Maxwell s Equations on a Yee Grid

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Instructor Dr. Ramond Rumpf (95) 77 6958 rcrumpf@utp.du 57 Computational lctromagntics (CM) Lctur # Mawll s quations on a Y Grid Lctur Ths nots ma contain coprightd matrial obtaind undr fair us ruls.

1 Instructor Dr. Ramond Rumpf (95) Computational lctromagntics (CM) Lctur # Mawll s quations on a Y Grid Lctur Ths nots ma contain coprightd matrial obtaind undr fair us ruls. Distribution of ths matrials is strictl prohibitd Slid Outlin Y Grid Mawll s quations on a Y grid Finit diffrnc approimations of Mawll s quations on a Y grid Matri form of Mawll s quations Numrical disprsion Gnraliation to full anisotropic matrials Altrnativ grids Bonus Lctur Slid

$in Applid Mathmatics from th Univrsit of California at Brl in 957, 958, and 96, rspctivl. did rsarch on lctromagntic diffraction whil mplod b Lochd Missil and Spac Co.$ (959 96). has bn associatd with th Lawrnc Livrmor Laborator sinc 96. At prsnt h is a profssor in mathmatics at Kansas Stat Univrsit.

(959 96). has bn associatd with th Lawrnc Livrmor Laborator sinc 96. At prsnt h is a profssor in mathmatics at Kansas Stat Univrsit.

2 Y Grid Lctur Slid Kan S. Y Kan S. Y was born in Canton, China on March 6, 9. rcivd a B.S..., M.S.. and Ph.D. in Applid Mathmatics from th Univrsit of California at Brl in 957, 958, and 96, rspctivl. did rsarch on lctromagntic diffraction whil mplod b Lochd Missil and Spac Co. (959 96). has bn associatd with th Lawrnc Livrmor Laborator sinc 96. At prsnt h is a profssor in mathmatics at Kansas Stat Univrsit. is main aras of intrst ar lctromagntics, hdrodnamics and numrical solution to partial diffrntial quations. K. S. Y, A Closd Form prssion for th nrg Dissipation in a Low Loss Transmission Lin, I Trans. Nuclar Scinc, vol., no., pp. 6 8, 97. Kan S. Y Lctur Slid

3 D Grids A thr dimnsional grid loos li this: On unit cll from th grid loos li this: N, N, N 5,, grid rsolution paramtrs Lctur Slid 5 Collocatd Grid Within th unit cll, whr should w plac,,,,, and? A straightforward approach would b to locat all of th fild componnts at a common point within in a grid cll; prhaps at th origin. Lctur Slid 6

4 Y Grid Instad, w ar going to staggr th position of ach fild componnt within th grid clls. K. S. Y, Numrical solution of th initial boundar valu problms involving Mawll s quations in isotropic mdia, I Trans. Microwav Thor and Tchniqus, vol., pp. 6 69, 966. Lctur Slid 7 Stro Imag of Y Cll To viw th Y cll if full D, loo past th imag abov so that th appar doubl. Whn th doubl imags ovrlap so that ou s thr Y clls, th middl imag will b thr dimnsional. Lctur Slid 8

5 Y Cll for D, D, and D Grids D Y Grids Mod Mod * Ths ar th sam for isotropic matrials. D Y Grids Mod Mod D Y Grid Lctur Slid 9 Rasons to Us th Y Grid. Divrgnc fr. lgant arrangmnt to approimat curl quations. Phsical boundar conditions ar naturall satisfid Lctur Slid 5

Consquncs of th Y Grid Th fild componnts ar at phsicall diffrnt positions. This has th following consquncs Fild componnts within th sam grid cll can rsid in diffrnt matrials.

6 Consquncs of th Y Grid Th fild componnts ar at phsicall diffrnt positions. This has th following consquncs Fild componnts within th sam grid cll can rsid in diffrnt matrials. multiplis so ths functions should b mad to ist at th sam positions across th grid. multiplis so ths functions should b mad to ist at th sam positions across th grid. multiplis so ths functions should b mad to ist at th sam positions across th grid., and ar uniqu arras and must b constructd sparatl. Fild componnts within th sam grid cll will b slightl out of phas. This must b accountd for whn constructing sourcs and whn postprocssing th fild data. Th grid causs numrical disprsion whr wavs propagat slowr than a phsical wav would. Lctur Slid Visualiing tndd Y Grids Grid Grid ( Mod) i j Lctur Slid 6

7 7 Lctur Slid Finit Diffrnc Approimations of Mawll s quations on a Y Grid Lctur Slid Starting Point W start with Mawll s quations in th following form. r w hav rtaind diagonall anisotropic matrial tnsors. This will b ndd to incorporat a prfctl matchd lar boundar condition. Ths quations ar valid indpndnt of th chosn sign convntion.

8 8 Lctur Slid 5 Normali th Grid Coordinats Th grid is normalid according to This absorbs th trm into th spatial drivativs and simplifis Mawll s quations to Lctur Slid 6 Finit Diffrnc quation for,, i j,, i j,, i j,, i j,, i j,,,,,,,,,,,, i j i j i j i j i j j i

9 Finit Diffrnc quation for i, i, Lctur Slid 7 Finit Diffrnc quation for,, i j i, i, j, j, Lctur Slid 8 9

10 Finit Diffrnc quation for j, j, Lctur Slid 9 Finit Diffrnc quation for i, i, Lctur Slid

11 Finit Diffrnc quation for i, j, i, j, Lctur Slid Summar of Finit Diffrnc Approimations of Mawll s quations j, i, i, j, j, i, i, j, Lctur Slid

12 Two Dimnsional Analsis For D problms, th dvic is uniform and infinit in th dirction and wav propagation is rstrictd to th plan. Lctur Slid Mawll s quations for D Analsis Lt th uniform dirction b th ais and wav propagation b rstrictd to th - plan. For this cas, and th finit diffrnc quations simplif. j, i, i, j, j,,,,,,,,,,,,, i j i j i j i j i j i j j, j, j, i, i, j, j, i, j, i, j, Lctur Slid

13 Two Mods for D Analsis Mawll s quations split into two sts of thr coupld quations. j, i, i, j, j, i, i, j, j, i, j, j, j, i, j, i, j, j, i, j, Mod Mod Lctur Slid 5 Matri Form of Mawll s quations Lctur Slid 6

14 Rcall That Filds ar Stord in Column Vctors D Sstms 5 5 D Sstms Lctur Slid 7 Matri Rprsntation of Point b Point Multiplication ( of ) 5 r r r r r 5 ri, i r r r r r5 5 r r r r ε r r r r r r5 5 r5 5 Lctur Slid 8

15 Matri Rprsntation of Point b Point Multiplication ( of ) 5 r r r r r 5 ri, i r r r r r5 5 r r r r ε r r r r r r5 5 r5 5 Lctur Slid 9 Drivativ Oprators for lctric Filds ( of ) 5 i i i D Lctur Slid 5

16 Drivativ Oprators for lctric Filds ( of ) 5 i i i D Lctur Slid Drivativ Oprators for Magntic Filds ( of ) 5 i i i 5 h Dh Lctur Slid 6

17 Drivativ Oprators for Magntic Filds ( of ) 5 i i i 5 h Dh Lctur Slid Simplst Boundar Conditions Dirichlt Boundar Conditions Assum Priodic Boundar Conditions D Assum 6 5 D Lctur Slid 7

18 Drivativ Oprators on a Grid Using Dirichlt Boundar Conditions D Dirichlt boundar conditions N D h D Dirichlt boundar conditions N h D Not: Ths matrics hav onl two diagonals so th ar vr as to construct! Lctur Slid 5 D Drivativ Oprators for D Grids Whn N= and N> D D Z D h and D h ro matri is standard for D grid D h D Whn N> and N= h D and D h D D Z is standard for D grid ro matri Not: W will do somthing diffrnt whn w account for obliqu angl of incidnc. h D Lctur Slid 6 D 8

19 Si of Drivativ Oprators D Grids (N ) If our grid has N points, our matrics will b N N with a total of N lmnts. D Grids (N ) If our grid has N N points, our matrics will b N N N N with a total of D Grids (N 6 ) NN NNN lmnts. If our grid has N N N points, our matrics will b N N N N N N with a total of lmnts. Lctur Slid 7 US SPARS MATRICS!!!!!!! Th drivativ oprators will b XTRMLY larg matrics. For a small grid that is just points: Total Numbr of Points:, Si of Drivat Oprators:,, Total lmnts in Matrics:,, Mmor to Stor On Full Matri: 6 Gb Mmor to Stor On Spars Matri: Mb NVR AT ANY POINT should ou us FULL MATRICS in th finit diffrnc mthod. Not vn for intrmdiat stps. NVR! Lctur Slid 8 9

20 Drivativ Oprators on a Grid Using Dirichlt Boundar Conditions D Dh D Dh Not: Ths matrics hav onl two diagonals so th ar vr as to construct! Lctur Slid 9 D and h D on a Y Grid D D h Lctur Slid

21 D and h D on a Y Grid D D h Lctur Slid D and h D on a Y Grid D D h Lctur Slid

22 Rlationship Btwn th Drivativ Oprators Transpos Opration T A A j i A T, A AT = transpos(a); AT = A. ; T A, A a a a T a A, a a A a a rmitian (Conjugat) Transpos Opration A * A j i * * a a a a A, * * a a A a a j j j j A, A j j j j AT = ctranspos(a); AT = A ; Rlationship Btwn th Drivativ Oprators h D D DX = -DX ; h D D DY = -DY ; This mans ou onl hav to construct drivativ oprators for th lctric fild. Th drivativ oprators for th magntic fild can b computd dirctl from th lctric fild drivativ oprators. This rlation dos not hold for som boundar conditions such as Numann. Lctur Slid What About th Scond Ordr Drivativs? Rcall from Lctur 5, Slid 7 that D D was a poor approimation of D. D D D What about th drivativ oprators drivd from th Y grid? h D D h DD Lctur Slid Th numbrs is this matri ma diffr slightl from th idal nd ordr drivat oprator du to th boundar conditions.

23 Mawll s quations in Matri Form j, i, i, j, D D μ h D D μ h D D μ h j, i, i, j, D h D h ε h h D h D h ε h h D h D h ε h h Lctur Slid 5 Summar of What W Did No chargs j j Normalid r r Normalid grid & diagonal tnsors Finit diffrnc approimation, j, i,, i j Matri form of Mawll s quations D D μ h D D μ h D D μh j, i, i, j, D h D h ε h h Dh Dh ε h h Dh Dh ε h h Lctur Slid 6

24 Numrical Disprsion Lctur Slid 7 Disprsion on a Y Grid Rcall th disprsion rlation for an isotropic matrial with paramtrs r and r. c r r Th analogous disprsion rlation on a frqunc domain Y grid filld with r and r is r r sin sin sin v In this quation, th spd of light c is writtn as v bcaus th vlocit changs du to th disprsion of th grid. Lctur 8

25 Drawbacs of Structurd Grids ( of ) Structurd grids hibit high anisotropic disprsion. Anisotropic Disprsion FDM Lctur Slid 9 Compnsation Factor ( of ) Th numrical disprsion quation is solvd for vlocit v. v rr sin sin sin In th absnc of grid disprsion, v should b actl th spd of light c. Du to th Y grid, wavs slow down b a factor. v c W can calculat this factor b combining th abov quations. c sin sin sin r r Lctur 5 5

26 Compnsation Factor ( of ) W can writ a simplr and mor usful prssion for. sin sin sin n n r r Lctur 5 Compnsating for Numrical Disprsion Givn that th wav slows down b factor in th dirction of, it follows that w can compnsat for th disprsion b artificiall spding up th wav. W do this b dcrasing th valus of r and r across th ntir grid b a factor of. r r r Nots:. W can onl compnsat for disprsion for on dirction.. W can onl compnsat for disprsion in on st of matrial valus r and r.. It is bst to choos avrag or dominant valus for ths paramtrs.. Choos =.5 if nothing ls is nown. r.5 Lctur 5 6

27 7 Lctur Slid 5 Gnraliation to Full Anisotropic Matrials Lctur Slid 5 Rtain Anisotropic Trms Our analtical quations with just diagonall anisotropic matrials wr For full anisotropic matrials, ths ar now

28 First Guss at Finit Diffrnc Approimations j, i, i, j,?? j,? i, j, i, j, j, j,?? i, i, j,? i,, i, j, j, i j, j, i, j, j, Lctur Slid 55 Th Problm j, i, i, j, j, j, i, i, j, j,,,,, j, j, i j i j ach trm in a finit diffrnc approimation MUST ist at th sam position in spac. Th bod trms ist at phsicall diffrnt locations than th othr trms. Lctur Slid 56 8

29 Th Corrction i, i, i, j, j, i, j, i, j, j, j, i, j, i, i, i, i, j, j, j, j, j, i,, j,, i j i i, i i j,,,, j, j i j,,,,, i, j i i i,,, i i,,, j i j, j, j, j, j, i, j, i, j, i, i, i,,, i j i, i, j, j, i, j, i, j, i, i,,, j,,, j, i, j, j, j, i, j, i, j, i, j, i, j, j, j, i, j, j, j, j, i, i, i j i j j, W ar forcd to intrpolat th problm trms so th ist at th sam positions as th othr trms in th finit diffrnc quations. W intrpolat th products and so that th fild and matrial valu bing multiplid ar at th sam points. Lctur Slid 57 Clos Up of j,,,,, i, i, i, i, j, j, j, i, j, i, j, i j i j i, j, i, j, i, j, i, i, j, j, i, j, i, j, j, j, j, j, j, j, j,,, j,,, j,, j i, j, i, i, i, i, j, i j i j i j,, Lctur Slid 58 9

30 Clos Up of j, j,,, j,,, i, j,, i, j, j, j, i, j, i, j, i, i, j, i j i j i j, i, j,,,,,,, i, j, i j j, i j i, j, i j i, j, j, j, j, i, j, i, i, i, i, j, j, j, j, j, Lctur Slid 59 Matri Form i, i, i, j, j, i, j, i, j, j, j, i, j, i, i, i, i, j, j, j, j, j, i,, j, i i i i i i i i,,, j, j i j,,,,, j,,,, i, j, i i i, i, i,,, i j, j, D D μ h R R μ h R R μ h D D R R μ h μ h R R μ h D D R R μ h R R μ h μ h j, j, j, i, j, i, j, i, i, i,,, i j i, i, j, j, i, j, i, j, i, i,,, j,,, j, i, j, i, i, i, i, i, j, j, j, j, j, j, i, i, i j i j j, D h D h ε R R ε R R ε h h D h D h R R ε ε R R ε h h D h D h R R ε R R ε ε h h Lctur Slid 6

31 Bloc Matri Form ( of ) D D μ h R R μ h R R μ h D D R R μ h μ h R R μ h D D R R μ h R R μ h μ h D D μ RRμ RRμ h D D RRμ μ RRμ h D D RRμ RRμ μ h D h D h ε R R ε R R ε h h D h D h R R ε ε R R ε h h D h D h R R ε R R ε ε h h D D h ε R R ε R R ε h h h h D Dh RRε ε RRε h D D h RRε RR ε ε Lctur Slid 6 Bloc Matri Form ( of ) W can writ our two bloc matri quations as h h h ε C μr C h h h h r Nots:. W can handl anisotropic matrials just b modifing th matrial matrics.. W do this b incorporating intrpolation matrics.. h C C h h D D D D h h C D D C D D h h D D D D ε RRε RRε μ RRμ RRμ ε r RRε ε RRε μ r RRμ μ RRμ RRε RRε ε RRμ RRμ μ Lctur Slid 6

32 Intrpolation Matrics ( of ) Th drivativ oprators wr constructd from a simpl finitdiffrnc approimation of th form f.5 f f Th intrpolation matrics ar constructd actl th sam wa, but uss th following quation for intrpolation: f.5 f f Th intrpolation matrics hav th following intrprtations: R R i i intrpolats along th i-ais using a valu from th nt cll along i intrpolats along th i-ais using a valu from th prvious cll along i Lctur Slid 6 Intrpolation Matrics ( of ) Thin of forming th intrpolation this wa R D R D R D Strictl spaing, this will not wor bcaus th opration bras som boundar conditions. This calculation approach dos wor for Dirichlt boundar conditions and for priodic boundar conditions that do not includ phas. Th intrpolation matrics ar rlatd through: R R R R R R Lctur Slid 6

33 Altrnativ Grids Lctur Slid 65 Drawbacs of Uniform Grids Uniform grids ar th asist to implmnt, but do not conform wll to arbitrar structurs and hibit high anisotropic disprsion. Anisotropic Disprsion (s Lctur ) Staircas Approimation (s Lctur 8) Lctur Slid 66

34 Drawbacs of Structurd Grids ( of ) Structurd grids ar th asist to implmnt, but do not conform wll to arbitrar gomtris. Structurd Grid Unstructurd Grid Lctur Slid 67 agonal Grids agonal grids ar good for minimiing anisotropic disprsion suffrd on Cartsian grids. This is vr usful whn tracting phas information. Phas Vlocit as a Function of Propagation Angl Y FDTD FDTD S Tt, pp.. Lctur Slid 68

Nonuniform Orthogonal Grids ( of ) Nonuniform orthogonal grids ar still rlativl simpl to implmnt and provid som abilit to rfin th grid at localid rgions. S Tt, pp. 6 7.

35 Nonuniform Orthogonal Grids ( of ) Nonuniform orthogonal grids ar still rlativl simpl to implmnt and provid som abilit to rfin th grid at localid rgions. S Tt, pp Lctur Slid 69 Nonuniform Orthogonal Grids ( of ) Uniform Grid Simulation 8 6 clls,8 clls Nonuniform Grid Simulation clls 77,8 clls Conclusion: Roughl 5% mmor and tim savings. Lctur Slid 7 5

36 Curvilinar Coordinats Mawll s quations can b transformd from curvilinar coordinats to Cartsian coordinats to conform to curvd boundaris of a dvic. S Tt, pp M. Fusco, FDTD Algorithm in Curvilinar Coordinats, I Trans. Ant. and Prop., vol. 8, no., pp , 99. Lctur Slid 7 Structurd Nonorthogonal Grids This is a particularl powrful approach for simulating priodic structurs with obliqu smmtris. M. Fusco, FDTD Algorithm in Curvilinar Coordinats, I Trans. Ant. and Prop., vol. 8, no., pp , 99. Lctur Slid 7 6

Irrgular Nonorthogonal Unstructurd Grids Unstructurd grids ar mor tdious to implmnt, but can conform to highl compl shaps whil maintaining good cll aspct ratios and global uniformit.

37 Irrgular Nonorthogonal Unstructurd Grids Unstructurd grids ar mor tdious to implmnt, but can conform to highl compl shaps whil maintaining good cll aspct ratios and global uniformit. Comparison of convrgnc rats ln P. arms, J. L, R. Mittra, A Stud of th Nonorthogonal FDTD Mthod Vrsus th Convntional FDTD Tchniqu for Computing Lctur Rsonant Frquncis of Clindrical Cavitis, I Trans. Microwav Thor and Tchniq., vol., no., pp. 7 76, 99. Slid 7 Bodis of Rvolution (Clindrical Smmtr) Thr dimnsional dvics with clindrical smmtr can b vr fficintl modld using clindrical coordinats. Dvics with clindrical smmtr hav filds that ar priodic around thir ais. Thrfor, th filds can b pandd into a Fourir sris in. vn m m m,,, cos, sin h m h m,,, cos, sin m vn odd odd Du to a singularit at r=, updat quations for filds on th ais ar drivd diffrntl. S Tt, Chaptr Lctur Slid 7 7

$Diffractiv$ Lnss Focusing

38 Som Dvics with Clindrical Smmtr Bnt Wavguids Clindrical Wavguids Dipol Antnnas Conical orn Antnna Diffractiv Lnss Focusing Antnnas Lctur Slid 75 8

Lecture Outline. Skin Depth Power Flow 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3e

Lecture Outline. Skin Depth Power Flow 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3e 8/7/018 Cours Instructor Dr. Raymond C. Rumpf Offic: A 337 Phon: (915) 747 6958 E Mail: rcrumpf@utp.du EE 4347 Applid Elctromagntics Topic 3 Skin Dpth & Powr Flow Skin Dpth Ths & Powr nots Flow may contain