, which can b put into a matix and solvd. Th Galkin tsting pocdu spciis that th tsting and basis unctions a chosn to b th sam. b.) Choic o Basis Functions W us th ollowing st o abitaily high-od basis unctions: uijk vijk wijk i u P ( v) P ( a i i j i P ( u) v P ( a k P ( u) P ( v) w a j k k u v w P P uijk vijk P wijk a u a v a w, 1 u, i u 1, i 1 Pi ( u), 1 u, w 1 i u 1, i 2, vn i u u, i 3, odd Wh a, a u v and aw a unitay vctos. Ths basis unctions hav th advantag o bing hiachical [3] (ach st o basis unctions is a subst o any high od st) as wll as bing cul-conoming (i.., this choic o basis unctions satisis th continuity o th lctic ilds tangntial componnt acoss th boundais btwn lmnts). It is impotant to not that th basis unctions a only dind ov a singl lmnt. Tho, th tsting intgals only nd to b valuatd ov th volum o a singl lmnt. Additionally, ths intgals will b zo i th basis and tsting unction cospond to unknowns om two dint lmnts. Th ist coupl o unctions P i (u) a shown in igu 1. Th ist two basis unctions povid continuity btwn th dint lmnts o th systm (w do this by quating ths unknowns btwn lmnts), whil th high od basis unctions povid a mo accuat appoximation to th lctic ild insid th lmnt. Expanding th ild appoximation with high od basis unctions allows us to incas th siz o ach lmnt, and ducs th numb o ovall unknowns. W a abl to modl th nti computational domain with a w high od lmnts ath than a lag amount o ist od lmnts.

9 9 Figu 1: Basis unctions o FEM (in 1 D) [4] c.) Gomtical Elmnts Whn th siz o ach lmnt incass, objcts must b caully constuctd om a st o ths lmnts. In th past, ttahdal and bick shapd lmnts hav bn usd. Ths choics a not xtmly lxibl o abitay shapd domains, spcially whn th siz o th lmnt incass. Instad, w opt to us a st o gnalizd hxahdal lmnts, which a mappd via Lagang-typ intpolation polynomials om th physical domain to th st o local coodinats u, w, ov which th basis unctions a dind. This mapping allows us to din a (thotically) abitay high gomtical od to th lmnt. This additional high od gomtic appoximation allows th lmnts to b it to mo complicatd gomtis, which a not ncssaily asy to modl with ttahdal o bick lmnts. Th mapping om th lmnt to th local coodinats is givn by [5]: ˆK L uvw, 1 u, w 1 M i i1 i Wh i din th intpolation nods usd to din th lmnt, and ˆK Li uvw a th polynomials dind by: ˆK K L L u K ( u) L v Kw 2 i m n ( v) Ll (, i 1 m n( Ku 1) l( Ku 1)( Kv 1), m, n K v, l K w 1 i M ( Ku 1)( K v 1)( K w 1) K u

10 1 K u, K v and K w a gomtical ods o th lmnt, and polynomials, givn by th omula L K m ( u) K j jm u u u m j u j K L m a Lagang-typ intpolating Wh u j a intpolating nods. Figu 2 shows an xampl o on o ths lmnts, mappd om th cubical domain ov which th basis unctions a dind, to th physical, hxahdal domain. Figu 2: Mapping om lmnt to local coodinat systm [1] d.) Anisotopic and Inhomognous Mdia In od to implmnt th pctly matchd lay, it s impotant to hav cod that can icintly and accuatly simulat anisotopic and inhomognous matials. Fotunatly, du to th lxibility o th gomtical lmnts w us, simulating ths complicatd mdia is vy doabl. Fo a gnal, anisotopic and inhomognous lmnt, w can pom th sam Lagang intpolation schm usd in th gomtical modling to xpand th pmittivity tnso in tms o a sis o Lagang polynomials, ach wightd by tnso at th intpolation nods [6]., xx, xy, xz M u M v M w M u M v M w, yx( u,, yy, yz, mnplm ( u) Ln ( v) Lp ( m n p, zx, zy, zz

11 11 With this, ou cod can povid an accuat continuous appoximation to any gnal matial. To put th usulnss o this atu in pspctiv, th industy standad ull wav simulato HFSS can only appoximat inhomognous matials with picwis appoximations. This quis a lag numb o lmnts and a much lag amount o unknowns. Th lxibility o using high od appoximations gatly dcass th computational tim quid. a.) Pupos II. Pctly Matchd Lay Th pctly matchd lay is cuntly th biggst ocus o this pojct. Sinc in th Finit Elmnt Mthod w cannot modl all o spac, w a quid to tminat th computational domain in such a way that dos not caus unphysical sults. Sval dint schms hav bn suggstd in od to do this. On xtmly common mthod is th ABC, as psntd in sval scintiic paps, as wll as txtbooks on lctomagntics (.g., [7]). Many vaiants on th ABC hav aisn, including th common scond-od ABC. Mo cntly, th PML bounday condition has suacd, which is abl to b caully dsignd so as to povid btt computational sults than th ABC [2]. As opposd to th ABC, which involvs applying a bounday condition at th dg o th domain, th PML actually involvs us modling a lay aound th physical computational domain, which will not caus unphysical lctions, but will attnuat any outgoing wavs. b.) Thoy Implmnting th PML involvs a omulation o th doubl cul vcto wav quation to a mo gnal quation [8]: s -1 s E k E, wh 2 s 1 s x 1 x s y 1 z s z z In gnal, ilds that satisy this quation do not satisy Maxwll s quations, unlss s x s y s z 1. I s x, s y and sz a chosn appopiatly (as complx numbs), it can b shown that th mdium will attnuat popagating wavs. Additionally, an intac btwn a mdium with ths paamts and a mdium with s x s y s z 1 will xhibit zo lction, as long as both mdia hav th sam lativ pmittivity and pmability and. In od o s x, s y and sz to poply attnuat outgoing wavs, thy must b chosn dintly dpnding on thi location within th computational domain. Fo instanc, i a cubical PML lay is modld, ach sid o th cub will hav dint valus o s x, s y and s z. Fo this ason, it is a mo diicult poblm to gnat an abitay PML that will attnuat any outgoing wav. Th vntual goal o this pojct is to implmnt a conomal PML that will accomplish xactly this task.

12 12 It can b shown that th so-calld sttching paamts, s, s x y and s z, can b absobd into and i thy a takn to b 3x3 complx tnsos instad. Th poblm thn dos not qui omulation o th cul opato, but mly modling o th pop anisotopic mdium to cospond to th dsid valus o s, s and s z. [7] Th appopiat quation is now just E k E Wh x y S ysz Sx SzS S y x S x S S z y I this tnso is absobd into th pmittivity and pmability tnsos, th pml quation is just th wav quation again. Figu 3: PML Diagam [8] c.) Wav Equation Romulation

13 13 In th taditional implmntation o scatting poblms, th xcitation o th scatt is don on th vy outsid o th computational domain. Howv, as th out lay o th computational domain (th PML) now xpincs attnuation, w cannot xcit th domain om th vy outsid. W qui a omulation o th oiginal computational poblm. W Total E cogniz that in a scatting poblm, th total lctic ild can b dcomposd into two Incidnt Scattd spaat ilds E and E [2]. Th incidnt lctic ild is gnally givn by a plan wav, which w know analytically. W can wit th doubl-cul wav quation in tms o ths two ilds spaatly. -1 E Scattd 2 k E Scattd -1 E Incidnt 2 k E With this nw omulation, w now hav two spaat domains, ach with thi own om o this quation. Rath than xpanding th nti lctic ild in tms o th basis unctions, w just xpand th scattd ild, sinc th incidnt ild is alady known. Insid th scatting objct (which has ith 1 o 1), th ight hand sid bcoms a vcto at applying th tsting pocdu. Th lt hand sid mains th sam as bo, only th unknowns a now colatd only to th scattd ild. At applying th tsting pocdu w aiv at: Incidnt V ( iˆ ˆjk ˆ -1 ) ( E Scattd ) dv k 2 V ijk ˆˆ ˆ E Scattd dv ijk ˆˆ ˆ D S -1 E Scattd ds V ( iˆ ˆjk ˆ -1 ) ( E Incidnt ) dv k 2 V ijk ˆˆ ˆ E Incidnt dv ijk ˆˆ ˆ Sc S -1 E Incidnt ds D H, S dnots th out bod o th computational domain, and th scatting objct. Sc S dnots th suac o In spac and insid th PML, th ight hand sid o this quation ducs to and w a only lt with th lt hand sid, th scattd ild omulation. At omulating th poblm at hand, th st o th task is mainly an issu o gnating modls with PML s appopiat to th paticula scatting poblm. III. Applications o Computational Elctomagntics On o th most impotant qustions aising duing a discussion o this pojct is what th pupos o computational lctomagntics ally is. This ild has a vast amount o applications, anging om militay applications, to mdical inomation, and communications systms. Elctomagntic intactions a a undamntal pat o any lctonics systm, and cannot b ignod, spcially as lctonics systms mov to high and high quncis.

14 14 a.) Mdical Otn a qustion o gat impotanc is how lctomagntic wavs intact with th human body. By gnating modls o human body pats, o nti human bodis, w can s how th human body acts to vaious quncis and stngths o lctomagntic ilds. This analysis can hlp nsu saty o al wold lctomagntic systms, and can also b usd to hlp dvlop lctomagntic wav basd oms o mdical imaging. Figus 3a and 3b shows an xampl o a modl mad o simulating scatting om a human bain. Appopiat choics o conductivity and lativ pmittivity w mad o bain matial, and thn a modl was mshd o us with th lctomagntics lab s VIE cod. Figu 4a: Modl o a Human Bain o Scatting Simulation in MoM [9] Figu 4b: Computd Elctic Fild (nomalizd) insid th Na Fild Plan at 9MHz

15 15 b.) Militay In militay applications, it is otn dsiabl to hav aicat which a had to dtct using standad ada tchniqus. Fo th ason, it could b ncssay to simulat th ada coss sction o an aicat. Rath than attmpting to build small modls in od to analyz scatting pattns, it is much chap, simpl, and quick to modl an objct on a comput and quickly comput its ctivnss at masking itsl om ada. Figu 5 shows a modl o stalth bomb, which could b usd to analyz ada scatting and dsign a mo ctiv stalth plan. c.) Communications Systms Figu 5: Stalth Bomb Modl usd o Rada Computation [1] CAD is usd commonly in th ild o antnna dsign, in which w wish to know how a paticula antnna dsign woks in accodanc with th st o an lctomagntic systm, o at a vaying quncy spctum. Any ild which quis lag amounts o data tansmission via adiation bnits om ctivly dsignd CAD tools. Figu 6 shows a mtallic cougatd hon antnna mshd o simulation with SIE cod. This is just an xampl o th multitud o antnna dvics that can b simulatd with CAD. Figu 6: Hon Antnna mshd o SIE computation [9]

16 16 IV. Tsting and Validation Aguably th most impotant phas o sotwa dvlopmnt is th tsting and validation phas. Any sotwa that is shippd to a custom is xpctd to b 1% obust and accuat. Similaly o ou sotwa, which is usd in a sach stting; i th sotwa is not woking compltly, w cannot xpct to apply it in solving majo sach poblms. Fo this ason, th sotwa must b continually tstd as it is ind. Sinc this sotwa was alady in opation pio to th bginning o this pojct, a lag potion o th tsting wok is b to simply assu that any changs w hav mad to th cod do not act th opation in a ngativ way. This coms into play whn optimizing old pats o th cod. Fo this ason, w hav a sis a bnchmak tsts, which w can continually un to nsu continual accuacy and xpctd opation o th cod. Ths bnchmaks hav known pop sults, whth om analytic solutions, sults psntd in past litatu, o compaison to commcial lctomagntic solvs. Figu 6 shows a sph mshd in WIPL-D v11, a commcial lctomagntic solv usd to simulat scatting poblms as wll as antnna adiation poblms. W typically us sotwa such as this as a good compaison point. It is impotant to hav bnchmaks o vy possibl sction o cod w chang. Fo th most pat in this pojct w a daling with scatting poblms. W must simulat multipl shaps and sizs o scatts to nsu th cod is obust /(a 2 ) [db] Mi's Sis 1st od ABC Polyn. od Unknowns N = a/ Figu 7: Sph mshd in commcial solv Figu 8: Analytic Solution o Scatting om Sph V. Conclusion and Futu Plans As o yt, w hav yt to poduc any viiabl sults as to whth o not th plannd PML implantation will b ctiv. Howv, xtnsiv litatu saching has bn pomd, and w a conidnt that th PML implmntation will b ctiv with th quation omulations discussd in chapt II.

17 17 On th oth hand, vaious oth impovmnts hav bn mad to th FEM cod including making th cod un ast and incasing th adability o th cod o dvlops, and w will continu to mak uth impovmnts in th coming smst. Additionally, documntation o th cods opation and stuctu hav bn bgun, uth incasing th as with which nw pojct mmbs can nt this othwis xtmly complicatd pojct. Continud dvlopmnt o th PML At dvlopmnt and implmntation o th basic cubical PML is compltd, it must b xhaustivly tstd using bnchmak scatting poblms. Fo dtails on tsting, s chapt IV. Paalllization In od to alistically us th nwly dvlopd cod o lag domain computational poblms, it must b ctivly paalllizd. Paalllization has bn don in th past, and w xpct to vntually mak su all additional pats o th pogam a paalllizd to maximiz computational icincy. Fo th puposs o th lctomagntics lab s sach, paalllization will allow cods to b un on th Cay XT6m high-pomanc computing systm, which gatly incass th siz and scop o poblms w can simulat. Scond Od and Conomal PMLs Th xists a omulation o th Pctly Matchd Lay which is consid scond od, in that it xhibits much btt convgnc to pop sults. Byond implmnting th simplst omulation o th PML talkd about h, w can choos mo advancd matial paamts such that ou PML will indd poduc vn mo accuat sults. Additionally, in al poblms, it is otn diicult to it a scatting objct to a cubical domain, as is quid to apply th most basic PML. Fo this ason, th conomal PML xists, which has a much mo complicatd thotical backgound than th basic cubical PML, but will it xactly to th domain o any simulation, and potntially duc th numb o unknowns quid to modl a poblm. In od to implmnt th conomal and scond od PML schms, much uth wok must b don in th coming smst, including additional litatu viws, and uth thotical divations. Applications Th PML has oth uss outsid o basic scatting and adiation poblms. Fo instanc, it can b usd to analyz wavguids o vaying dilctic composition (such as in (nc)). Th PML will allow o th gnal advancmnt o th lctomagntics lab s sach by incasing th scop o simulations w can do. At implmntation and xhaustiv tsting, w may bgin applying th PML to novl sach poblms.

18 18 Rncs [1] Ilić, M. High od hxahdal init lmnts o lctomagntic modling, Univsity o Massachustts Datmouth, May 23. [2] J.-M. Jin, W. Chw, Combining PML and ABC o th init-lmnt analysis o scatting poblms, Micowav and Optical Tchnology Ltts 12 [3] Ilić, M. M., A. Ž. Ilić, and B. M. Notaoš, "High od lag-domain FEM modling o 3-D multipot wavguid stuctus with abitay discontinuitis," IEEE Tansactions on Micowav Thoy and Tchniqus, Vol. 52, No. 6, , Jun 24. [4] Nada Skljic, Milan Ilic Analiza 3D еlktomagntskog poblma mtodom konačnih lmnata sa hijahijskim bazisnim unkcijama, Univsity o Blgad, Apil 28 [5] M. M. Ilic and B. M. Notaos, High od lag-domain hiachical FEM tchniqu o lctomagntic modling using Lgnd basis unctions on gnalizd hxahda, Elctomagntics, vol. 26, no. 7, pp , Oct. 26. [6] A. B. Manić, S. B. Manić, M. M. Ilić, and B. M. Notaoš, Lag anisotopic inhomognous high od hiachical gnalizd hxahdal init lmnts o 3-D lctomagntic modling o scatting and wavguid stuctus, Micow. Opt. Tchnol. Ltt. 54, (212) [7] Jin, J. M. and D. J. Rily, Finit Elmnt Analysis o Antnnas and Aays, John Wily & Sons, Nw Yok, 28. [8] Jin, J. M., Thoy and Computation o Elctomagntic Filds, Wily, 21. [9] Eln Chobanyan, PhD Disstation Dns Psntation, Dcmb 1, 214 [1] Gyongsang National Univsity Aospac Computational Modling Laboatoy [11] M. M. Ilic, S. V. Savic, and B. M. Notaos, Fist Od Absobing Bounday Condition in Lag-Domain Finit Elmnt Analysis o Elctomagntic Scatts, Poc. 1th Intnational Connc on Tlcommunications in Modn Satllit, Cabl and Boadcasting Svics TELSIKS 211, Octob 5-8, 211, Nis, Sbia.

19 19 Bibliogaphy [1] J.P. Bng, A pctly matchd lay o th absoption o lctomagntic wavs, J. Comput. Phys. 114 (1994), [2] W. C. Chw and W. H. Wdon, "A 3D pctly matchd mdium om modiid Maxwll's quations with sttchd coodinats," Micowav and Optical Tchnology Ltts, vol. 7, pp , Sptmb 1994.

20 2 Appndix A: Abbviations ABC: Absobing Bounday Condition CAD: Comput Aidd Dsign IEEE: Institut o Elctical and Elctonics Engins FEM: Finit Elmnt Mthod MoM: Mthod o Momnts PEC: Pct Elctic Conducto PML: Pctly Matchd Lay VIE: Volum Intgal Equation Appndix B: Budgt Sinc this pojct consists solly o litatu sach and sotwa dvlopmnt, vy littl is ndd on th ont o inancing. Aound $1 o xpns is xpctd in th sping smst to pay o psntation matials. Th only costs in dvlopmnt o this sotwa a in th licnsing o dvlopmnt and tsting sotwa, which is takn ca o by th lctomagntics lab.

21 21 Oiginal Pojct Plan Appndix C: Pojct Plan Evolution Bgin Dat End Dat Tasks Dlivabls 9/1/14 9/15/14 Pliminay Rsach Rading PML Rading FEM Rading Cuvilina Coodinat Systms 9/16/14 9/3/14 Undstanding Cod Documnt o basic nd-toknow Cod Oganization pats o cod o nw Rading FORTRAN studnts, as wll as ditd and Languag oganizd cod (Aaon, Ana). 1/1/14 11/31/14 Implmntation o cubical Rpot o cubical PML, as PML wll as inishd cod with Tsting o cubical PML woking cubical PML, and Additional cod optimization tsting sults with Rading Conomal PML compaisons to outsid 12/1/14 12/31/14 Implmntation o sphical PML Tsting o sphical PML soucs (Aaon, Ana, Sanja). Rpot on sphical PML, as wll as cod, documntation, divations and omulations (Aaon, Ana). 1/1/15 3/31/15 Implmntation o conomal Woking cod with conomal PML PML and quations/divations o conomal PML, with list o good litatu soucs (Aaon, Ana, Sanja). 3/31/15 End Tsting conomal PML Documnt with computational sults, compad to xtnal soucs (Aaon, Ana, Sanja). List o uth changs to cod, itms to impov (Aaon, Ana).

22 22 Rvisd Pojct Plan Bgin Dat End Dat Tasks Dlivabls 9/1/14 9/15/14 Pliminay Rsach Rading PML Rading FEM Rading Cuvilina Coodinat Systms 9/16/14 9/3/14 Undstanding Cod Documnt o basic nd-toknow Cod Oganization pats o cod o nw Rading FORTRAN studnts, as wll as ditd and Languag oganizd cod (Aaon, Ana). 1/1/14 12/31/14 Implmntation o cubical Rpot o cubical PML, as PML wll as inishd cod with Tsting o cubical PML woking cubical PML, and Additional cod optimization tsting sults with Rading Conomal PML compaisons to outsid soucs (Aaon, Ana, Sanja). 1/1/15 3/31/15 Implmntation o conomal Woking cod with conomal PML PML and quations/divations o conomal PML, with list o good litatu soucs (Aaon, Ana, Sanja). 3/31/15 End Tsting conomal PML Documnt with computational sults, compad to xtnal soucs (Aaon, Ana, Sanja). List o uth changs to cod, itms to impov (Aaon, Ana). As th pojct has pogssd, it was dmd that implmnting and tsting a spciically sphical PML was unncssay. Rath than attmpting to do this, sinc it would not b vy usul in th long un, w dcidd to spnd additional tim on claning and optimization o th cod. This was a vy impotant task, not to b ovlookd. In act, this task will likly b in pogss o th nti duation o th pojct. It is an ongoing pocss, not somthing that can b don ntily in on un.

23 23 ACKNOWLEDGEMENTS I would lik to thank my supviso, D. Banislav Notaos, o his invaluabl advic and assistanc in my acadmic dvlopmnt, and in gtting m statd on this pojct. Additionally, I would lik to thank my tam mmbs, Ana Manic and Sanja Manic, as wll as th oth gaduat studnts o th lctomagntics lab, Eln Chobanyan, and Nada Skljic, o thi advic and guidanc in my lanings.