Finite Difference Time Domain

Transcription

1 7/1/17 Insrucor r. Raond Rupf (915) Copuaional lecroagneics (CM) Lecure #16 Finie ifference ie oain Lecure 16 hese noes a conain coprighed aerial obained under fair use rules. isribuion of hese aerials is sricl prohibied Slide 1 Ouline Inroducion o F Concep of he updae equaion ie doain UPML erivaion of he updae equaions oal field/scaered field source Calculaing ransission and reflecion Bloc diagra of F Sequence of code developen Lecure 16 Slide 1

2 7/1/17 Inroducion o Finie ifference ie oain Lecure 16 Slide 3 Flow of Mawell s quaions B A circulaing field induces a change in he B field a he cener of circulaion. B A B field induces an field in proporion o he pereabili. A field induces an field in proporion o he periivi. A circulaing field induces a change in he field a he cener of circulaion. Lecure 16 Slide 4

3 7/1/17 Flow of Mawell s quaions Inside Linear, Isoropic and Non ispersive Maerials In aerials ha are linear, isoropic and non dispersive we have In his case, he flow of Mawell s equaions reduces o A circulaing field induces a change in he field a he cener of circulaion in proporion o he pereabili. A circulaing field induces a change in he field a he cener of circulaion in proporion o he periivi. Lecure 16 Slide 5 FF Vs. F Yee Grid FF F he Yee grid, finiedifferences, and nuerical behavior are alos idenical for FF and F. FF assebles he large se of finie difference equaions ino a single ari equaion and solves he siulaneousl. A b 1 A b = A\b; F loops hrough he large se of finie difference equaions and updaes he fields in sall ie seps. for = 1 : IM for n = 1 : N for n = 1 : N (n,n) = (n,n)... + ((n+1,n) - (n,n))/d... - ((n,n+1) - (n,n))/d; end end Lecure 16 end Slide 6 3

4 7/1/17 FF Vs. F his is wha FF calculaes. his is wha F calculaes. FF obains a soluion a a single frequenc. F inherenl siulaes a broad range of frequencies so a ransien response is alwas observed. Lecure 1 Slide 7 aple Siulaion: Pulsed Radar Lecure 16 Slide 8 4

5 7/1/17 aple Siulaion: Phoonic Crsal Waveguide Bend Lecure 16 Slide 9 aple Siulaion: Self Colliaion Clindrical CW Source Source in a Nonfuncional Laice Source in a Self Colliaing Laice Lecure 16 Slide 1 5

6 7/1/17 ighl Resonan evices F is ver slow for highl resonan devices. nerg ges suc in he device. F has o eep ieraing unil ha energ escapes. Lecure 16 Slide 11 Concep of he Updae quaion Lecure 16 Slide 1 6

7 7/1/17 Approiaing he ie erivaive (1 of 3) An inuiive firs guess a approiaing he ie derivaives in Mawell s equaions is: iss a iss a iss a iss a his is an unsable forulaion. Lecure 16 Slide 13 Approiaing he ie erivaive ( of 3) We adjus he finie difference equaions so ha each er eiss a he sae poin in ie. hese equaions will ge ess if we include inerpolaions. Is here a sipler approach? Lecure 16 Slide 14 7

8 7/1/17 Approiaing he ie erivaive (3 of 3) We sagger and in ie so ha eiss a ineger ie seps (,,, ) and eiss a half ie seps (/, +/, +/, ). he spaial derivaives are handled eacl lie he are in FF. Lecure 16 Slide 15 erivaion of he Updae quaions he updae equaions are he equaions used inside he ain F loop o calculae he field values a he ne ie sep. he are derived b solving Mawell s equaions for he field a he fuure ie value. Lecure 16 Slide 16 8

9 7/1/17 Anao of he F Updae quaion Updae coefficien o speed siulaion, we calculae hese before he ain loop. Field a he ne ie sep. Field a he previous ie sep. Curl of he oher field a an inerediae ie sep Lecure 16 Slide 17 ie oain UPML Lecure 16 Slide 18 9

10 7/1/17 Recall he Uniaial PML he 3 PML can be visualied his wa s ss s ss s ss s s 1 s 1 s 1 s 1 s 1 s 1 Lecure 16 Slide 19 Ficiious Conduciviies he perfecl ached laer (PML) is an absorbing boundar condiion (ABC) where he ipedance is perfecl ached o he proble space. Reflecions enering he loss regions are prevened because ipedance is ached. Reflecions fro he grid boundar are prevened because he ougoing waves are absorbed. 1 PML PML Proble Space PML 1 1 PML 1 Lecure 16 Slide 1

11 7/1/17 Calculaing he PML Loss ers For bes perforance, he loss ers should increase graduall ino he PMLs. s s s 1 j 1 j 1 j 3 L 3 L 3 L Lecure 16 Slide 1 L? lengh of he PML in he? direcion Incorporaing a UPML ino Mawell s quaions Before incorporaing a UPML, Mawell s equaions in he frequenc doain are j r r j We can incorporae a UPML independen of he acual aerials on he grid as follows: j r s r j s Lecure 16 Slide 11

12 7/1/17 Noralie Mawell s quaions We noralie he elecric field quaniies according o 1 1 c Mawell s equaions wih he UPML and noralied fields are r j s c j r s c Lecure 16 Slide 3 Vecor pansion of Mawell s quaions r j s c j s c r ss j c s ss j c s ss j c s s s c s j j ss c s j s s c s Lecure 16 Slide 4 1

13 7/1/17 Mawell s quaions wih a UPML Neglecing he loss er, we have r j s c 1 c j1 1 1 j j j 1 c j j j j c j j j j j s c 1 j1 1 1 c j j j 1 j j j j1 1 1 c j j j j 1 c r Lecure 16 Slide 5 Sae quaions in he ie oain c c d d c c d d c c d d c d c d c d c d c d c d Lecure 16 Slide 6 13

14 7/1/17 Updae quaions Lecure 16 Slide 7 Finie ifference Approiaions Puing all he ers ogeher, he equaion is c C d d C c 4 c i, j, C c C Lecure 16 Slide 8 14

15 7/1/17 Suar of All Nuerical quaions c,,,, i j i j,,,,,,,, 1 i j i j i j c i j C 4 c, j,,,,,,, 1 i j i j i j c i, j 4 C, C i,,,, i j i j 1 c c C 4 i C, j, i, j,,,,,,,,, i j i j i j i j 1,,,, i j i j c c C 4 C,,,,,,,,,,,, i j i j i j i j i j i j 1,,,, i j i j c c C C 4, j, i,,,, i j i j 1,,,, i j i j c c C 4 C Lecure 16 Slide 9 C Solve for a he Fuure ie Sep Solving our nuerical equaion for he field a +/ ields c,,,, i j i j,,,,,,,, 1 i j i j i j c i j C 4 C i, j, c,,,, i j i j C,,,, c i j i j i j i j i j i j 4,,,, C +,,,,,, i j 4 Lecure 16 Slide 3 15

16 7/1/17 Final For of he Updae quaion for he updae coefficiens are copued before he ain F loop. i j c 1 c,,,,,, i j 1 i j i j 3 4,, 1 he inegraion ers are copued inside he ain F loop, bu before he updae equaion. i, j1, 1,,,, i j,,,, i j i j i j C I C I C he updae equaion is copued inside he ain F loop iediael afer he inegraion ers are updaed C IC I Lecure 16 Slide 31 Final For of he Updae quaion for he updae coefficiens are copued before he ain F loop. i j c 1 c,,,,,, i j 1 i j i j 3 4,, 1 he inegraion ers are copued inside he ain F loop, bu before he updae equaion. i, j, 1 i, j, i 1, j, i, j, C I C I C he updae equaion is copued inside he ain F loop iediael afer he inegraion ers are updaed i, j, C IC I Lecure 16 Slide 3 16

17 7/1/17 Final For of he Updae quaion for he updae coefficiens are copued before he ain F loop. i j c 1 c,,,,,, i j 1 i j i j 3 4,, 1 i, j, he inegraion ers are copued inside he ain F loop, bu before he updae equaion. i 1, j, i, j, i, j 1, i, j,,,,, i j,,,, i j i j i j C I C I C he updae equaion is copued inside he ain F loop iediael afer he inegraion ers are updaed i, j, C IC I Lecure 16 Slide 33 Final For of he Updae quaion for he updae coefficiens are copued before he ain F loop. i j c 1 c 1,, 4 i j,, 1 3 he inegraion ers are copued inside he ain F loop, bu before he updae equaion. i, j1, 1,,,, i j i j C I C I C he updae equaion is copued inside he ain F loop iediael afer he inegraion ers are updaed. 1 3 C + 4 i, j, i, j, i, j, i, j, i, j, i, j, C I I Lecure 16 Slide 34 17

18 7/1/17 Final For of he Updae quaion for he updae coefficiens are copued before he ain F loop. i j c 1 c 1,, 4 i j,, 3 he inegraion ers are copued inside he ain F loop, bu before he updae equaion. 1 i1, j,,, i j C I C I C he updae equaion is copued inside he ain F loop iediael afer he inegraion ers are updaed. 1 3 C + 4 i, j, i, j, i, j, i, j, i, j, i, j, C I I Lecure 16 Slide 35 Final For of he Updae quaion for he updae coefficiens are copued before he ain F loop. i j c 1 c 1,, 4 i j,, 1 3 he inegraion ers are copued inside he ain F loop, bu before he updae equaion. i1, j, i, j1,,,,,,,,, i j i j i j i j C I C I C he updae equaion is copued inside he ain F loop iediael afer he inegraion ers are updaed. 1 3 C + 4 C I I Lecure 16 Slide 36 18

19 7/1/17 Final Updae quaions for,, and he updae coefficiens are copued before he ain F loop ,, 1 i j 1 he updae equaions are copued inside he ain F loop Lecure 16 Slide 37 oal Field/Scaered Field Source Lecure 16 Slide 38 19

20 7/1/17 Source pes: Siple ard Source he siple hard source is he easies o ipleen, bu he locaion where he source is injeced reflecs waves 1%. I is difficul o conrol he apliude of his wave since power i injeced in all direcions. Lecure 16 Slide 39 Source pes: Siple Sof Source he siple sof source is alos as eas o ipleen and he waves pass copleel hrough he locaion where he source is injeced. I is sill difficul o conrol he apliude of his wave since power i injeced in all direcions. Lecure 16 Slide 4

21 7/1/17 Source pes: F/SF Sof Source he F/SF sof source is ore difficul o ipleen, bu is a sof source ha is ransparen o waves. his ehod provides coplee conrol over he apliude of he source. Lecure 16 Slide 41 F/SF Fraewor Reflecion Plane F/SF Planes Spacer Region Uni cell of real device Spacer Region Scaered Field oal Field scaered field oal field ransission Plane Lecure 16 Slide 4 1

22 7/1/17 Grid Schee for oubl Periodic evices For doubl periodic devices, a PML is onl needed a he ais boundaries. Reflecion Plane F/SF Planes Spacer Region Uni cell of real device odel valid in here Spacer Region ransission Plane F grid real device Lecure 16 Slide 43 Correcions o Finie ifference qs. Scaered Field Reflecion Plane F/SF Planes ver cell along he F/SF inerface requires a correcion er. Spacer Region Uni cell of real device oal Field Spacer Region ransission Plane Lecure 16 Slide 44

23 7/1/17 pical View of 3 F wih F/SF MALAB s slice() funcion was used o generae his plo. Gaussian Pulse Lecure 16 Slide 45 Calculaing Sead Sae Fields Lecure 16 Slide 46 3

24 7/1/17 Brue Force Fourier ransfors he easies, bu leas eor efficien, ehod o copue a Fourier ransfor is o perfor a siulaion and record he desired field as a funcion of ie. Afer he siulaion is finished, hese funcions can be Fourier ransfored using an FF. refleced field ransied field source Lecure 16 Slide 47 fficien Fourier ransfor (1 of ) he sandard Fourier ransfor is defined as j F f f e f If he funcion f() is onl nown a discree poins, he Fourier ransfor can be approiaed nuericall as M j f F f f e 1 his can be wrien in a slighl differen for. M j f F f e f 1 Lecure 16 Slide 48 4

25 7/1/17 fficien Fourier ransfor ( of ) he final for on he previous slide suggess an efficien ipleenaion. he Fourier ransfor is updaed ever ieraion so b he end of he ain loop: M j f e f F f 1 his uliplicaion can be done afer he ain F loop in a posprocessing sep. e j f his is sipl he field value of ineres a he curren ie sep. his ernel can be copued prior o he ain F loop for each frequenc of ineres. he ernels can be sored in a 1 arra. Lecure 16 Slide 49 fficien Fourier ransfor Algorih Copue Kernels j fi K e i Iniialie Fourier ransfors F i one? es no Updae fro Finalie Fourier ransfors F F i i Finished! Updae fro Updae Fourier ransfors,, F F K i j i i i Lecure 16 Slide 5 5

26 7/1/17 Calculaing ransission and Reflecion Lecure 16 Slide 51 he Fourier ransfors We picall sar he copuaion of power b Fourier ransforing he refleced and ransied field using one of he ehods described previousl. pical F siulaion resuls loo lie his src FF src A roll off is observed in he frequenc responses. his occurs sipl because here is less power in he source a he higher frequencies. I does no ean he device is less reflecive or ransissive. ref rn FF FF ref rn Lecure 16 Slide 5 6

27 7/1/17 Noralie he Fourier ransfors We us noralie he specra o calculae ransiance and reflecance. We do his b dividing he reflecion and ransission specru b he source specru. R f FF FF ref src R f f C f f FF FF rn src I is ALWAYS good pracice o chec for energ conservaion b adding he reflecance and ransiance and ensuring he su equals 1% (assuing no loss or gain in our device). Lecure 16 Slide 53 Bloc iagra of F Lecure 16 Slide 54 7

28 7/1/17 Bloc iagra for Mode (1 of ) efine evice Paraeers Iniialiaion efine F Paraeers Copue Grid Paraeers Build evice on Grid Copue ie Sep Copue Source Copue PML Paraeers Copue Updae Coefficiens 1,, 3 1,, 3 1,, 4 1 Iniialie Fields,,, Iniialie Curl Arras C, C, C Iniialie Inegraion Arras IC, IC, I Lecure 16 Slide 55 Bloc iagra for Mode ( of ) es one? Finished! Main loop BC s no Copue Curl of C, C Updae Inegraions IC = IC + C IC = IC + C Copue Curl of C Updae Inegraions I = I + Updae = 1.* +.*C + 4.*I; Updae Field = 1.* +.*C + 3.*IC; = 1.* +.*C + 3.*IC; Injec Source (ns,ns) = (ns,ns) + g(); Updae = 1.*; Visualie Fields Lecure 16 Slide 56 8

29 7/1/17 Bloc iagra for Mode (1 of ) efine evice Paraeers efine F Paraeers Copue Grid Paraeers Copue ie Sep Copue Source Copue PML Paraeers Iniialiaion Copue Updae Coefficiens 1,, 4 1,, 3 1,, 3 1, 1 Iniialie Fields,,,, Iniialie Curl Arras C, C, C Iniialie Inegraion Arras IC, IC, I Lecure 16 Slide 57 Bloc iagra for Mode ( of ) one? es Finished! Main loop no BC s Copue Curl of C Updae Inegraions I = I + Updae Field = 1.* +.*C + 4.*I; Injec Source (ns,ns) = (ns,ns) + g(); Visualie Fields Copue Curl of C, C Updae Inegraions IC = IC + C; IC = IC + C; Updae Field = 1.* +.*C + 3.*IC; = 1.* +.*C + 3.*IC; Updae Field = 1.*; = 1.*; Lecure 16 Slide 58 9

30 7/1/17 Sequence of Code evelopen Lecure 16 Slide 59 Sep 1 Basic Updae quaions he basic updae equaions are ipleened along wih siple irichle boundar condiions. irichle Boundar Condiion irichle Boundar Condiion irichle Boundar Condiion irichle Boundar Condiion Lecure 16 Slide 6 3

31 7/1/17 Sep Incorporae Periodic Boundaries Periodic boundar condiions are incorporaed so ha a wave leaving he grid reeners he grid a he oher side. Periodic Boundar Condiion Periodic Boundar Condiion Periodic Boundar Condiion Periodic Boundar Condiion Lecure 16 Slide 61 Sep 3 Incorporae a PML he perfecl ached laer (PML) absorbing boundar condiion is incorporaed o absorb ougoing waves. lo PML lo PML hi PML hi PML Lecure 16 Slide 6 31

32 7/1/17 Sep 4 oal Field/Scaered Field Mos periodic elecroagneic devices are odeled b using periodic boundaries for he horional ais and a PML for he verical ais. We hen ipleen F/SF a he verical cener of he grid for esing. Periodic Boundar Condiion Periodic Boundar Condiion Lecure 16 Slide 63 Sep 5 Calculae RN, RF, and CON We ove he F/SF inerface o a uni cell or wo ouside of he op PML. We include code o calculae Fourier ransfors and o calculae ransiance, reflecance, and conservaion of power. Lecure 16 Slide 64 3

33 7/1/17 Sep 6 Model a evice o Benchar We build a device on he grid ha has a nown soluion. We run he siulaion and duplicae he nown resuls o benchar our new code. Lecure 16 Slide 65 Suar of Code evelopen Sequence Sep 1 Basic Updae + irichle Sep Basic Updae + Periodic BC Sep 3 Add PML Sep 4 F/SF Sep 5 Calculae Response Sep 6 Add a evice and Benchar Lecure 16 Slide 66 33