Generalized Alternating-Direction Implicit Finite-Difference Time-Domain Method in Curvilinear Coordinate System*

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J. Electromagetic Aalsis & Applicatios, 00, : 34-33 doi:0.436/emaa.00.504 Published Olie Ma 00 (http://www.scirp.

1 J. Electromagetic Aalsis & Applicatios, 00, : doi:0.436/emaa Published Olie Ma 00 ( Geeralied Alteratig-Directio Implicit Fiite-Differece Time-Domai Method i Curviliear Coordiate Sstem* Wei Sog, Yag Hao Ceter for Electromagetic Simulatio, School of Iformatio Sciece ad Techolog, Beiig Istitute of Techolog, Beiig, Chia; Departmet of Electroic Egieerig, Quee Mar, Uiversit of Lodo, Lodo, Uited Kigdom. wsog@bit.edu.c, ag.hao@elec.qmul.ac.uk Received Februar st, 00; revised March 6 th, 00; accepted April 3 rd, 00. ABSTRACT I this paper, a ovel approach is itroduced towards a efficiet Fiite-Differece Time-Domai (FDTD) algorithm b icorporatig the Alteratig Directio Implicit (ADI) techique to the Noogoal FDTD (NFDTD) method. This scheme ca be regarded as a etesio of the covetioal ADI-FDTD scheme ito a geeralied curviliear coordiate sstem. The improvemet o accurac ad the umerical efficiec of the ADI-NFDTD over the covetioal oogoal ad the ADI-FDTD algorithms is carried out b umerical eperimets. The applicatio i the modellig of the Electromagetic Badgap (EBG) structure has further demostrated the advatage of the proposed method. Kewords: Alteratig Directio Implicit Techique, Numerical Istabilit, Noogoal FDTD. Itroductio The Fiite-Differece Time-Domai (FDTD) Method [] has bee prove to be a effective algorithm i computatioal electromagetics. The first FDTD algorithm was itroduced b Yee [] i 966. Sice the, etesios of this method [3-8] have bee made cotiuousl. The oogoal FDTD (NFDTD) method [9-] is a stadard geeralied FDTD algorithm based o the curviliear coordiate sstem. As far as a curved geometr is cocered, the NFDTD algorithm has demostrated improved efficiec over the covetioal Yee s algorithm [], due to the fact that cosiderabl fewer cells are eeded i the former b usig coformal meshes istead of emploig staircase approimatio. As is the case i the Cartesia FDTD algorithm, the time iterval ma used i the NFDTD method is costraied b the Courat-Friedrich-Lev (CFL) stabilit coditio [3]. Cosider a two-dimesioal NFDTD cell show i Figure, ξ i ad η deote the skewed coordiates, ad θ i (i ad are cell idices) deotes the rotatig agle from ξ i to η. The Courat criterio for this particular cell (i, ) is doated as mai,. The time icremet *This work is supported b the NSFC (Chia) uder Grat ad the 973 Program (Chia) uder Grat 005 CB370. used i the NFDTD simulatio should ot eceed the Courat criterio ma give b (), which is the miimum value of mai,, i order to guaratee that the CFL coditio is satisfied for ever cell i the NFDTD meshes. The ogoal CFL coditio is a special case whe θ = 90º. For ogoal meshes, it is straightforward to calculate ma. However, with oogoal meshes, it is ot trivial to fid the miimum value of mai, as diverse cells eist. More to the poit, whe the meshes cotai globall large but locall ver small or skewed cells, the usig of the smallest mai, ma result i low efficiec i the NFDTD simulatios. sii, ma mi ma i, mi c i I additio, it has bee prove that the NFDTD approach suffers late-time umerical istabilities [3-5]. Various efforts have bee made to alleviate this problem [6-8]. The alteratig directio implicit (ADI) techique has bee successfull applied to the ogoal FDTD istead of the Yee s leapfroggig scheme, resultig i the () Copright 00 SciRes.

2 Geeralied Alteratig-Directio Implicit Fiite-Differece Time-Domai Method i Curviliear Coordiate Sstem 35 Figure. A two-dimesioal oogoal FDTD cell removal of the CFL stabilit coditio ad a improvemet i the efficiec of FDTD method. I [9], the ADI priciple as applied i [6] ad [7] is eteded to the curviliear coordiates based o Lee s NFDTD scheme ad a ovel NFDTD method that is free of the CFL stabilit coditio is briefl itroduced. Similar idea was proposed based o Hollad s NFDTD formulatio i [0]. I this paper, the formulatio of the geeralied ADI- NFDTD as i [9] is preseted i details ad its umerical efficiec ad accurac is further studied. Compared to the covetioal NFDTD method, the umerical efficiec ca be improved b usig a icreased, ad the late time istabilit of the NFDTD scheme has bee greatl reduced. Compared with the covetioal ogoal ADI-FDTD, this geeralied scheme is ot ucoditioall stable. However, it shows improved umerical efficiec i modellig curved structures b requirig less computer memor ad computer ru time, because coarser coformal grids are emploed. I order to demostrate its efficiec, the proposed ADI-NFDTD is applied i calculatig the badgap diagram of the Electromagetic Badgap structure (EBGs).. Formulatio The Mawell curl equatios ca be writte as the followig for a source-free space: D H () t H E (3) t where μ is the medium permeabilit. Both of the equatios ca be casted ito partial differetial equatios i the curviliear coordiates as show i [5]. A -D TE wave module is used to illustrate the scheme. The formulatio for the TM module ca be obtaied i a similar wa. Two procedures are used to solve the differetial equatios. The first procedure is based o (4)-(6), ad the secod procedure is based o (7)-(9). ) The First Procedure: D i, D i, g i, (4) H i, H i, D i, D i, gi, (5) H, i H i, H i, H i, i, gi (6) E i, E i, E i, E i, ) The Secod Procedure: D i, D i, g i, (7) H i, H i, D i, D i, gi, (8) H, i H i, H i, H i, E i, (9) i, gi, E i, E i, E i, where E, E, H are the covariat electric ad magetic field compoets which represet the flow of field alog the grid, while D, D ad H are the cotravariat electric flues ad magetic field compoet which represet the flow goig through the facets of the grid. μ is the medium permeabilit for calculatig H ad g is the metric tesor computed from the local curviliear coordiates. The covariat field compoets (E ad E ) must be calculated from the cotravariat flu (D ad D ) b iterpolatig equatios as itroduced i the covetioal NFDTD scheme [9-0]. As the covariat field compo- Copright 00 SciRes.

3 36 Geeralied Alteratig-Directio Implicit Fiite-Differece Time-Domai Method i Curviliear Coordiate Sstem ets should be valued i the same positio with their cotravariat pairs, the eighbourig averagig proectio scheme is emploed. For a -D model, g ad g are eros, ad g equals to uit for ever cell. Hece we obtai (0)-(4). g i, D, i, g i E i, i, 4 i, D i, D i, D i, D i, (0) g,, i D i g i, E i, i, 4 i, D i, D i, D i, D i, () g i, D, i g i, E i, i, 4 i, D i, D i, D i, D i, () H i, H i, (3) H i, H i, (4) where ε i (i =, ) is the material permittivit i the - or - directio. B substitutig (0), () ad (3) ito (6) ad the resultig epressio for H ito (5), oe ca obtai (5). From (5) the cell idices (i,) are used i the equatios for a D i, is used istead of eat presetatio, e.g., D i,. With D readil obtaied b (4), (5) is a tri-diagoal sstem of equatio that ca be easil solved. Thus D is updated. The H ca be calculated b (6). The field compoets i the secod procedure ca be updated i a similar wa, b the use of (), (), (4) ad (7)-(9). The proposed ADI-NFDTD algorithm is a etesio of the ADI-FDTD i the curviliear coordiate sstem. As a result, if the grid is uiform ad ogoal, this algorithm will reduce to the covetioal ogoal ADI-FDTD, as illustrated i (6)-(). We appl the followig relatios to the derived ADI- NFDTD equatios: D i, g ( i, ) E i, E i, E i, g (, i ) H i, H i, where E i,, E i, ad, i, (6) H i deote g i,, g i D i, 4 i, i, g( i, ) g( i, ) 4 i, i, g( i, ) g( i, ) g i, g i, (, ) D i D ( i, ) 4 i, i, g( i, ) 4 i, i, g( i, ),,,, E i E i E i E i D i, H i, H i, - g i, 4 gi (, ) i, gi (, ) i, gi (, ) a i ; g, b i g, i D a, b 6 i, g( i, ) i, ai; i, b ai; b a i ; b a, b ai; ai; g, b, b i g i D a, b D a, b 6, (, ) (, ) i, a i ; i, i g i g i a i ; b b (5) D Copright 00 SciRes.

4 Geeralied Alteratig-Directio Implicit Fiite-Differece Time-Domai Method i Curviliear Coordiate Sstem 37 the correspodig electric ad magetic field compoets i the Cartesia coordiates respectivel. The g tesors have the relatioship of (7) uder -D ogoal meshes. g(, i ) g (, i ) g (, i ) g (, i ) 0 (7) g (, i ) d(, i ) g (, i ) d(, i ) Substitutig (6)-(7) ito (5) ields (8), which is the ADI-FDTD updatig equatio uder arbitrar ogoal meshes. Give that the FDTD grid is uiform, ad deotig the spatial icremets alog the - ad - aes as Δ ad Δ, respectivel, (9) are ielded from (7)., g i,, g i,,ad g i (9) The substitutig (9) ito (8) ad multiplig 4 i, o both side of the equatio ields (0). Equatio (0) is the geeral equatio for ogoal ADI-FDTD scheme with uiform grid. Equatios (8) ad (0) ca treat homogeeous or ihomogeeous, isotropic or aisotropic media. I a medium where μ (i, ) = μ (i-, ), (0) further reduces to the formula give i [6]. 3. Numerical Results 3. Compariso of the Late Time Istabilit ad the Numerical Accurac of the ADI-NFDTD Algorithm with the Covetioal NFDTD Algorithm The resoat modes of a clidrical perfect electric coductor (PEC) cavit resoator are calculated b usig both the covetioal NFDTD ad the proposed ADI- NFDTD schemes. The radius of the resoator is 0.5m ad the structure is ifiitel log. The cavit is filled with vacuum of permittivit ε r =. The outer material is PEC, ad the whole computatioal regio is 0.56 m 0.56 m meshed b 5 6 cells. A sie modulated cosie pulse () is ecited iside the cavit to provide a wide bad ecitatio to ecite all the possible TE modes: m si cos Source A f f () where A m relates to the amplitude of the sigal, f is the frequec parameter, is the time icremet ad is the iteratio ide. The temporal sigatures of the magetic fields iside the resoator are recorded. The first 8,000 time steps results from both the covetioal ad the ADI- NFDTD algorithms are ormalied b their ow peak values ad are plotted i Figure. More simulatios were performed usig differet values to stud the stabilit ad the accurac of the ADI- NFDTD scheme. It is show that ulike the covetioal i, d i, d i, i, 4 i, d i, d i, d i, d i, 4 i, d i, d i, d i, E i, E i, E i, 4 i, d i, d i, 4 i, d i, d i, E i, H i, H i, d i, d i, 4 d i, d i, E i, E i, E i, E i, (,) i d i, ( i,) d i, i i, 4 i, i,, E i E i E i i, i, (-,),, 4 i, i, i, E i, H i, H i, i, E i, E i, or i, E i, E i, th (8) (0) Copright 00 SciRes.

38 Geeralied Alteratig-Directio Implicit Fiite-Differece Time-Domai Method i Curviliear Coordiate Sstem (a) Figure.

Istabilit i the temporal results is observed whe small (i.e., comparable with or smaller tha the ma required i the NFDTD scheme) is used.

This is demostrated i Figure 3. The removal of the CFL stabilit coditio o i the ADI-NFDTD method is also observed.

5 38 Geeralied Alteratig-Directio Implicit Fiite-Differece Time-Domai Method i Curviliear Coordiate Sstem (a) Figure. The ormalied H field results of the first 8000 time steps from the two schemes ADI-FDTD, this geeralied ADI-NFDTD scheme is ot a ucoditioall stable scheme. Istabilit i the temporal results is observed whe small (i.e., comparable with or smaller tha the ma required i the NFDTD scheme) is used. However, compared to the NFDTD scheme, the ustable result i the ADI-NFDTD occurs much later ad the spurious eerg grows much slower tha that i the covetioal NFDTD scheme. This is demostrated i Figure 3. The removal of the CFL stabilit coditio o i the ADI-NFDTD method is also observed. Whe reaches 6ps i the simulatios, the covetioal NFDTD caot provide reasoable results, idicatig that the CFL stabilit coditio is violated, while ADI-NFDTD scheme ca do so, eve usig a greater value. I order to compare the accurac of the ADI-NFDTD with the covetioal NFDTD algorithm, the resoat frequecies are calculated usig the Fast Fourier Trasformatio (FFT) ad are compared with the aaltical results. Sice both the NFDTD ad the ADI-NFDTD methods suffer umerical istabilit i late time steps, the temporal data are trucated to 40,000 ad 00,000 time steps respectivel. Such umbers are chose i order to obtai the best frequec resolutios, ad avoid too much spurious eerg i the spectra at the same time. Sice = ps, the frequec resolutios of the covetioal ad the ADI-NFDTD results are 0.05 GH ad 0.0 GH respectivel. The calculated resoat frequec spectra are plotted i Figure 4. As ca be see, it is possible to distiguish two closelspaced frequec spectra from the results obtaied b usig the ADI-NFDTD, while it is hard to do so i the covetioal NFDTD due to the limited frequec resolutio. Moreover, the oise level of the ADI-NFDTD is lower tha that of the covetioal NFDTD, if used i the two simulatios are comparable. (b) Figure 3. H field temporal results of (a) the covetioal NFDTD scheme, ad (b) the ADI-NFDTD scheme, with time iterval = ps Figure 4. Resoat frequec spectra of the cavit resoator calculated from the two schemes, with = ps. After [9] I order to compare the accurac of the two NFDTD schemes, the averaged relative error rate of the resoat frequec as a fuctio of the relative time iterval is itroduced. This error rate is calculated i the followig maer. Firstl, the absolute value of the differece of Copright 00 SciRes.

6 Geeralied Alteratig-Directio Implicit Fiite-Differece Time-Domai Method i Curviliear Coordiate Sstem 39 the calculated resoat frequec with the theoretical oe is divided b the latter, ad epressed i percetage, formig the relative error rate for each resoat mode. The, the stadard deviatio of the relative error rates of all the modes withi frequec bad of iterest (0-3 GH) are calculated as the averaged relative error rate of the umerical results. The curve of the averaged relative error rate as the fuctio of time iterval is plotted i Figure 5. I theor, the ADI-NFDTD ma ot ecessaril provide better accurac tha the covetioal NFDTD. However, to the best of the authors kowledge, the former ca alwas provide more stable temporal results tha the latter. As the FFT beig performed with a sufficiet frequec resolutio, the ADI-NFDTD shows smaller relative error rate at some values i Figure 5. The removal of the CFL coditio o NFDTD algorithm ca be see whe is greater tha 6 ps. However, accordig to the authors simulatio eperiece, should ot eceed the correspodig Courat criterio ma i order to maitai the accurac i ADI-NFDTD simulatios. B usig the same, the computatioal efficiec is decreased i the ADI-NFDTD simulatio tha i the NFDTD oe. 3. Compariso of the Numerical Efficiec ad Accurac of the ADI-NFDTD Algorithm with the ADI-FDTD Algorithm A two-dimesioal cavit resoator is modelled usig both the ADI-NFDTD ad the covetioal ogoal ADI-FDTD algorithms, i order to demostrate the umerical efficiec improvemet of the former scheme. The radius of the cavit is 0.5 m. The cavit is filled with vacuum of relative permittivit ε r =. The outer material is copper, with relative permittivit ε r = ad coductivit σ = S/m. The whole computatioal regio is 0.6 m 0.6 m, meshed b the ADI-NFDTD method usig cells, ad b the ogoal ADI-FDTD method usig 60 60, ad cells respectivel. The time step is chose to be ps ad 5,000 iteratios are ru i each simulatio. A modulated Gaussia pulse as show i () is ecited iside the cavit. The temporal resposes of probed poits iside the cavit are recorded ad Fourier Trasformed, after which the resoat frequecies is idetified as peaks i the frequec spectra. dela Source Am ep f ep () where A m is the maimum amplitude, f is the frequec parameter, is the iteratio ide, is the time icremet, dela f is the dela of the pulse i time step, ad 8f is the pulse half-duratio at the /e poit. I the simulatio, A m = ad f = 0.5 GH. Figure 5. The averaged relative error rate of the resoat frequec calculated b the covetioal NFDTD ad the proposed ADI-NFDTD schemes as the fuctio of relative time iterval /0 (0 = ps). After [9] To calculate the relative error rates (RERs), the frequecies of all the geuie modes are compared with the theoretical results [], ad the results are show i Table. Table presets a compariso of umerical errors, computer resources used i all simulatios i the same computer uder the same programmig eviromet of Matlab 6. The averaged relative error rate calculated from the geuie ad the spurious modes ad that from ol the geuie modes are deoted as ARER ad ARER-G, respectivel. Table. Compariso of the umerical accurac i terms of the geuie modes calculated b the ADI-NFDTD ad ogoal ADI-FDTD algorithms uder differet spatial resolutios Theoretical results [] (GH) Frequec of the geuie mode (GH)/RER(%) ADI-NFDTD Orthogoal ADI-FDTD / / / / / / / / /0.0.6/4.90.8/3.6.6/ /4.70.3/.78.34/0..38/ /0.64/.96.6/4.4.64/ /.76.7/0.78/4.7.7/ /.30.06/0.83./.79.0/ /0.9./.69.6/../ /.4.6/.6.8/.06./ / / /.05.36/ /.07.58/.06.5/.07.54/ / /.3.7/ / /4.3.66/.63.7/ / /.56.98/0.8.8/4.60.9/. Copright 00 SciRes.

7 330 Geeralied Alteratig-Directio Implicit Fiite-Differece Time-Domai Method i Curviliear Coordiate Sstem Table. Compariso of the accurac ad computer resources of the ADI-NFDTD ad the ogoal ADI-FDTD i calculatig the resoat modes of the copper cavit ADI-NFDTD OrthogoalADI-FDTD Mesh sie (ps) ma Frequec Resolutio(GH) Number of Spurious Modes ARER-G (%) ARER (%) Computer Memor (MB) Simulatio Ru Time (miutes) It ca be see that the ADI-NFDTD method ca provide results with less umerical error while usig less computer memor ad less simulatio ru time. This demostrates the accurac ad computig efficiec improvemet of the ADI-NFDTD algorithm over the ogoal ADI-FDTD. However, compared to the ogoal ADI-FDTD algorithm, the ADI-NFDTD method is ot a ucoditioall stable method. The late time istabilit iherited from the NFDTD scheme is a maor drawback of the ADI-NFDTD method. 3.3 Applicatio of ADI-NFDTD Algorithm i Badgap Structure Simulatios I this sectio, the ADI-NFDTD scheme is applied to model a electromagetic badgap (EBG) structure ad the simulatio results are compared with those from the NFDTD modellig. The two-dimesioal EBG structure comprises of metallic rods periodicall loaded i square lattice i free space. The uit cell approach [3] is used i this model. This EBG structure is ifiite i the - ad the -directio with a lattice costat (period) a. I the -directio, the rods are ifiitel log. Each rod is made from copper with relative permittivit ε r = ad coductivit σ = S/m. The ratio of the radius r to the lattice costat a is chose to be r/a = 0.. The uit cell, the Brilloui Zoe ad the oogoal mesh file used i both the ADI-NFDTD ad NFDTD algorithms are show i Figure 6. The dispersio diagram calculated b NFDTD is plotted i Figure 7. I the NFDTD simulatio, while calculatig the frequec spectra for vectors ear vector k = ΓM, the time domai sigals ehibit istabilit quickl. This will cause difficult i obtaiig the dispersio diagram. However, this problem ca be easil solved b usig the ADI-NFDTD algorithm. (a) (b) (c) Figure 6. (a) The uit cell of the EBGs; (b) the Brilloui Zoe i the reciprocal lattice with k = k show b a red vector; (c) the coformal mesh of the real lattice Take the eige-mode calculatio o oe wave vector k = k (Figure 6(b)) for eample. The temporal results at the same probe positio obtaied b usig the ADI- NFDTD ad the covetioal NFDTD are compared i Figure 8. As ca be see from Figure 8(b), the ADI- NFDTD result shows resemblace with the NFDTD oe at the begiig of simulatios. It is oted that, after the first 50 aosecods, the results obtaied via the ADI-NFDTD still remai stable while those from the NFDTD eperiece a epoetial growth sigalig umerical istabilit. Copright 00 SciRes.

8 Geeralied Alteratig-Directio Implicit Fiite-Differece Time-Domai Method i Curviliear Coordiate Sstem 33 Figure 7. The dispersio diagram of the square lattice EBG, TE mode results, calculated usig the NFDTD method Figure 9. The compariso of the frequec spectra i calculatig k = k usig the covetioal ad the ADI-NFDTD algorithms (a) Figure 8. (a) The compariso of the temporal resposes whe calculatig k = k ; (b) The elarged view of the first 50 aosecod sigals of (a) Cosequetl, the frequec spectra obtaied usig the ADI-NFDTD show a reduced oise level ad hece lead to more accurate solutios for idetifig eigemodes i the dispersio diagram, as show i Figure 9. For istace, the mode at a ormalied frequec of.36 (highlighted usig the datatip i Figure 7) ca be hardl detected from the NFDTD results as the NFDTD spectrum is heavil buried b the oise. However, this mode is clearl show as a distict peak i the ADI-NFDTD spectra. 4. Coclusios I this paper, the ADI-FDTD scheme is eteded ito the oogoal coordiate sstem to achieve a ADI- NFDTD algorithm. The stabilit, accurac ad efficiec of the proposed scheme are validated b the clidrical resoator simulatios. It is show that the CFL coditio of NFDTD is removed b the use of the ADI scheme. Secodl, although the ADI-NFDTD is ot ucoditioall stable, the ustable result occurs at a much later stage compared with the covetioal NFDTD algorithm, which is a sigificat improvemet i terms of late time istabilit. I this wa, the ADI-NFDTD is able to provide more precise frequec spectra with a smaller FFT resolutio ad a lower oise level tha those obtaied i the covetioal NFDTD. This is of great beefit i distiguishig resoat or eige-modes closel located i the frequec spectra. As a result, the ADI-NFDTD is quite suitable i the modellig of resoatig or periodic structures, i which the eerg is hardl dissipated, ad i which the resoat or the eige-modes are ke factors to be eamied i the stud. Without havig to emplo staircasig approimatio, the curved structures or oblique surfaces are more accuratel ad efficietl modelled usig the ADI-NFDTD scheme tha usig the ogoal ADI-FDTD. Although is ot limited b the CFL coditio ad the coformal meshes ca be coarser tha those i the ogoal FDTD, the icrease of value ad the decrease of the spatial resolutio will result i decreased accurac i the ADI-NFDTD results. Therefore, i ADI-NFDTD, the value ad the mesh profile should be carefull chose for the combied cosideratio of the efficiec ad accurac. REFERENCES [] A. Taflove, Computatioal Electrodamics: the Fiite- Copright 00 SciRes.

9 33 Geeralied Alteratig-Directio Implicit Fiite-Differece Time-Domai Method i Curviliear Coordiate Sstem Differece Time-Domai Method, d Editio, Artech House, Norwood, MA, 996. [] K. S. Yee, Numerical Solutio of Iitial Boudar Value Problems Ivolvig Mawells Equatios i Isotropic Media, IEEE Trasactios o Ateas Propagatio, Vol. 4, No. 3, Ma 966, pp [3] C. J. Railto ad J. B. Scheider, A Aaltical ad Numerical Aalsis of Several Locall Coformal FDTD Schemes, IEEE Trasactios o Microwave Theor ad Techiques, Vol. 47, No., Jauar 999, pp [4] S. S. Zivaovic, K. S. Yee ad K. K. Mei, A subgriddig Method for the Time-Domai Fiite-Differece Method to Solve Mawell s Equatios, IEEE Trasactios o Microwave Theor ad Techiques, Vol. 39, No.3, March 99, pp [5] Y. Zhao, P. Belov ad Y. Hao, Spatiall Dispersive Fiite-Differece Time-Domai Aalsis of Sub-Wavelegth Imagig b the Wire Medium Slabs, Optics Epress, Vol. 4, No., Jue 006, pp [6] F. Zheg, Z. Che ad J. Zhag, Toward the Developmet of a Three-Dimesioal Ucoditioall Stable Fiite-Differece Time-Domai Method, IEEE Trasactios o Microwave Theor ad Techiques, Vol. 48, No. 9, September 000, pp [7] T. Namiki, A New FDTD Algorithm Based o Alteratig-Directio Implicit Method, IEEE Trasactios o Microwave Theor ad Techiques, Vol. 47, No. 0, October 999, pp [8] N. V. Katartis, T. T. Zgiridis ad T. D. Tsiboukis, A Ucoditioall Stable Higher Order ADI-FDTD Techique for the Dispersioless Aalsis of Geeralied 3-D EMC Structures, IEEE Trasactios o Magetics, Vol. 40, No., March 004, pp [9] R. Hollad, Fiite-Differece solutio of Mawell s Equatios i Geeralied Noogoal Coordiates, IEEE Trasactios o Nuclear Sciece, Vol. 30, No. 6, December 983, pp [0] J. F. Lee, R. Paladech ad R. Mittra, Modelig Three-Dimesioal Discotiuities i Waveguides Usig Noogoal FDTD Algorithm, IEEE Trasactios o Microwave Theor ad Techiques, Vol. 40, No., Februar 99, pp [] A. J. Ward ad J. B. Pedr, Calculatig Photoic Grees Fuctios Usig a Noogoal Fiite-Differece Time-Domai Method, Phsical Review B, Vol. 58, No., September 998, pp [] W. Sog, Y. Hao ad C. G. Parii, Calculatig the Dispersio Diagram Usig the Noogoal FDTD Method, The Istitutio of Egieerig ad Techolog (IET) Semiar o Metamaterials for Microwave ad (Sub) Millimetrewave Applicatios: Electromagetic Badgap ad Double Negative Desigs, Structures, Devices ad Eperimetal Validatio, Lodo, September 006. [3] S. L. Ra, Numerical Dispersio ad Stabilit Characteristics of Time-Domai Methods o Noogoal Meshes, IEEE Trasactios o Ateas ad Propagatio, Vol. 4, No., Februar 993, pp [4] R. Schuhma ad T. Weilad, Stabilit of the FDTD Algorithm o Noogoal Grids Related to the Spatial Iterpolatio Scheme, IEEE Trasactios o Magetics, Vol. 34, No. 5, September 998, pp [5] M. Fusco, FDTD Algorithm i Curviliear Coordiates, IEEE Trasactios o Ateas ad Propagatio, Vol. 38, No., Jauar 990, pp [6] Y. Hao ad C. J. Railto, Aalig Electromagetic Structures with Curved Boudaries o Cartesia FDTD Meshes, IEEE Trasactios o Microwave Theor ad Techiques, Vol. 46, No., Jauar 998, pp [7] V. Douvalis, Y. Hao ad C. G. Parii, A Stable No- Orthogoal FDTD Method, Electroics Letters, Vol. 40, No. 4, Jul 004, pp [8] Y. Hao, V. Douvalis ad C. G. Parii, Reductio of Late Time Istabilities of the Fiite Differece Time Domai Method i Curviliear Coordiates, IEE Proceedigs, Part A, Sciece, Measuremet ad Techolog, Vol. 49, No. 5, September 00, pp [9] W. Sog, Y. Hao ad C. G. Parii, ADI-FDTD Algorithm i Curviliear Co-Ordiates, Electroics Letters, Vol. 4, No. 3, November 005, pp [0] H. Zheg, 3-D Noogoal ADI-FDTD Algorithm for the Full-Wave Aalsis of Microstrip Structure, IEEE Ateas ad Propagatio Societ Iteratioal Smposium 006, Albuquerque, Jul 006, pp [] W. Sog, Y. Hao ad C.G. Parii, A ADI-FDTD Algorithm i Curviliear Co-Ordiates, Asia-Pacific Microwave Coferece Proceedigs, Suhou, Vol. 3, December 005. [] Y. Hao, The Developmet ad Characteriatio of a Coformal FDTD Method for Oblique Electromagetic Structures, PhD. Thesis, Uiversit of Bristol, November 998. [3] J. D. Joaopoulos, R. D. Mead ad J. N. Wi, Photoic Crstals: Moldig the Flow of Light, Priceto, Priceto Uiversit Press, 995. Copright 00 SciRes.

U8L1: Sec Equations of Lines in R 2

U8L1: Sec Equations of Lines in R 2 MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie