EE757 Numerical Techniques in Electromagnetics Lecture 8

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1 757 Numerical Techiques i lecromageics Lecure 8

2 2 757, 206, Dr. Mohamed Bakr 2D FDTD e i J e i J e i J T TM

3 3 757, 206, Dr. Mohamed Bakr T Case wo elecric field compoes ad oe mageic compoe e i J e i J

4 T Case (Co d) 2 2 (, ) (, ) i j i j 2 ( i, j) Cee ( i, j) ( i, j) Ceh ( i, j) Ji ( i, j) 2 2 (, ) (, ) i j i j 2 ( i, j) Cee ( i, j) ( i, j) Ceh ( i, j) Ji ( i, j) (, ) (, ) (, ) (, ) 2 2 i j i j i j i j ( i, j) ( i, j) Che ( i, j) (/) 0 i 3D updae equaios 757, 206, Dr. Mohamed Bakr 4

5 5 757, 206, Dr. Mohamed Bakr TM Case e i J

6 TM Case (Co d) ( i, j) C ( i, j) ( i, j) ee ( i, j) ( i, j) (, ) (, ) i j i j 2 Ceh ( i, j) Ji ( i, j) (, ) (, ) 2 2 i j i j (, ) (, ) he(, ) i j i j C i j (, ) (, ) 2 2 i j i j he ( i, j) ( i, j) C ( i, j), (/) 0 i 3D updae equaios 757, 206, Dr. Mohamed Bakr 6

7 7 757, 206, Dr. Mohamed Bakr D Mawell s quaios, e i J, e i J. +ve -ve

8 D FDTD 2 2 ( ) ( ) i i 2 ( i) Cee ( i) ( i) Ceh ( i) Ji ( i), ( ) ( ) 2 2 i i ( i) ( i) Che ( i) se (/)0 ad (/)0 i 3D updae equaios 757, 206, Dr. Mohamed Bakr 8

9 D FDTD (co d) 2 2 ( ) ( ) i i 2 ( i) Cee ( i) ( i) Ceh ( i) Ji ( i) ( ) ( ) 2 2 i i ( i) ( i) Che ( i) se (/)0 ad (/)0 i 3D updae equaios 757, 206, Dr. Mohamed Bakr 9

10 The Coura-Friedrich-Lev (CFL) Limi c if h, h c 3 he FDTD ime-marchig scheme becomes usable if he ime sep eceeds he Coura limi usuall, we choose 0.9 CFL CFL for 2D ad D FDTD? 757, 206, Dr. Mohamed Bakr 0

11 Boudar Codiios PC PMC Absorbig Boudar Codiios Mur s Firs-order boudar codiio Mur s Secod-order boudar codiio Liao s boudar codiio Iroducio o PML 757, 206, Dr. Mohamed Bakr

12 PC: T Case se all ageial -field compoes a he boudar o ero for all ime seps FDTD updae equaios are applied ol o ierior elecric ad mageic field compoes PC 757, 206, Dr. Mohamed Bakr 2

13 PC: TM Case se he compoes a he elecrical wall o ero 757, 206, Dr. Mohamed Bakr 3

14 PMC where o pu he boudar mageic walls? 757, 206, Dr. Mohamed Bakr 4

15 PMC (Co d) half a cell awa from he boudar PMC 757, 206, Dr. Mohamed Bakr 5

16 Mur s Boudar Codiios iiial work : B. gquis ad A. Majda, Absorbig boudar codiios for he umerical simulaio of waves, Mahemaics of Compuaio, vol. 3, 977, pp sarig from he 3D wave equaio f f f f c L Lf c c wih respec o ever dimesio, he operaor L is decomposed io wo operaors. 757, 206, Dr. Mohamed Bakr 6

17 Mur s Boudar (Co d) wih respec o he -dimesio, he wave operaor is decomposed o LfL + L - f, where L S 2 c S, c S 2 2 L 2 2 c c he operaors L + ad L - are pseudo-differeial operaors ad cao be applied direcl o a fucio L - (f)0 represes a wave ravelig alog L + (f)0 represes a wave ravelig alog + Talor epasio is used o approimae hese operaors 757, 206, Dr. Mohamed Bakr 7

18 Firs-order Mur Boudar Codiio For a firs-order approimaio we use 2 he parial derivaives wih respec o ad are assumed ver small his is he case for a ormall icide plae wave L c, L c S a 0, we impose he codiio f f c 0 a ma, we impose he codiio f f c 0 757, 206, Dr. Mohamed Bakr 8

19 Illusraio of s order Mur s ABC for D a he lef boudar, we impose he oe-wa codiio ( ) ( ) ( c ) ( ) ( ) (0) () () (0) ( c ) he +ve wave operaor is used o derive he boudar codiio a ma 757, 206, Dr. Mohamed Bakr 9

20 Secod-order Mur s boudar codiios for he secod order Mur, we use he approimaio L L L S S c S c c c 2 c c c 2 2 c f+ f f f - f+ 0.5c f+ f 0 757, 206, Dr. Mohamed Bakr 20

21 Secod-order Mur (Co d) (0,j,k+) (0,j,k) (0,j+,k) (,j,k) 2 f 2 f f, j, k f 0, j, k f, j, k f 0, j, k /2, jk, 2 f 2f f f 2f f , j, k 0, j, k 0, j, k, j, k, j, k, j, k /2, jk, 757, 206, Dr. Mohamed Bakr 2

22 Secod-Order Mur (Co d) ad derivaives ca be igored o ield a simpler epressio 757, 206, Dr. Mohamed Bakr 22

23 Refereces [] A. lsherbei ad V. Demir, The Fiie Differece Time Domai Mehod for lecromageics wih MATLAB Simulaios, ACS Series o Compuaioal lecromageics ad gieerig, SciTech Publishig Ic. a Impri of he IT, Secod diio, diso, NJ, 205. [2] R.C. Booo, Compuaioal Mehods for lecromageicsad Microwaves, Wile, 992, pp [3] M.N.O. Sadiku, Numerical Techiques i lecromageics, CRC Press, 200, pp [4] W. Yu e al., Parallel Fiie Differece Time Domai Mehod, Arech, , 206, Dr. Mohamed Bakr 23

24 Refereces [5] A. Taflove, Compuaioal lerodamics: he Fiie- Differece Time-Domai Mehod, Arech, 995 [6]A. Taflove ad S.C. agess, same as above,2ded., Arech, 2000 [7] K. Ku ad R. Luebbers, Fiie-Differece Time-Domai Mehod for lecromageics, CRC Press, 993 [8] K.S. Yee, Numerical soluio of iiial boudar value problems ivolvig Mawell s equaios i isoropic media, I Tras. Aeas Propaga., vol. AP-4, No. 3, pp , Ma , 206, Dr. Mohamed Bakr 24

An Efficient Method to Reduce the Numerical Dispersion in the HIE-FDTD Scheme

An Efficient Method to Reduce the Numerical Dispersion in the HIE-FDTD Scheme Wireless Egieerig ad Techolog, 0,, 30-36 doi:0.436/we.0.005 Published Olie Jauar 0 (hp://www.scirp.org/joural/we) A Efficie Mehod o Reduce he umerical Dispersio i he IE- Scheme Jua Che, Aue Zhag School