FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

Transcription

1 FINIT DIFFRNC TIM DOMAIN MTOD (FDTD) The FDTD method, proposed b Yee, 1966, is aother umerical method, used widel for the solutio of M problems. It is used to solve ope-regio scatterig, radiatio, diffusio, microwave circuit modellig, ad biomedical etc. problems. Oe of the most importat cocers of the FDTD method is the requiremet of the artificial mesh trucatio (boudar) coditios. These coditios are used to trucate the solutio domai ad the are kow as absorbig boudar coditios (ABCs), as the theoreticall absorb fields. The space domai icludes the object ad it is termiated b Absorbig Boudar Coditios (ABCs). Imperfect ABCs create reflectios ad the accurac of the FDTD method depeds o the accurac of the ABCs. The followig advatages make the FDTD method popular: It s a direct solutio of Mawell s equatios, o itegral equatios are required ad o matri iversios are ecessar. Its implemetatio is eas ad it is coceptuall simple. It ca be applied to the three-dimesioal, arbitrar geometries. It ca be applied to materials with a coductivit. The FDTD method has also the followig disadvatages: Whe the FDTD method is applied, the object ad its surroudigs must be defied. Sice computatioal meshes are rectagular i shape it is difficult to appl the method to the curved scatterers. It has low order of accurac ad stabilit uless fie mesh is used.

2 Basic Fiite-Differece Time-Domai Algorithm I this method the coupled Mawell s curl equatios i the differetial form are discretized, approimatig the derivatives with two poit cetred differece approimatios i both time ad space domais. The si scalar compoets of electric ad magetic fields are obtaied i a time-stepped maer. B X X J. D. B 0 v D I liear, isotropic ad homogeeous materials D ad B are related to ad with the followig costitutive relatios: D B Also J is related to as, J Substitutig the costitutive relatios ito the Mawell s equatios, we ca write si scalar equatios i the Cartesia coordiate sstem.

3 lectric Field Itesit: 1 z t z 1 z t z z 1 z t Magetic field itesit: 1 z t z 1 z t z z 1 t Yee Algorithm Yee algorithm solves for both electric ad magetic fields i time ad space usig the coupled Mawell s curl equatios. The Yee algorithm ceters its ad field compoets i threedimesioal space so that ever compoet is surrouded b four circulatig compoets, ad ever compoet is surrouded b four circulatig compoets.

4 Yee s Uit Space Lattice Cell: z z z z The computatioal domai is divided ito a umber of rectagular uit cells. t Accordig to Yee algorithm ad field compoets are separated b i time. 2

5 t=0 =0 = =2 =3 t=0.5 t= Fiite Differeces (Discretizatio) A space poit i a uiform rectagular lattice is deoted as: ( i, j, k) ( i, j, kz) ere, ad z are the lattice space icremets i the,, ad z coordiate directios respectivel ad i, j, ad k are itegers. If a scalar fuctio of space ad time evaluated at a discrete poit i the grid ad at a discrete poit i time is deoted b u, the; u( i, j, kz, ) u i, j, k Usig cetral, fiite differece approimatio i space, i.e. w.r.t. : u u u i j k z t O 1,, 1,, 2 i j k i j k 2 2,,,

6 Usig cetral, fiite differece approimatio i time: i, j, k i, j, k,,, Scalar equatios are discretized as: lectric Field Itesit: u u u i j k z t O t t z i, j, k i, j, k z i, j 1, k z i, j, k i, j, k 1 i, j, k z i, j, k i, j, k i, j, k 1 i, j, k z i 1, j, k z i, j, k z i, j, k z i, j, k i 1, j, k i, j, k i, j 1, k i, j, k 2 Magetic Field Itesit: z i, j, k i, j, k i, j, k i, j, k 1 z i, j, k z i, j 1, k z i, j, k i, j, k z i, j, k z i 1, j, k i, j, k i, j, k z i, j, k z i, j, k i, j, k i, j 1, k i, j, k i 1, j, k Oe dimesioal free space formulatio:

7 Assume a plae wave with the electric filed itesit havig compoet, magetic field itesit havig compoet ad travelig i the z directio. Mawell s quatios become: z 0 z Takig cetral differece approimatio for both temporal ad spectral derivatives: 1 ( k) ( k) 1 0 ( k 1) ( k) 1/2 1/2 1/2 1/2 ( k) ( k) 1 ( k) ( k 1) z 0 z Update equatios: ( k) ( k) ( k 1) ( k) 1 1/2 1/2 0z ( k) ( k) ( k) ( k 1) 1/2 1/2 0z Notice that:

8 meas a time t t, The calculatios are iterleaved i both time ad space. For eample the ew value of is calculated from the previous value of ad the most recet values of. Writig the epressios of ad i Matlab computer code: e(k)=e(k)-(dt/(eps0*dz))*(h(k+1)-h(k)); h(k)=h(k)-(dt/(mu0*dz))*(e(k)-e(k-1)); Note that, +1/2, -1/2 superscripts are igored. Also ote that k+1/2 ad k-1/2 are rouded off i order to specif a positio i a arra i the program. Writig a Matlab program code: Calculate field b usig a loop. Calculate the source. i. e. the iitial coditio. Geerate a Gaussia pulse i the ceter of the problem space. Appl the B.C. to fid the values at the boudaries of the problem space. Calculate the fields b usig a loop. Geeratio of the Gaussia pulse: pulse=ep(-((*dt-t0)^2)/(t1^2)); e(ks)=pulse+ e(ks);

9 The stabilit criteria: c where is the dimesio umber of the simulatio. For oe dimesioal simulatio: c clear all K=200; c=3e8; for k=1:k e(k)=0; h(k)=0; ed dz=0.1; % Stabilit criteria dt=dz/(c); % Iitialize the costats eps0=8.85e-12; mu0= e-6; Stop=250; dtedz=dt/(eps0*dz); dtmdz=dt/(mu0*dz); % time steppig for =1:Stop

10 % save the field at k=25 ad k=50 for all time steps e25()=e(25); e50()=e(50); %update electric field for k=1:k-1 e(k)=e(k)-dtedz*(h(k+1)-h(k)); % save the field at =10 for the whole space domai if(==10) ed ed et(k)=e(k); %source %Appl sie wave ecitataio %e(1)=si(2*3.14*3*10^7**dt); % Appl Gaussia pulse ecitataio e(1)=ep(-(-30)^2/100); %update magetic field for k=2:k h(k)=h(k)-dtmdz*(e(k)-e(k-1)); ed ed

11 Refereces [1] M.N.O. Sadiku, Numerical Techiques i lectromagetics, CRC Press, 2001, pp [2] Taflove, Computatioal lertodamics: the Fiite- DifferecTime-Domai Method, Artech, 1995 [3] Taflove, S.C. agess, same as above, 2d ed., Artech, 2000 [4] K. Kuz ad R. Luebbers, Fiite-Differece Time-Domai Method for lectromagetics, CRC Press, 1993 [5] Kae S. Yee, Numerical solutio of iitial boudar value problems ivolvig Mawell s equatios i isotropic media, I Tras. Ateas Propagat., vol. AP-14, No. 3, pp , Ma,1966.