MULTIRESOLUTION TIME-DOMAIN ANALYSIS OF MICROWAVE STRUCTURES USING MATLAB. Ondřej Franek, Zbyněk Raida

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1 MULTIRSOLUTION TIM-DOMAIN ANALYSIS OF MICROWAV STRUCTURS USING MATLAB Odřej Frek, Zběk Rid Deprtmet of Rdio lectroics, Bro Uiversit of Techolog Abstrct I this pper, umericl lsis of microwve structures usig multiresolutio time-domi (MRTD) method i MATLAB eviromet is preseted. Commol used sclig d wvelet orthogol sstems re itroduced. Correspodig lgorithm d its implemettio i the form of m-file re described. Fill, ccurc d computtiol demds of the proposed progrm re discussed. Itroductio Nowds, the most exploited method for time-domi umericl lsis of microwve structures is the well-kow fiite-differece time-domi (FDTD) method [1]. It is ccurte eough for wide vriet of problems d t the sme time strightforwrd to implemet, which ws successfull demostrted i []. This techique hs, however, certi limittios i describig fst vritios of the electromgetic field i the computtiol domi. I such cses, sufficiet degree of ccurc is reched ol b utilizig ver fie meshes, which leds to high computer memor demds. The multiresolutio time-domi (MRTD) method uses expsio of field distributio ito sclig fuctios d wvelet fuctios of the sme d higher resolutio levels [3]. Usig the wvelet bsis ebles us to properl describe the res where the field chges rpidl. Nevertheless, the computer burde does ot icrese sigifictl, s most of the wvelet coefficiets re smll eough to be eglected. The implemettio of the MRTD bsic lgorithm i MATLAB eviromet is object of this pper. Theor The electromgetic field i the computtiol domi, represeted b pproximted b geerll ifiite series of sclig d wvelet fuctios, f (x), c be f ( x) = = m 0 ( x) dmψ m( x) m= m0 1 = c φ, (1) where φ d ψ represet the sclig d wvelet fuctios respectivel, with s coordite of trsltio d m s wvelet resolutio level ( m 0 is the primr resolutio level). From (1) it c be see tht the pproprite choice of the sclig d wvelet fuctios ffects the precisio of field pproximtio. The tpicl sclig d wvelet orthogol sstems re described below. r sstem sclig fuctio i spce or time domi is pulse fuctio, while the Fourier trsform is the well-kow sic fuctio. This bsis is loclized i spce or time but ot i the spectrum, which ields egtive effects t discretiztio. Usig r sclig fuctios ol we obti the FDTD method, which is ppretl specil cse of MRTD. Sho sstem is the opposite of r sstem, s its spectrum is formed b pulse fuctio d the origil domi expsio is doe b the sic. Sho bsis is perfectl smooth but ot loclized i spce or time.

2 Bttle-Lemrie sstem is compromise betwee the two precedig sstems: it is firl loclized i both spce/time d frequec domis. I the followig, lgorithm bsed o Bttle-Lemrie orthogol sstem will be described. Ol the sclig fuctios were icorported for simplicit. Further theoreticl bckgroud of the MRTD method c be foud i [3]. Formultio The formultio of MRTD is bsed o the first two Mxwell equtios, where the derivtives re expressed from the expsio (1). The Bttle-Lemrie sstem ws used for expsio i spce domi, while pulse fuctios were utilized i time domi, which sves the mout of storge eeded. For exmple, the two-dimesiol cse with TM mode is cosidered. The multiresolutio time-domi equtio sstem for lossless medi c the be writte s [3] t j 1 x =, 1/ x ) i j i, j 1/ µ = j z i,, () i 1/, j = 1/ i 1/, j t µ i 1 i' = i i') z i', j, (b) i 1 j t = z i') i, j ε i' 1/, j i' = i = j 1 1 z ) i, j x i, 1/. (c) ere, x, d z re the ivolved mgetic d electric field compoets, d t re, respectivel, the spce d time steps d µ with ε re the mteril prmeters (permebilit d permittivit). Idices i, j deote the spce coordites d deotes the time coordite. Fill, the α coefficiets relte to the sclig fuctios expsio with s mximum of the terms icorported i the lgorithm (idell ifiite umber). Applictio The described lgorithm ws implemeted i the form of MATLAB fuctio. The progrm ws highl vectorized i order to chieve the best performce i MATLAB. The progrm kerel for D TM mode () is writte s follows: % mtrix idices preprocessig id = 1:(xmx*mx); id = reshpe(id,xmx,mx);... for t = 1:Nt, % time-mrchig ccle % idices iitiliztio id1 = [id(1:xmx-1,:) ; -id(xmx-1,:)]; id = [id(1,:) ; -id(1:xmx-1,:)]; id3 = [id(:,1:mx-1) -id(:,mx-1)]; id4 = [id(:,1) -id(:,1:mx-1)]; for i = 1:, % lph ccle z = z C1(i).*( (bs(id1)) - (bs(id))... - x(bs(id3)) x(bs(id4)) );

3 ed; id1 = id1 (id1>0); msk = ismember(bs(id1),pc); id1(msk) = -id1(msk); id1 = id1 (id1<0); id = id (id>0); msk = ismember(bs(id),pc); id(msk) = -id(msk); id = id (id<0); id3 = id3 xmx.*(id3>0); msk = ismember(bs(id3),pc); id3(msk) = -id3(msk); id3 = id3 xmx.*(id3<0); id4 = id4 xmx.*(id4>0); msk = ismember(bs(id4),pc); id4(msk) = -id4(msk); id4 = id4 xmx.*(id4<0); z(pc) = 0; % PC boudr coditio... % idices iitiliztio id1 = [id(:xmx-1,:) ; -id([xmx xmx],:)]; id = [id(1,:) ; -id(:xmx,:)]; id3 = [id(:,:mx-1) -id(:,[mx mx])]; id4 = [id(:,1) -id(:,:mx)]; for i = 1:; % lph ccle ed; ed; = C(i).*( sig(id1).*z(bs(id1))... sig(id).*z(bs(id)) ); x = x - C(i).*( sig(id3).*z(bs(id3))... sig(id4).*z(bs(id4)) ); id1 = id1 1; msk = ismember(bs(id1),pc); id1(msk) = -id1(msk); id = id 1; msk = ismember(bs(id),pc); id(msk) = -id(msk); id3 = id3 xmx; msk = ismember(bs(id3),pc); id3(msk) = -id3(msk); id4 = id4 xmx; msk = ismember(bs(id4),pc); id4(msk) = -id4(msk);

4 ere, the C1 d C coefficiets correspod to the lph coefficiets combied with time d spce steps d mteril properties (), which miimizes the mout of rithmeticl opertios. The idices chge with the coefficiets to properl model the boudr coditio mde of perfect electric coductor (lout icluded i PC), ccordig to the imge priciple. Mximum memor svigs re chieved b chgig the idices i ccle d usig their sig to idicte the sese of icremettio d polrit of imge. ve more storge c be reserved b usig less extesive dt tpes such s it16, lthough the progrm is the firl complicted. Results The costructed D code ws used to lze the wvemode frequecies of microwve trsmissio lie, prticulrl the shielded ir-strip lie lzed i []. Cross sectio of the structure is displed i Fig mm 1.7 mm 1.7 mm 1.7 mm Fig. 1: Cross sectio of the lzed shielded ir-strip lie The correspodig mgitude spectrum obtied through the Fourier trsform (prticulrl b the FFT) is show i Fig.. The re of trsverse sectios ws discretized with 0 cells per side. The problem ws solved i frequec rge from 0 to 50 Gz, with 10 Mz resolutio f [Gz] Fig. : Frequec spectrum for shielded ir-strip lie, 0 cells per side

5 The obtied results ppered to be more ccurte th with the origil FDTD method used for compriso, but, o the other hd, the ruig time ws scrificed. The totl computtio time ws mi., wheres the secod order FDTD method mged to compute the sme structure withi 3 sec, eve with higher frequec resolutio d o slower mchie. Coclusio This pper follows the work [] i provig tht eve the multiresolutio time-domi method c be implemeted i MATLAB with dvtge. This fct hs bee demostrted o the two-dimesiol TM mode cse, the T mode or the full 3D cse c be obtied similrl, however. The lgorithm c lso be slightl modified to ccommodte the wvelet fuctios d, i this w, to improve the ccurc where eeded. Ackowledgemet This work ws ficill supported b the reserch progrms MSM d MSM 6000, b the grts of the Czech Grt Agec o. 10/01/0571, 10/01/0573 d 10/03/086, d b the grt of the Czech Miistr of ductio o. 195/003. Referece [1] TAFLOV, A. Computtiol lectrodmics: The Fiite-Differece Time-Domi Method. Bosto: Artech ouse, [] FRANK, O., RAIDA, Z. Time-Domi Alsis of Microwve Structures Usig MATLAB, Proceedigs of the 10 th Coferece MATLAB 00. Prh: VŠCT, 00, pp [3] TAFLOV, A. Advces i Computtiol lectrodmics: The Fiite-Differece Time- Domi Method. Bosto, Lodo: Artech ouse, Cotct frek@feec.vutbr.cz rid@feec.vutbr.cz Deprtmet of Rdio lectroics Bro Uiversit of Techolog Purkňov Bro

SOLUTION OF DIFFERENTIAL EQUATION FOR THE EULER-BERNOULLI BEAM

SOLUTION OF DIFFERENTIAL EQUATION FOR THE EULER-BERNOULLI BEAM Jourl of Applied Mthemtics d Computtiol Mechics () 57-6 SOUION O DIERENIA EQUAION OR HE EUER-ERNOUI EAM Izbel Zmorsk Istitute of Mthemtics Czestochow Uiversit of echolog Częstochow Pold izbel.zmorsk@im.pcz.pl