Finite Differences. Centered vs. forward/backwards. ( x) Consider the Taylor expansion of f(x+h) around x:

Transcription

1 FD Bass C 66 Nua Mhods Phoos F Dffs Cd vs. fowad/bawads Cosd h Tao paso of f(h) aoud : h f ( h) f h f f K Fo whh w g h fowad fs dff ( h) f f f h f ( h) f f h h h O( h) Now pad boh f(h) ad f() aoud f(h/) ad suba h h f h 8 h h 8 3 h h 48 h f f f f K To g h d fs dff ( h) f O( h ) h f f h ( h) f ( h) O( h ) h f 3 h 48 h f K f f Aa oda vso h ( h) f f ( h) f f h No v aua O h Cd d dff Mo aua Rob R. MLod Uvs of Cooado Copuao Mhods fo M & Mowavs Booo 4

2 5 C 66 Nua Mhods Phoos Rob R. MLod Uvs of Cooado FDTD fo h D wav quao Dvao of h -adva foua FDTD FD Bass L L L ± D saa wav quao Fao opao o Fowad ad bawads popagao opaos Subsu d d dff opaos Sov fo o g p -ahg quao S: No ha sohg sg happs wh

3 6 C 66 Nua Mhods Phoos Rob R. MLod Uvs of Cooado FDTD FD Bass FDTD fo h D wav quao Mag sp L h adva quao Subsu gsouos o opaos L ± f B B L f F F L o h adva quao abov [ ] [ ] [ ] [ ] F f f f f F F F F ad sa B B whh a h a souos. O possb D hoogous spas bu good fo udsadg.

4 7 C 66 Nua Mhods Phoos Rob R. MLod Uvs of Cooado FDTD fo h D wav quao Dspso h sapd spa FDTD FD Bass Pug ouous pa wav o d od wav quao Phas ad goup vo foow fo dspso ao v p v g Pug ds pa wav o -ahg quao ~ ~ ~ Ca ou oo s o g [ ] ~ os os whh ds dspso ao fo ds spa:

5 8 C 66 Nua Mhods Phoos Rob R. MLod Uvs of Cooado [ ] os os ~ v p π λ FDTD fo h D wav quao Dspso h sapd spa (o) FDTD FD Bass ~ Ch ha hs dus o ouous as ~ Wha abou ag sp? [ ] ~ os os ~ I ga (sov fo ) [ ] os os ~ Fo whh w a w phas vo of ds spa ~

6 FD Bass C 66 Nua Mhods Phoos FDTD fo h D wav quao Choosg ad Podu: Rqu ha g s boudd hough h odo. Fd hs odo b wg sppg quao as a quao ad sovg fo gvaus. Rsu: λ / v p /.75 / Lg dspso (phas o) qus D Coua sab I o 3D ad/o pobs wh a d w a opa gh a hs whh ads o ua dspso: v p λ λ Nqus Rob R. MLod Uvs of Cooado 9

7 agoh C 66 Nua Mhods Phoos 3D FDTD Th hod W ou Maw u quaos Ad df ds gd ( ) ( ) ad ag fd gds a offs b ½ a 3D o p d fsdffs of h u quaos. Rob R. MLod Uvs of Cooado K. S. I Tas. o Aas ad Popagao. Vo. 4 pp Ma 966 3

8 agoh C 66 Nua Mhods Phoos 3D FDTD T adva quaos Ap s Law gvs adva of fds a dgs of o g sps: D J Faada s Law gvs adva of ag fds a fas of o haf-g sps: B Rob R. MLod Uvs of Cooado 3

9 agoh C 66 Nua Mhods Phoos 3D FDTD Dvao fo ga quaos S B ds () C d [ BX BX ] [ ] Faada s aw ga fo Tasa o dff quao w/ d dffs: Dvd b ad aag o g sa upda quao Rob R. MLod Uvs of Cooado 3

10 agoh C 66 Nua Mhods Phoos Dos sh ob Gauss Laws? O usd Ap s ad Faada s aws o dv. Dos sh ob Gauss Laws? Sa wh ga fo sou f go: S D ds 6 D S S V ds 6 d C D ds ρ dv? Gauss Law Dffa wh sp o App o fas of Us Ap s Law ah dg of ub s usd w wh oppos sg so S D ds Q osa so f a hag f (Q) sas hag f ad Gauss Law s sasfd. Rob R. MLod Uvs of Cooado 33

11 agoh C 66 Nua Mhods Phoos Sab ad dspso 3D Coua odo ad 3D dspso ao Foowg pag 7: ( ) 3 Ipao (daw fo D): / 3D Coua sab Fo uba s T sp us b sho ha shos popagao bw ada ods whh s o h dagoa of h. As w saw D dspso s d b opag as os as possb o hs. Ufoua 3D hs s qu fa fo h o-dspso odo Dspso ao 3D: ~ ~ ~ s s s s Dpds o do so ds spa s asoop. Rob R. MLod Uvs of Cooado 34

12 Bouda odos C 66 Nua Mhods Phoos Bouda odos PC ad PMC PC: Fo ap s ad : B D PMC: Fo ap s ad : B B dfo (odd s) B Gauss Law (v s) B Faada s Law (odd s) B Ap s Law (v s) B dfo (odd s) B Gauss Law (v s) D B Ap s Law (odd s) B Faada s Law (v s) Rob R. MLod Uvs of Cooado 35

13 Bouda odos C 66 Nua Mhods Phoos Radao bouda odo gqus ad Mada -wa wav q. I 3D h o-wa wav opaos fo a wav avg do (fo ap) bo: ± L ( ) ( ) Usg a bouda Good ws: gqus ad Mada showd ha hs w absob a wav a popagag a a ag. Bad ws: Th squa oo as qu dffu o p. Souo (Mu): Appoa squa oo as Tao ss. Rob R. MLod Uvs of Cooado B. gqus ad A. Mada Mah. Copu. vo. 3 pp

14 Bouda odos C 66 Nua Mhods Phoos Radao bouda odo Fs-od Mu odo Th fs h Tao paso of h squa oo s dg h fs-od Mu opao L ± v p Opao fo wav avg ± If w w phas vo of wav hs woud b a. Bu hs woud qu h b o o pa wav d o h bouda. So assu wav s oa (v p ). Dff quao: ( ) ( ) ( ) ( ) Sov fo ( ) Cd dffs aoud / / Mus so hs vau Ca opu hs vau fo Rob R. MLod Uvs of Cooado G. Mu I Tas. o oag Copab Vo. MC-3 Novb 98 pp

15 38 C 66 Nua Mhods Phoos Rob R. MLod Uvs of Cooado Radao bouda odo Sod-od Mu odo Kp Tao ss: Ta dvav Ad subsu fo ow souo o L opao S Mu pap fo dsao. Rqus soag of wo pvous sps p bouda od o p sod dvav. Aso o ha asvs dvavs a o b pd a os so us v o a fs-od odo h. FDTD Bouda odos

16 Bouda odos C 66 Nua Mhods Phoos Radao bouda odo Pfoa of Mu s ad d Po sou a (55) of a sh. Dashd s od sod d od. T sp 4. Sa as abov bu sou a (33). Rob R. MLod Uvs of Cooado G. Mu I Tas. o oag Copab Vo. MC-3 Novb 98 pp

17 4 C 66 Nua Mhods Phoos Rob R. MLod Uvs of Cooado Radao bouda odo Rfv vs. ag FDTD Bouda odos s os os s s os os s Mu To fd fv assu a d ad fd pa wav o h bouda : Pug o s od Mu upda quao ad sov fo : Foowg sa podu fo sod-od BC: s s os os s s b a b a b a Mu K.S. Ku R. J. Lubbs FDTD Mhod fo M CRC Pss 993

18 Bouda odos C 66 Nua Mhods Phoos Radao bouda odo Tpa fv D»».... Fs vs. sod-od a o s Fs-od Sod-od λ / / D»»... λ / 5 λ / λ / Sod-od vs. s.. / D Rob R. MLod Uvs of Cooado 4

19 Sous C 66 Nua Mhods Phoos Sous Tpoa shap S dspso dpds o sas λ us ags pob. Gaussa: σ σ π σ Moduad Gaussa: σ s( π f ) σ π π f σ Sg fqu s( π f ) δ ( π f ) σ > Rob R. MLod Uvs of Cooado 4

20 Sous C 66 Nua Mhods Phoos Toa/sad fd foas Bagoud Vaous po sous wh h sh a b dv a sg gd pos. Tpa ops w wa a dd sou d fo f spa. Df: To To sa sa Df ha d fd popagas f-spa: o I oa fd go us oa hod fo oa fd sa o sa I f-spa sad fd go popaga o sad fds o o So s obs Maw Rob R. MLod Uvs of Cooado 43

21 Sous C 66 Nua Mhods Phoos Toa/sad fd foas Us Suoud o go of oa fd wh sad fd go. Id fds as o bouda RBC RBC RBC Toa fd go Sad fd go RBC Puss: RBCs o da wh sad (ougog) fds Of wa o ow sad fds so podus aua Ca us sa upda quao houghou gd h o ba ad add/suba d fd o T/S bouda. Muss: Id fd us b popagad ds spa o ah ua dspso Sgfa aou of soag s usd sad go Rob R. MLod Uvs of Cooado 44

22 Sous C 66 Nua Mhods Phoos Toa/sad fd foas Id fd popagao Id wav us b phas-ahd o h asoop dspso of h 3D ds gd. S poa pus shap us oa f badwdh us ah dspso a bad of fqus. A as wa o do hs s o popaga h d fd s ow D FDTD spa. Bg wh dspso aos fo D ad 3D: ~ s s D ~ ~ ~ s s s s If d wav s aog pa as a pf ah b sg D s qua o s popagao do. Ohws sov fo agud of ~ a fqu ad us hs o p h D s. ~ ˆ ~ ˆ s s s s ~ D s s D ~ ˆ Rob R. MLod Uvs of Cooado 45

23 46 C 66 Nua Mhods Phoos Rob R. MLod Uvs of Cooado [ ] [ ] o sa X sa o sa X sa X sa Toa/sad fd foas Ipao FDTD Sous sa sa sa sa Cas : A fds sad o: 3 4 Cas 3: Fd sad o poo of s oa o Cas : A fds oa o: o o o o [ ] [ ] [ ] [ ] X o X o sa o o X o X o sa o o o µ µ µ µ µ µ µ µ µ µ Cas 4: Fd oa o poo of s sad o Upda quaos sa houghou gd. Th o ba ad add/suba d fd o sad/oa fd bouda. Sad Toa

24 Maas C 66 Nua Mhods Phoos Asoop aas () D Two pobs: σ σ σ Cosuv ao Subsu o Ap s Law σ opo. Off dagoa opos of oup h opos of Ap s Law so a sp sov fo upda quao. Do hav a opos of a pop ad spa s ( s s bo). Rob R. MLod Uvs of Cooado J. B. Shd ad S. udso I Tas. Aas Popaga. vo. 4 pp Ju

25 48 C 66 Nua Mhods Phoos Rob R. MLod Uvs of Cooado Asoop aas () X X X σ σ σ Fo s o RS vovg oduv (whh a sapd a h wog sp) a avag o g a ½. App h oa d dffs fo s o RS vovg. X X σ σ σ σ σ σ Rpa podu fo oh wo opos of Ap s Law d hs sa oao (½ ½). W ow hav h quaos h uows whh w a hfo sov. W o p h s s a h gh oao. To g quaos fo ad h dvao a h appoa oao fo hos fds. FDTD Maas ad X Ioduo of sos oups a fd opos suh ha h sp gddg of d dffs fudaa bas. W s wh h gd baus of a h oh good ous bu hs w b dous h. Th sps wa o udsad h podu s o osd h s o fo ah opo. Th foowg s fo h opo. Rpa h podu fo ad. Bg b dsg a h opos of Ap s aw a h oao ½ ½. Th opo fo ap woud b:

26 49 C 66 Nua Mhods Phoos Rob R. MLod Uvs of Cooado Asoop aas (3) FDTD Maas [ ] X X X X σ σ Th su fo h podu d a ½ s: Good ws: Ca ow w s of quas a pvous s. Bad ws: Na v fd opo o h RS s a h wog spaa sapg oao. Souo: Ipoa as ssa. Fo ap: X 4 Rpa h podu fo o dv h oh wo -fd upda quaos. No ha h s s ow spaa ag whh w pa h wh bouda odos. Tpa souo do us asoop aas a boudas. ad Ths opo s a h gh oao. Th ohs a.

27 5 C 66 Nua Mhods Phoos Rob R. MLod Uvs of Cooado Dspsv aas () FDTD Maas R. Lubbs a. I Tas. o M Copa V 3 N 3 Aug 99 χ δ τ) dτ ( χ(τ) () D() W h saa av pv as a saaous (odspsv) ad dspsv : Whh odfs h osuv ao To dff Ap s Law w d pssos fo D a sps ad : τ τ χ τ τ χ d D d D Ds χ b dfg D D χ χ χ Ta h dff o g h dvav of D τ τ χ χ d

28 5 C 66 Nua Mhods Phoos Rob R. MLod Uvs of Cooado Dspsv aas () FDTD Maas χ χ χ χ Df χ χ χ Ad subsu o h ds Ap s Law o fd upda quao fo wh Σ ½ pss h su of h fou fds h ds su of of h oop ga. d Good ws: av psso fo updag. Bad ws: Appas o qu soag of hso of as og as h va of χ Souo: Dff h dffa quao fo P χ : s χ U s χ s χ P P P χ χ s χ Poa d Ds fo Dfo Upda quao

29 Maas C 66 Nua Mhods Phoos Noa aas Sp hod Fo a saaous K oa ( 3) 3 δ χ δ χ D δ Avod pobs wh oa quao b assug s o oo dff fo h auao of h oa pv D [ α( ) α ( ) α ( ) ] X X X Nos. Noao ad so fo of oa foow [] bu oud b gad o op hd-od so oas hough h us of a a vso sp sa o Shd ad udso. Rqus poao o g h fd opos a h pop oaos 3. O vad fo saaous oas 4. W qu v f sps o a assupo vad. Ths w u qu v f spa sps o avod dspso. Thus pa o usfu fo D pobs. Rob R. MLod Uvs of Cooado G. W. hg ad K. S. Ch I. J. Ifad M. Wavs vo. 3 o. 8 pp

30 Maas C 66 Nua Mhods Phoos D Noa aas Mo op hod To da wh boh dspsv ad so dspsv oas df P L P NL Ta a dspssv pa as bfo. Foowg ha appoah fo a paua phsa ss ha gas h oa w h dffa quao ha dsbs h vouo of P NL ad us FDs o fd a upda quao fo P NL. Fo ap (s f) K Raa Th Raa fo ap foows h foowg D Rob R. MLod Uvs of Cooado P. M. Gooa ad A. Tafov Ops L. B 7 pp. 8-8 Fb

31 Maw-Boh C 66 Nua Mhods Phoos Maw-Boh quaos D FDTD suo Fd Aos of gsas > > Maw Cu quaos: µ o o o P Shodg quao: h ˆ ( ( )) p ˆ µ Coupg: ˆ µ ˆ P Np Rob R. MLod Uvs of Cooado Ma Sow NMP 4 fa po 54

32 Maw-Boh C 66 Nua Mhods Phoos Ao Poaao Quau Mhas Shodg quao: h ˆ ( ( )) w/ sa spa { } ds souos of fo: ( ) ~ ~ F ao: / : [ ] h ( ) > > [ ] No os oss : [ ] os Osas a ao fqu Rob R. MLod Uvs of Cooado 55

33 Maw-Boh C 66 Nua Mhods Phoos ap dog o wavfuos Goud sa: - Pa S ( ) a 3/ / a d sa: a o - Pa p P 3/ / a 3 ( ) os( θ ) a a 3 π a o Osag dpo o do: p ˆ os Rob R. MLod Uvs of Cooado 56

34 Maw-Boh C 66 Nua Mhods Phoos Louv quao ow o had sb of aos Laos o Shodg quao: Dos od a sb of aos Ca o as ud doh No spoaous sso Ds Ma: ρ ˆ ρ ρ ρ ρ Louv quao: sb avags * * * * pao vau of h posv o gav fqu opo of p Popuao pobabs goud ad d sas & ρ [ ] ( q ) ˆ ρ γ ρ ρ ˆ h Fo Shodg Phoooga da s Rob R. MLod Uvs of Cooado 57

35 Maw-Boh C 66 Nua Mhods Phoos Dvao of Boh quaos & ρ µ [ ρ ρ] γ ρ Popuao sa h γ & ρ [ [ ] h ρ µ ρ ρ ρ h * ρ ρ Coh bw & ρ Wg ou ah of h Louv quaos ρ Boh vo psao: Subsu: R[ ] ρ I[ ] ρ ρ 3 ρ Lag a hag of ohs Cosvao of popuao & & & Boh quaos T µ ( T h q µ 3 3 ( T h ) 3 ) 3 Rob R. MLod Uvs of Cooado 58

36 Maw-Boh C 66 Nua Mhods Phoos Fa s of Ds Subsu poaao Maw: p Nµ & & & & µ ( ) q ( ) h & µ o o T T T Nµ T o h µ Nµ 3 o Boh sph: 3 [ ] Rob R. MLod Uvs of Cooado 59

37 Maw-Boh C 66 Nua Mhods Phoos Nua pob Nd o adva sa vo b d: C & 3 C C T & T w/ µ & 3 T h q µ & T h 3 FDTD upda s: 3 3 / 3/ Rob R. MLod Uvs of Cooado 6

38 Maw-Boh C 66 Nua Mhods Phoos Soo pus sus 8.8 π Sh Pus Fd R Poso (µ) π Sh Pus Fd R Poso (µ) Rob R. MLod Uvs of Cooado 6

39 Vadao agas ow sus C 66 Nua Mhods Phoos Vadao of a FDTD fa-fd poo od Copaso wh hod of os Copaso wh sp aaa as Rob R. MLod Uvs of Cooado M.J. Bah R. R. MLod R. W. oows J. M Wavs ad Apps V6 N 99 6

40 Vadao agas ow sus C 66 Nua Mhods Phoos Vadao wh M sag fo sph No ha FDTD pds s ad agud of whspg ga od bu o shap du o ds sufa. Copaso vs. Copuaoa spa usg PC ad PMC bouda odos as s pas. Copaso vs. ag a o Rob R. MLod Uvs of Cooado 63

41 Vadao agas ow sus C 66 Nua Mhods Phoos Vadao wh p Sag fo ub. FDTD pd o do w hs as s ob fs agua gd. Rob R. MLod Uvs of Cooado 64

42 65 C 66 Nua Mhods Phoos Rob R. MLod Uvs of Cooado Sab ow o p h od fo podg A ga hod o d h sab s s o subsu a assud souo ha uds h sab h sov fo h odos whh h sab dos o gow (vo Nua aass). Fo ap sag wh ou FD psao of h 3D wav quao (gad fo pag 7): Assu a souo of h fo [ ] α Subsu o h FD quao ad dvd b [ ] [ ] [ ] [ ] { } [ ] [ ] [ ] [ ] os os os osh osh os os os α α α α Th apfao w a boudd o f s agud s. O a ds gd π / ad hus h odo fo sab s whh s h Coua odo fo pag 34. Vadao Sab

43 Vadao Covg C 66 Nua Mhods Phoos Coss & Covg ow w ow h asws a gh Ra ou pssos fo d fs ad sod-od FDs (pag 4). ( h) f ( h) O( h ) f f h ( h) f f ( h) O( h ) f f h Tuao o A FD quao s oss wh h oga PD quao wh h uao o gos o o as h sh s gos o o. A ap of a oss FD sh s a d-dff vso of h ha dffuso quao (DuFo-Fa sh): u u u u u u u u B sp wg h Tao ss fo h FDs w fd h adg s of h uao o a: 4 u 4 u u... La quva ho Gv a pop posd a a vau pob ad a f dff appoao o ha sasfs h oss odo sab s h ssa ad suff odo fo ovg. Dos go o o f / s a osa! A FD appoao s sad o ovg wh h souo of h FD quao ovgs o h souo of h PD as h sh s gos o o. Rob R. MLod Uvs of Cooado PARTIAL DIFFRNTIAL QUATIONS 66