EE 5303 Elecromagneic Analsis Using Finie Difference Time Domain Lecure #5 Advanced FDTD Algorihms Lecure 5 These noes ma conain coprighed maerial obained under fair use rules. Disribuion of hese maerials is sricl prohibied Slide 1 Lecure Ouline Alernaing Direcion Implici (ADI) Algorihm Pseudospecral Time Domain (PSTD) M4 Algorihm Inroducion Formulaion Performance Improvemen Lecure 5 Slide 1
Alernaing Direcion Implici Algorihm Lecure 5 Slide 3 Some Limiaions of Ordinar FDTD Recall he Couran Sabili Condiion c 1 min 1 1 1 c0 3 x 0 Problem If he cell sie is much less han he wavelengh, hen a prohibiivel large number of ieraions will be required due o he exremel small ime sep ha ensures sabili. Low frequenc bioelecromagneics Simulaion of VLSI circuis We would like o exceed he Couran limi b more han 10. How? Lecure 5 Slide 4
New Sabili Condiion In an alernaing direcion implici (ADI) algorihm, we no longer have o consider he grid resoluion. We need onl look a he ccle ime of he highes frequenc. min min N N fmax 1 0 We can ge awa wih exremel fine grid resoluion wihou having o reduce for sabili! ADI FDTD is uncondiionall sable, bu his does no mean uncondiionall accurae. Lecure 5 Slide 5 Alernaing Direcion Implici Mehod Suppose we have he following PDE: x u u u u i, i, n1 n We have so far solved his using he Crank Nicolson scheme i i i i un1 un un1 un u x,,,, Insead, we can spli his ino wo ime seps, each of duraion /. u u u u nn1 : x i, i, i, i, n1 n n1 n u u u u n1n1: x i, i, i, i, n1 n1 n1 n1 Lecure 5 Slide 6 3
Zheng/Chen/Zhang ADI Algorihm Spaial Derivaives: Fields are saggered on an ordinar Yee grid. Time Derivaives: Fields are collocaed in ime. Original Finie Difference Equaion: E 1, 1, 1, 1, 1,, 1 1,, 1 1,, 1,, i k i k i k i k i k i k x E 1 x 1 H H H n 1 n 1 H n 1 n n n1 ADI Finie Difference Equaions (now wo seps): E E H H i1,, k i1,, k i1, 1, k i1, 1, k i1,, k1 i1,, k1 x n 1 x n 1 H H n1 n1 n n i1,, k i1,, k i1, 1, k i1, 1, k i1,, k1 Ex E n 1 x n 1 1 H H n 1 H n 1 H n1 i1,, k1 n1 Never calculaed No calculaed e Lecure 5 Slide 7 Complee Se of Spli Finie Difference Equaions Subieraion #1 Subieraion # Lecure 5 Slide 8 4
Derivaion of ADI Updae Equaions (1 of ) Subieraion #1 We subsiue Eq. (4.100) ino Eq. (4.99) o eliminae he H fields a he n+1/ ime seps. We sill reain Eq. (4.100). Lecure 5 Slide 9 Derivaion of ADI Updae Equaions ( of ) Subieraion # We subsiue Eq. (4.10) ino Eq. (4.101) o eliminae he H fields a he n+1 ime seps. We sill reain Eq. (4.10). Lecure 5 Slide 10 5
ADI Finie Difference Equaions Subieraion #1 Subieraion # Noe: H field updae equaions remain unchanged. Lecure 5 Slide 11 Soluion o ADI Finie Difference Equaions (1 of ) Subieraion #1 This equaion is wrien once for each occurrence of E x a a consan posiion. This se of equaions has he form of a ridiagonal marix and is easil solved. This equaion is wrien once for each occurrence of E a a consan posiion k. This se of equaions has he form of a ridiagonal marix and is easil solved. This equaion is wrien once for each occurrence of E a a consan posiion i. This se of equaions has he form of a ridiagonal marix and is easil solved. Lecure 5 Slide 1 6
Soluion o ADI Finie Difference Equaions ( of ) This equaion is wrien once for each occurrence of E x a a consan posiion k. This se of equaions has he form of a ridiagonal marix and is easil solved. Subieraion # This equaion is wrien once for each occurrence of E a a consan posiion i. This se of equaions has he form of a ridiagonal marix and is easil solved. This equaion is wrien once for each occurrence of E a a consan posiion. This se of equaions has he form of a ridiagonal marix and is easil solved. Lecure 5 Slide 13 Noes in ADI FDTD ADI FDTD is uncondiionall sable for all so he Couran sabili condiion no longer applies. ADI FDTD has accurac issues. Dispersion error increases seadil above he Couran sabili condiion. Increasing error wih increasing. ADI FDTD no well suied for elecricall large simulaions. Bes applied o elecricall small problems requiring ver fine grids. Lecure 5 Slide 14 7
Pseudospecral Time Domain Lecure 5 Slide 15 Purpose of PSTD Numerical dispersion is a serious problem ha is paricularl severe in elecricall large simulaions. I arises due o he numerical error arising from approximaing he spaial derivaives in Maxwell s equaions. Specral accurac is achieved when he fields are represened b rigonomeric funcions or Chebshev polnomials. This means numerical dispersion decreases exponeniall wih sampling densi. Lecure 5 Slide 16 8
Opions for Approximaing Spaial Derivaives Finie Difference Approximaion dfi fi 1 fi 1 dx x Requires a minimum of 10 o 0 poins per wavelengh. Fourier (Trigonomeric) Approximaion df FFT dx Nx 1 nfft f Requires a minimum of poins per wavelengh. Chebshev Approximaion Requires a minimum of poins per wavelengh. Lecure 5 Slide 17 Achieving Specral Accurac Single Domain PSTD Inernal medium mus be coninuousl inhomogeneous. coninuousl inhomogeneous piecewise inhomogeneous Mulidomain PSTD When he inernal medium is piecewise inhomogeneous, singledomain is applied o each subdomain and hen mached a he boundaries. Lecure 5 Slide 18 9
Noes Wraparound Effec When rigonomeric funcions are used, he grid becomes inherenl periodic. This can be miigaed b using a PML a he boundaries. Gibb s Phenomenon When he field has disconinuiies, like a a boundar of an obec, a significan overshoo and ringing is inroduced in he vicini of he boundar. Lecure 5 Slide 19 M4 Algorihm Daa and diagrams in his secion were borrowed from M. F. Hadi, M. Pike Ma, A Modified FDTD (,4) Scheme for Modeling Elecricall Large Srucures wih High Phase Accurac, IEEE Trans. on An. and Prop., Vol. 45, No., pp. 54 64, 1997. Lecure 5 Slide 0 10
Wh M4? Problem excessive phase error ha accumulaes during an FDTD simulaion. Waves on a grid propagae differenl han phsical waves. Paricularl severe for large srucures. (,4) scheme means nd order differences in ime and 4 h order differences in space. (4,4) scheme means 4 h order differences in ime and 4 h order differences in space. These higher order schemes suffer from insabili and more complicaed boundar condiions. Lecure 5 Slide 1 Noaion L# 1 # L algorihm (S=sandard, M=modified) # 1 order of accurac in ime # order of accurac in space S Sandard FDTD wih nd order differences in ime and nd order differences in space. This is wha we learned his semeser. S4, S44 Improved formulaions, bu wih some problems. M4 Modified FDTD wih nd order differences in ime and 4 h order differences in space. Currenl sae of he ar. Lecure 5 Slide 11
S4 Updae Equaion (1 of ) Recall our S updae equaion for E. i, i, i, i1, i, i, 1 E H H H x H E x c 0 i, x We can wrie a similar equaion, bu wih 4 h order accurae finiedifferences. i1, i, i1, i, H 7 H 7 H H,, i i E E c 0 4 x i, i, 1 i, i, 1 i, Hx 7 Hx 7 H x H x 4 Lecure 5 Slide 3 Rearrange S4 Updae Equaion For simplici, le h = x = i, i, i1, i, i1, i, i, 1 H 7 H 7 H H E E c, 1,, 1, 0 4 h i i i i + Hx 7 Hx 7 H x H x The righ hand side can be rearranged as follows i, i, i, i, 1 i, i1, i, i, i, 1 i, i1, E 7 7 7 7 E Hx Hx H H H x Hx H H c 0 4h 4h i, i, i, E E 9 i, i, 1 i, i1, 1 i, 1 i, i1, i, H x c 0 8h Hx H H Hx Hx H H 4h,, 1 i, i1, i, i, i, E E 9 i i 1 i, 1 i, i1, i, hh 3 x hh x h H h H hh x 3hH x 3 3 c hh hh 0 8h 89h Lecure 5 Slide 4 1
S4 Conains Closed Conour Line Inegrals We recognie ha he righ hand side of our finie difference equaion has wo expressions in he form of closed conour line inegrals. E 9 1 H d H d c0 8h 89h C1,, 1 i, i1,, 1, i1, i, hh x hh x h H h H 3 3 3 3 hh x hh x hh hh i, i, i, E E 9 i i 1 i i c 0 8h 89h C Lecure 5 Slide 5 Maxwell s Equaions in Inegral Form Recall Maxwell s equaions in inegral form B E Ed Bds L S D H H d Dds L S These equaions le us calculae he line inegrals as surface inegrals. Lecure 5 Slide 6 13
Use Surface Inegral Insead of Line Inegral We calculae he line inegrals b insead calculaing he he surface inegrals over he area enclosed b each conour. H d Dds C1 S1 S1 E E ds S1 S1 E h E h E ds ds E H d ds C S E 3h E 9 h Lecure 5 Slide 7 Compile New Equaion We sar wih our S4 equaion derived wih line inegrals. E 9 1 H d H d c0 8h 89h C C1 We replace he line inegrals wih our new surface inegrals. E 9 E 1 E h 9 h c h 0 8 C 89h 1 C Now we simplif. E 9 E 1 E c 0 8 8 FDTD C 1 C We see his is us a weighed sum of wo applicaions of Ampere s circui law. The coefficiens add up o uni (-1/8 + 9/8 = 1) so ha he inegri of Maxwell s equaions is preserved. Lecure 5 Slide 8 14
Spli The Ouer Loop Here, we spli he ouer loop ino wo disinc loops. Noe, half of he erms are included in he firs ouer loop and he remaining are included in he second ouer loop. E 9 E 1 E c0 8 8 FDTD C 1 C E 9 E 1 E 1 E c 0 8 16 16 FDTD C 1 C C3 Lecure 5 Slide 9 Assign Arbirar Weighs We need more degrees of freedom in order o reduce numerical error. To do his, we assign arbirar weighs o he erms in our equaion. E E E E K K K c 3 1 0 FDTD C 1 C C3 Noe ha in order o preserver he inegri of Maxwell s equaions, we require ha K 1 + K + K 3 = 1. To enforce his, our equaion is wrien as E E E E 1K K K K c 1 1 0 FDTD C 1 C C3 Lecure 5 Slide 30 15
M4 Updae Equaion for E (1 of ) Saring wih E E E E 1K K K K c 1 1 0 FDTD C 1 C C3 Each erm on he righ is calculaed as E 1,, 1 i, i 1, i i Hx Hx H H C h 1 E 1, 1, i 1, i, i i Hx Hx H H 3 C h E 1 i1, i1, i1, 1 i1, 1 i1, 1 i1, 1 i, 1 i, 1 Hx Hx H x Hx H H H H 6 C h 3 Lecure 5 Slide 31 M4 Updae Equaion for E ( of ) So he overall updae equaion is now i, i, i, E E 1,, 1 i, i 1, K1 K i i Hx Hx H H c 0 h K 1,, 1 i, 1 i, i i H x H x H H 3h H H H K 6h H H H H i1, i1, i1, 1 i1, 1 x x x H x i1, 1 i1, 1 i, 1 i, 1 Lecure 5 Slide 3 16
Opimum Values for K 1 and K NRES ki Global Phase Error 1 ki k i k d i 0 ki ki phsical wave number k numerical wave number i angle of wave hrough grid Lecure 5 Slide 33 Global Phase Error Vs. Frequenc Lecure 5 Slide 34 17
Global Phase Error Vs. NRES NRES Lecure 5 Slide 35 Memor Requiremens NRES min Lecure 5 Slide 36 18