Performance advantages of CPML over UPML absorbing boundary conditions in FDTD algorithm

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1 Joural of ELECTRICAL ENGINEERING, VOL 68 (17, NO1, Performace advatages of CPML over UPML absorbig boudary coditios i FDTD algorithm Brako D. Gvozdic, Dusa Z. Djurdjevic Implemetatio of absorbig boudary coditio (ABC has a very importat role i simulatio performace ad accuracy i fiite differece time domai (FDTD method. The perfectly matched layer (PML is the most efficiet type of ABC. The aim of this paper is to give detailed isight i ad discussio of boudary coditios ad hece to simplify the choice of PML used for termiatio of computatioal domai i FDTD method. I particular, we demostrate that usig the covolutioal PML (CPML has sigificat advatages i terms of implemetatio i FDTD method ad reducig computer resources tha usig uiaxial PML (UPML. A extesive umber of umerical experimets has bee performed ad results have show that CPML is more efficiet i electromagetic waves absorptio. Numerical code is prepared, several problems are aalyzed ad relative error is calculated ad preseted. Keywords: fiite differece time domai, FDTD, perfectly matched layer, PML, covolutioal PML, CPML, uiaxial PML, UPML 1 Itroductio With the developmet of techology ad rapid icrease of computer resources, the fiite differece time domai (FDTD method became oe of the most popular umerical method i today s computatioal electromagetics (CEM. FDTD method is primarily used for atea ad microwave circuits desig, electromagetic wave ad radio propagatio simulatio ad aalysis, i photoics. There is ofte a ecessity to simulate ifiite space or spatially ubouded systems i FDTD simulatios. Implemetatio of absorbig boudary coditios (ABC at the computatioal boudaries is used i ifiite space FDTD simulatios. Oe of the most importat challeges i FDTD method is to efficietly ad accurately implemetabcsadsotosimulatetheextesioofthefdtd lattice to ifiity. The perfectly matched layer (PML [1] is well kow ABC for efficiet absorptio of electromagetic waves of arbitrary polarizatio, agle of icidece ad frequecy. PML had proved efficiecy for homogeeous, ihomogeeous, liear, oliear, dispersive ad aisotropic domais. PML defied i [1] is based o o-physical field splittig of Maxwell s equatios which produce a sigificat amout of discretizatio error i discrete FDTD lattice. PML with the uiaxial aisotropic medium based o electric ad magetic permittivity tesors is proposed i [] ad implemeted i [3]. Uiaxial PML (UPML [] has the same efficiecy as the split-field PML [4,5], while the discretizatio error is decreased. After the validatio of this cocept [6,7], may modificatios of PML were proposed [8, 9]. Stretched coordiate (SC formulatio of Maxwell s equatio exteded the use of the PML ito other orthogoal coordiate systems [1,11] ad ito geeral curviliear coordiate systems [1, 13], but it had weak causality. The complex frequecy shifted (CFS tesor coefficiets used for PML parameters gaied the causality of PML [14,15]. Very effective implemetatio of PML based o SC, CFS ad recursive covolutio techique [16] is derived i [17]. Obtaied covolutioal PML (CPML is etirely idepedet of the host medium ad without the eed of ay modificatios whe applied i ihomogeeous, lossless, lossy, dispersive, oliear ad aisotropic media, CPML is superior to the other PMLs. Improved CPMLs were recetly derived i [15, 18, 19]. I [] comprehesive study about the choice of PML i fiite differece frequecy domai (FDFD ad i fiite-elemet method (FEM, is preseted. I this paper, umerical experimets are performed to ivestigate absorptio of electromagetic waves ad implemetatio of UMPL ad CPML i FDTD method. I particular, 3D FDTD simulatio of a differetiated Gaussia pulse propagatig i free space is used for compariso of absorptio for UPML ad CPML ABC. Additioally, PML absorptio for electromagetic scatterig from dipole atea geeratig electromagetic wave ad PEC sphere is calculated i order to simulate complex wave propagatio ad scatterig i 3D FDTD domai. Relative error for electric field is calculated for both case studies ad with differet thickesses. Implemetatio advatages i favor of CPML are show i Sec.. Numerical results preseted i Sec. 3 demostrate that absorptio characteristics of CPML are three orders of magitude better tha of UPML. Faculty of Techical Scieces, Keza Milosa 7, Kosovska Mitrovica, 38, Serbia, brako.gvozdic@pr.ac.rs DOI: /jee-17-6, Prit (till 15 ISSN , O-lie ISSN X c 17 FEI STU

2 48 B. D. Gvozdic, D. Z. Djurdjevic: PERFORMANCE ADVANTAGES OF CPML OVER UPML ABSORBING BOUNDARY... UPML ad CPML Implemetatio i FDTD After Bereger s pioeerig work of split-field PML[1], usplit form with SC formulatio of Maxwell s equatio is proposed i [1], ad idepedetly i [11]. SC formulatio eabled mappig of Maxwell s equatio ito complex coordiate space. Assumig that the PML parameters s w = 1+σ w /jωε are cotiuous alog its correspodig axis (w = x,y,z, σ w coductivity, ε permittivity, stretched coordiate space derivatives are defied as [1] x = 1 s x x, ȳ = 1 s y y, z = 1 s z z. (1 Stretched coordiates i the complex form of Ampere s law i free space therefore are ( 1 jωε E = ˆx s y y H z 1 s z z H y ( 1 ŷ s z z H x 1 ( 1 s x x H z +ẑ ad after time domai coversio t ( ŷ + ( ε E = ˆx( s y y H z s z z H y + s z z H x s x x H z s x x H y 1 s y y H x, ( ( +ẑ s x x H y s y y H x (3 where represets covolutio as a cosequece of frequecy depedece of SC metrics ad s w is the iverse Fourier trasform of sw 1. Neither split-field PML or SC PML are physical medium. A aisotropic, physical model composed of electric ad magetic permittivity tesors is formulated i [] ad [3] ad it is referred as UPML. Thus, the geeral form of UPML implemetatio i Ampere s law i free space is jωε s ys z s x s xs z s y s xs y s z E = H (4 where s x,y,z = k x,y,z +σ x,y,z /jωε are tesor coefficiets for geeral media ad k x,y,z 1 is real stretchig coefficiet cotributig to a effective scalig of the mesh i the PML regio. The split-field PML ad UPML have the same reflectio properties ad propagatio characteristics [4]. However, both are ot efficiet i absorbig evaescet waves ad ca cause large reflectios at low frequecies due to the weak causality of PML [4,5]. A causal form of the PML is proposed ad derived i [14], based o shiftig the pole of s w ito the upper-half of complex plae. Complex frequecy shifted (CFS tesor coefficiets from [14] are s w = k w + σ w α w +jωε. (5 I (5 α w is complex frequecy shift parameter, with a property of homogeous coductivity. To implemet CFS i time domai, oe requires Fourrier trasform of s 1 w s w (t = F 1( 1 k w + σw α w+jωε = δ(t σ ( w k w ε kw e σw ε + αw k w ε h(t = δ(t +η w (t (6 k w where δ(t is the uit impulse fuctio, ad h(t is the uit step fuctio. Isertig (6 ito (3 yields time domai expressio t (ε E = ( 1 ˆx k y y H z 1 k z z H y +η y y H z η z z H y ( 1 +ŷ k z z H x 1 k x x H z +η z z H x η x x H z ( 1 +ẑ k x x H y 1 k y y H x +η x x H y η y y H x. (7 Improper implemetatio of covolutio pairs o the right-had side of (7 i computer algorithm leads to the usage of a huge amout of computer resources. Approaches to resolvig this situatio use the recursive covolutio (RC [17]. The discrete impulse respose of η w ad recursive covolutio relatio gives with ψ w,v ( = b w ψ w,v ( 1+c w w H v( (8 ( t b w = e σw ε kw +αw ε, σ w c w = [ b w 1 ]. k w (σ w +α w k w (9 I (9 coefficiets are ozero oly i PML regio ad computed alog with scaled tesor parameters σ w, α w ad k w ( = i; w = x,y,z. By implemetig this form of Ψ w,v ( good efficiecy of time advacemet i FDTD algorithm is achieved. SC, CFS ad RC implemeted asi [17] resultsfdtd domai with CPML ABCs. FDTD time ad space discretizatio of Ampere s law with CPML yields explicit update of E x expressed as 1= C a 1 E 1 x 1+C b 1 E x + 1 H z 1,j+ 1,k Hz 1,j 1,k k yj y ψ Ex,y 1 ψ E x,z H y 1+1 H y k zk z. (1

3 Joural of ELECTRICAL ENGINEERING 68 (17, NO1 49 I (1 ψ Ex,y, ψ Ex,z are PML coefficiets existig oly i PML regio, updated as follows ψ Ex,y 1= b y j ψ Ex,y 1 1 ( Hz +c 1,j+1,k H z yj y 1,j 1,k, (11 ψ Ex,z 1= b z k ψ Ex,z 1 1+ c zk H y 1 H y (1 z Coefficiets C a ad C b areused forupdate of E x field ad they are calculated as 1 C a 1 = C b 1 = σ 1 t ε 1 1+ σ 1 t ε 1 t ε 1 1+ σ 1 t ε 1,. (13 Similar expressios are derived for five remaiig field compoets (E y, E z, H x, H y ad H z for 3D FDTD domai, with the adequate replacemet of (i ad (x,y,z. Efficiecy of CPML is maily depedet o the proper choice of parameters. Parameters ca be spatially graded i differet ways, but two the most successful are polyomial ad geometric gradig. I this paper, the polyomial gradig is used. PML parameters are scaled as follows [3, 1] k w (l = 1+ ( k w,max 1 ( l m, (14 d σ w (l = σ w,max ( l d m, (15 ( d l ma, α w (l = α w,max l d (16 d where l is PML loss depth, d is PML thickess, m ad m a are the scalig orders. Coductivity σ w is scaled to be at the PML surface(l = ad σ w,max at the PM outer boudary (l = d. Stretchig coefficiet k w is 1 at the begiig of PML ad k w,max at the ed of PML. Complex frequecy shift parameter α w has a maximum at the frot of PML, thereby decreasig reflectio error of evaescet modes. Iside the PML, α w is decreased to a miimum i order to appropriately decay low frequecies of the wave propagatig [17]. The proper choice of PML parameters is decisive for PML efficiecy. Trade-off betwee reflectio error from the PML outer boudary ad discretizatio error from the frot PML iterface have to be properly balaced. If σ w,max is too small, reflectio error from the back of PML is domiat, while for large σ w,max discretizatio error is sigificatly icreased. I [17] optimal choice for polyomial graded σ w,max is proposed, derived for geeral media as σ w,opt =.8(m+1 Z w εr,eff µ r,eff, (17 where Z isimpedaceoffreespace, w isspatialstepi w = x,y,z directio, ε r,eff ad µ reff are effective relative permittivity ad permeability, respectively. Optimal CPML parameters are [8,17]:.75σ w,opt < σ w,max < 1.4σ w,opt, 7 < k w,max <, ad.15 < α w <.3. Scalig orders are i rages: 3 m 4, ad m a 1. It ca be see that CPML is simpler to implemet resultig also i more storage-efficiet algorithm tha UPML implemetatio. I particular, UPML is quite simple to implemet i existig FDTD codes, but with the cost of doublig memory requiremets through etire FDTD domai. Usage of triple-ested loops for the fields iside the computatioal domai, ad idividual loops i UPML regio saves the memory, but it icreases the complexity of programmig. IthecaseofCPMLimplemetatioiFDTD,CPML variables are stored oly i PML regio, hece the superior memory efficiecy over the UPML. Furthermore, CPML implemetatio remais uchaged i the case of homogeeous, ihomogeeous, lossy ad dispersive medium. O the cotrary, UPML requires additioal two variables per field compoet i all those mediums. I order to estimate the advatages of CPML over the UPML i FDTD, simulatios are performed ad umerical results are obtaied for two differet electromagetic problems. The first problem is 3D FDTD simulatio of electromagetic wave propagatio i free space with a differetiated Gaussia pulse as a source. The secod problem is 3D FDTD electromagetic scatterig from PEC sphere ad dipole atea cetered i the computatioal domai as a example of complex FDTD case. For both cases, relative error at two probe poits is calculated comparig absorptio for two PMLs. The explicit FDTD algorithm is used ad calculated with origial C++ codes. Numerical results of the electromagetic field ad relative error graphs are plotted with the commad-lie drive Guplot graphig utility. 3 Numerical Results ad Discussios 3D Simulatio of Gaussia Pulse i Free Space Propagatio of differetiated Gaussia pulse i free space i 3D FDTD domai is simulated i space lattice, with 1-mm-square cells ad time-step of dt = ps (.99 times of Courat limit. Duratio of simulatio is 6 time-steps ( s. Gaussia

4 5 B. D. Gvozdic, D. Z. Djurdjevic: PERFORMANCE ADVANTAGES OF CPML OVER UPML ABSORBING BOUNDARY Fig. 1. E z field compoet for 1 cell-thick UPML after 3 timesteps, xy plae Fig.. E z field compoet for 1 cell-thick CPML after 3 timesteps, xy plae Fig. 3. E z field compoet for 1 cell-thick UPML after 4 timesteps, xy plae Fig. 4. E z field compoet for 1 cell-thick CPML after 4 timesteps, xy plae B PML Test domai Source A Referece domai Fig. 5. Illustratio of test ad referece FDTD domai for relative error calculatio for Gaussia pulse pulse is placed i the ceter of the computatioal domai with time fuctio J (x,y,z,t = [(t t d /g w ]e [(t t d/g w], (18 where g w = 3 ps is half-width of Gaussia pulse ad t d = 4g w is a time delay. For compariso purposes, FDTD domai is termiated with 1-cell thick UPML ad CPML, with polyomial gradig defied i (14, (15 ad (16. Numerical results show i Fig. 1 preset E z field distributio for 1-cell thick UPML after 3 time steps, over the xy plae, with m = 3, σ w,max =.75σ w,opt (with σ w,opt from (17, k w,max = 15 ad α w =, yieldig the properties of UPML. Figure shows the E z field distributio for 1-cell thick CPML after 3 time steps, over the xy plae, with the same PML parameters exceptfor α w =.4 ad m a = 1,yieldigtheproperties of CPML. A sigificat amout of umerical dispersio reflectig from the computatioal domai outer boudary ca be see i Fig. 1, i compariso with results show i Fig.. Numerical results preseted i Fig. show fie absorptio of E z field compoet, without reflectig ay field compoets back to the computatioal domai. Numerical results show i Fig. 3 ad Fig. 4 preset E z field distributio for 1-cell thick UPML ad CPML after 4 time steps, over the xy plae, respectively. The icrease of umerical dispersio of E z field compoet, reflectig back from UPML ito the computatioal domai, ca be see i Fig. 3. Such umerical artifacts are itolerable i simulatios where the precise calculatio is required. Cosequetly, efficiet CPML absorptio of electromagetic wave ca be see i Fig. 4 (steady-state. I order to demostrate beefits of CPML over UPML ABCs, the relative error is calculated for electric field E at poits A ad B, as show i Fig. 5. Test domai with cell grid ad referece domai with cell grid are used for relative error calculatio,

5 Joural of ELECTRICAL ENGINEERING 68 (17, NO1 51 Relative error UPML poit A UPML poit B CPML poit A CPML poit B ca be see o UMPL graphs as a cosequece of lowfrequecy evaescet fields iteractio with PML layers. Relative error for calculated E field at poits A ad B i the case of 5-cell thick UPML ad CPML is show i Fig. 7. Observig the CPML graphs, the early time error peaks are due to discretizatio error, which slowly decay after time-steppig icrease. Nevertheless, it is show that, compared with UPML, eve the 5-cell thick CPML exhibits three orders of magitude of error reductio t (s Fig. 6. Relative error for 1 cell-thick PMLs with Gaussia pulse Relative error CPML poit B UPML poit B CPML poit A UPML poit A t (s Fig. 7. Relative error for 5 cell-thick PMLs with Gaussia pulse with: R E i E ref i = E refmax i i. (19 I (19, E is electric field at probe poit ad i time-step i test domai, E ref is electric field at i probe poit ad time-step i referece domai ad E refmax is the maximum amplitude of the referece field at probe poit over the time-steppig rage i of iterest. Referece domai is kept sufficietly large to avoid reflectio from the walls of FDTD domai durig 1 time-steps of iterest. The same source fuctio as for Gaussia pulse propagatio i free space is used, with g w = 5 ps, t d = 4g w, i test ad referece domai. Idetical source locatio (cetered i FDTD grid is used for both domais ad probe poits are at the same positio relative to the source. Poit A (,, ad poit B (38,,38 i test domai correspod to poit A (18,18,18 ad poit B (18,,18 i referece domai. Relative error for 1-cell thick ad 5-cell thick differet PMLs are obtaied, with the same parameters like i the case for umerical results. The relative error for the calculated E field at two probe poits for 1-cell thick UPML ad CPML is plotted i Fig. 6. Comparig UPML ad CPML graphs it is clearly visible that CPML provides error reductio for more tha three orders of magitude o a logarithmic scale. Late time reflectio error with very slow decay 3D simulatio of dipole atea with PEC sphere The electromagetic wave scatterig from PEC sphere i 3D FDTD domai is simulated i space lattice, with 1-mm-square cells ad time-step of dt = ps (.99 times of Courat limit. The source of the electromagetic wave is dipole atea placed i the ceter of FDTD computatioal domai. PEC sphere is made from alumium ad it is placed like i Fig. 1. Simulatio time was 1 time-steps. The z-directed dipole atea is drive with differetiated Gaussia pulse as source fuctio with a time sigature of (15 ad g w = 3 ps, t d = 4g w. FDTD domai is surrouded with 1-cell thick PML ABCs with polyomial gradig defied i (14, (15 ad (16. Numerical results preseted i Fig. 8 show E z field compoet over the xy plae, at 5 th time-step for 1- cell thick UPML, with m = 3, σ w,max =.75σ w,opt (with σ w,opt from (17, k w,max = 15 ad α w =, yieldig the properties of UPML. E z field after 5 time steps for 1- cell thick CPML, over the xy plae, with the same PML parameters except for α w =.4 ad m a = 1, yieldig the properties of CPML, is show i Fig. 9. Results preseted i Fig. 8 show that UPML layers reflect icidet field compoets as well as scatterig field compoets from PEC sphere, hece completely udermiig the iterpretatio of the umerical results. Results preseted i Fig. 9 cotrary show that CPML layers liearly absorb all impigig field compoets. E z field compoet is give i Fig. 1 i the case of UPML ABCs, over the xy plae, close to the ed of the simulatio, after 7 time-steps. The plot clearly idicates that the reflected field compoets are propagated back to the dipole, cetered i the computatioal domai, cofrotig the late time icidet field compoets from thesource.ifig.11,itisshowthatafter7time-steps i the case of CPML, oly late time icidet field compoets are visible close to the dipole, slowly approachig to steady state. Results idicate that CPML permits much smaller FDTD space lattice to be employed while retaiig accuracy. Relative error i the case of electromagetic scatterig with dipole ad sphere is calculated as well, by usig the relatio (19. Test domai with dimesios: cells ad referece domai with FDTD lattice are used. Probe poit A is at the same positio as i the previous example, while poit B is at (38,, i

6 5 B. D. Gvozdic, D. Z. Djurdjevic: PERFORMANCE ADVANTAGES OF CPML OVER UPML ABSORBING BOUNDARY Fig. 8. Ez field compoet for 1 cell-thick UPML after 5 timesteps, xy plae 3 Fig. 9. Ez field compoet for 1 cell-thick CPML after 5 timesteps, xy plae Fig. 1. Ez field compoet for 1 cell-thick UPML after 7 time-steps, xy pla Referece domai Test domai 1 3 Fig. 11. Ez field compoet for 1 cell-thick CPML after 7 time-steps, xy plae 1 Relative error UPML poit A B Source 1-3 PEC 1 UPML poit B -4 CPML poit A A 1-5 PML 1-6 Fig. 1. Illustratio of test ad referece FDTD domai for relative error calculatio for dipole ad sphere CPML poit B t (s. Fig. 13. Relative error for 1 cell-thick PMLs with dipole ad sphere a test domai ad (18, 18, i referece domai, as illustrated i Fig Coclusios Simulatio parameters, source parameters ad UPML ad CPML parameters are the same as for dipole ad sphere umerical results. The error is calculated for 1cell thick PMLs ad show i Fig. 13. Two CPML graphs o Fig. 13 illustrate superior absorptio properties of CPML over the UPML i complex FDTD case. I this paper, we compared two mai algorithms of PML ABC used today i FDTD method: UPML ad CPML. Basic theory ad algorithm steps cocerig UPML ad CPML i FDTD method are briefly explaied ad discussed. Extesive umerical FDTD simulatios were performed by usig the origial computer

7 Joural of ELECTRICAL ENGINEERING 68 (17, NO1 53 code. Preseted umerical results clearly demostrate advatages of CPML absorptio of electromagetic waves over UPML i FDTD algorithm. The mai advatages of CPML boudary coditios are highlighted: they are much simpler to implemet i FDTD code ad more computatioally efficiet tha UPML. Refereces [1] J. P. Bereger, A Perfectly Matched Layer for the Absorptio of Electromagetic Waves, Joural of Computatioal Physics, vol.114, pp.185, [] Z. S. Sacks, D. M. Kigslad, R. Lee ad J. F. Lee, A Perfectly Matched Aisotropic Absorber for Use as a Absorbig Boudary Coditio, IEEE Tras. Ateas Propagat., vol. 43, , [3] S. D. Gedey, A Aisotropic Perfectly Matched Layer Absorbig Media for the Trucatio of FDTD Lattices, IEEE Tras. Ateas Propagat., vol. 44, , [4] J. P. Bereger, Numerical Reflectio from FDTD PMLs: A Compariso of the Split PML with the Usplit ad CFS PMLs, IEEE Tras. Ateas Propagat., vol. 5, 58 65,. [5] D. Correia ad J. M. Ji, Performace of Regular PML, CFS-PML, ad Secod-Order PML for Waveguide Problems, Microwave ad Optical Techology Letters, vol. 48, 11 16, 6. [6] D. S. Katz, E. T. Thiele ad A. Taflove, Validatio ad Extesio to Three-Dimesios of the Bereger PML Absorbig Boudary Coditio for FDTD Meshes, IEEE Microwave Guided Wave Lett., vol. 4, 68 7, [7] J. Demoerloose ad M. A. Stuchley, Reflectio Aalysis of PML ABC s for Low-Frequecy Applicatios, IEEE Microwave Gui -ded Wave Lett., vol. 6, , [8] J. P. Bereger, Improved PML for the FDTD Solutio of Wave-Structure Iteractio Problems, IEEE Tras. Ateas Propagat., vol. 45, , [9] D. M. Sulliva, A Simplified PML for Use with the FDTD Method, IEEE Microwave Guided Wave Lett., vol. 6, 97 99, [1] W. C. Chew ad W. H. Weedo, A 3D Perfectly Matched Medium from Modified Maxwell s Equatios with Stretched Coordiates, IEEE Microwave Guided Wave Lett., vol. 7, , [11] C. M. Rappaport, Perfectly Matched Absorbig Boudary Coditios based o Aisotropic Lossy Mappig of Space, IEEE Microwave Guided Wave Lett., vol. 5, 9 9, [1] F. L. Teixeira ad w. C. Chew, PML-FDTD i Cylidrical ad Spherical Coordiates, IEEE Microwave Guided Wave Lett., vol. 7, 85 87, [13] F. L. Teixeira, K. P. Hwag, W. C. Chew ad J. M. Ji, Coformal PML-FDTD Schemes for Electromagetic Field Simulatios: A Dyamic Stability Study, IEEE Tras. Ateas Propagat., vol. 49, 9 97, 1. [14] M. Kuzuoglu ad, R. Mitra, Frequecy Depedece of the Costitutive Parameters of Causal Perfectly Matched Aisotropic Absorbers, IEEE Microwave Guided Wave Lett., vol. 6, , [15] J. P. Bereger, A Optimized CFS-PML for Wave-Structure Iteractio Problems, IEEE Trasactios o Electromagetic Compatibility, vol. 54, , 1. [16] R. J. Luebbers ad F. Husberger, FDTD for Nth-Order Dispersive Media, IEEE Tras. Ateas Propagat., vol. 4, , 199. [17] J. A. Rode ad S. D. Gedey, Covolutioal PML (CPML: A Efficiet FDTD Implemetatio of the CFS-PML for Arbitrary Media, Microwave Optical Tech. Lett., vol. 7, ,. [18] I. Giaakis ad A. Giaopoulos, Time-Sychroized Covolutioal Perfectly Matched Layer for Improved Absorbig Performace i FDTD, IEEE Ateas ad Wireless Propagatio Letters, vol. 14, , 15. [19] Z. H. Li ad Q. H. Huag, Applicatio of the Complex Frequecy Shifed Perfectly Matched Layer Absorbig Boudary Coditios i Trasiet Electromagetic Method Modellig, Chiese J. Geophys, vol. 57, , 14. [] W. Shi ad S. Fa, Choice of the Perfectly Matched Layer Boudary Coditio for Frequecy-Domai Maxwell s Equatios Solvers, Joural of Computatioal Physics, vol. 31.8, , 1. [1] A. Taflove ad S. C. Hagess,, Computatioal electrodyamics: The Fiite-Differece Time-Domai Method, 3rd ed., Norwood, USA, Artech House, 5. Received 9 November 16 Brako Gvozdic was bor i Kosovska Mitrovica, Serbia, i He received BSc ad MSc degrees i electrical ad computer egieerig, i 8 ad 1, from the Faculty of Techical Scieces i Kosovska Mitrovica, Uiversity of Pristia, Serbia. He is PhD cadidate i the Faculty of Techical Scieces i Kosovska Mitrovica, Uiversity of Pristia, Serbia. I 1 he became a teachig assistat at the Faculty of Techical Scieces at the Uiversity of Pristia. Areas of research iclude fiite differece time domai method, modelig of electromagetics, photoics desig. Dusa Djurdjevic was bor i Pristia, Serbia, i 196. He received BE degree i electrical egieerig from the Uiversity of Pristia i 1983, MS degree from the Uiversity of Nis i 199, ad PhD degree (Doctorate i Techical Scieces from the Uiversity of Belgrade i From to 4 he joied the School of Electrical Egieerig at the Nottigham Uiversity, Uited Kigdom, as a research associate. I 4 ad 5 he joied the Dept. of Electrical ad Computer Egieerig at the Uiversity of Massachusetts, Dartmouth, USA, as a visitig scholar. I 1997 he became a Lecturer ad i 11 associate professor, teachig Electromagetics, Theory of electric circuits, Ateas ad propagatios ad Microwaves. His research iterests iclude aalytical ad umerical modelig of electromagetics i atea ad photoics desig.