Geri Dön

Zamanda sonlu farklar yöntemi ve yutucu sınır koşulları

Başlık çevirisi mevcut değil.

  1. Tez No: 75593
  2. Yazar: FUNDA AKLEMAN
  3. Danışmanlar: DOÇ. DR. LEVENT SEVGİ
  4. Tez Türü: Yüksek Lisans
  5. Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
  6. Anahtar Kelimeler: Sonlu farklar yöntemi, Uzay, Finite differences method, Space
  7. Yıl: 1998
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Elektromanyetik Alanlar ve Mikrodalga Tekniği Ana Bilim Dalı
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: Belirtilmemiş.

Özet

ÖZET Bu çalışmada, kısaca FDTD olarak tanımlanan Zamanda Sonlu Farklar (Finite- Difference Time-Domain) yöntemi ayrıntılı olarak incelenmiştir. FDTD yöntemi FD olarak bilinen sonlu farklar yönteminin 1966 yılında Yee tarafından Maxwell denklemlerine uyacak şekilde zaman domeni için genişletilmesiyle birlikte ortaya atılmış, özellikle 1980'lerin ortalarında bilgisayarların hız ve kapasitelerindeki hızlı artışla birlikte elektromagnetik problemler için en çok kullanılan yöntemlerden biri haline gelmiştir. Çalışma sırasında öncelikle FDTD yönteminde ortam modelleme, sayısal dispersiyon, zamanda ve konumda ayrıklaştırma, hata analizi gibi konular incelenmiş, daha sonra da konumda sınırlı sayıda hücre kullanılması nedeniyle oluşan ya pay sınırlardaki yansımaların giderilmesi için uygulanan sınır koşulları üzerinde durulmuştur. Sonsuza giden sınırları simüle eden birinci-derece ve ikinci-derece Mur, Higdon, DBC (Dağıtıcı Sınır Koşulu), Liao ekstrapolasyonu gibi açık sınır koşullan teorik olarak açıklandıktan sonra, 1994 yılında Berenger tarafından ortaya atılan PML (Mükemmel uyumlu Tabaka) yöntemi anlatılmıştır. Uygulamalar sırasında mikroşerit hatlardaki karakteristik empedans, efektif dielektrik sabiti ve yansıma katsayısı FDTD yöntemi ile sayısal olarak bu lunmuş ve bu değerler yardımıyla açık sınır koşullarının etkileri birbiri ile karşılaştırılmıştır. Bu işlemlerin sonucunda gerekli optimizasyonlar yapıldığı taktirde PML yönteminin en güvenilir sonuçları verdiği görülmüştür. Ancak PML yöntemi FDTD algoritmasında ihtiyaç duyulan hafıza miktarını arttırmakta ve programın çalışmasını da yavaşlatmaktadır. Bu yüzden, çok hassas analizlere gerek duyulmuyorsa, incelenen yapıya ve probleme uygun olan başka bir açık sınır koşulu kullanılabilir. En son olarak FDTD yönteminin farklı problemlere ve yapılara ne şekilde uygulanacağına dair örnek olması için alçak geçiren filtre, çeyrek dalga empedans transformatörü, mikroşerit küple devresi gibi düzlemsel mikroşerit yapılar FDTD ile incelenmiştir. v iii

Özet (Çeviri)

SUMMARY THE FINITE-DIFFERENCE TIME-DOMAIN METHOD AND ABSORBING BOUNDARY SIMULATIONS In this study the Finite-Difference Time-Domain (FDTD) method is investigated. FDTD is widely regarded as one of the most popular full-wave computational electromagnetics (EM) algorithm. It was first investigated in 1966 by Yee [1] and only a few studies have appeared until the end of 1970s because;. Large computer storage and high speed requirements have not been supplied until the mid of 1980s.. FDTD, as introduced by Yee was not suitable for most of the EM problems such as antenna simulations and RCS calculations [27] etc.. Source implementations and open boundary simulations have not been han dled in early FDTD, as introduced by Yee. Yee [1] cliscretized Maxwell's two curl equations directly in time and spatial do mains and put them in iterative form. In Yee formulation the physical volume of interest is divided into cubic reference cells where characreristics of the medium are defined by three parameters; permittivity (e), conductivity (cr) and permeabil ity (fi). Within the reference cell three electric and three magnetic field compo nents are located at different locations in such a way to minimize the computation duty after the discretization of two curl equations by using central-difference ap proach (or by taking up to the second order terms in their Taylor's expansion). Besides the differences in the locations of six field components, there is also a half time step difference between electric and magnetic field components. This is called a leap-frog computation. During FDTD simulation electric field com ponents are calculated at each cell at time instants t = 0,At,2At,3At,... etc., but magnetic field components are calculated at t = At/2,3At/2, 5At/2,... etc., having a At/2 time difference between electric, and magnetic field components. Therefore, in electromagnetic analysis, synchronization in both time and spatial domains are needed for E and H field or V and / calculations at a fixed point in FDTD volume. This is fortunately accomplished by simple cell and time averag ing processes. Since 1980, hundreds of publications about FDTD have appeared related to the algorithmic improvements as well as its applications to broad range of complex electromagnetic problems. Major improvements may be groupped in: ix. Narrow and broad band source simulations and their injections in given FDTD cells.. Open boundary simulations which extend the ability of FDTD algorithms to handle antenna radiation simulations and radar cross-section (RCS) cal culations.. Both frequency and time domain near- to- far field transformations based on Huygen's equivalent source principle. where effords are still needed for further modifications. Some of the applications may also be groupped as;. The analysis of planar microstrip structures,. Specific Absorption Rate (SAR) calculations in human tissues near electro magnetic sources,. Mutual effects of hand-held receivers and human head,. Antenna simulations and RCS calculations,. Analysis of waveguiding structures,. The simulation of ground-penetrating radars,. The simulation of microwave ovens. In this study, PML which is the most, efficient and available ABC, is implemented numerically for 3D rectangular FDTD volume. Although PML is effective in absorbing guided waves of all angular distribution, its implementation is quite difficult. Only a few studies related to the application of PML to 3D-FDTD [18], [17], [19], [22] have appeared in the literature up to day. For this reason, this study aims to focus on not only FDTD analysis of complex structures, but also 3D PML implementations. Three dimensional (3D) FDTD mesh suggested by Yee [1] is shown in Section 2.1 (see figure 2.1). In a lossy source-free medium, Maxwell's curl equations are given as 9H - - ^o-jT = -V xE-crE (la) BE.^. = Vx// (lb)where £q, (iç,, a are permittivity, permeability and conductivity respectively, and they are discretized as A/ H?(iJ,k) = H^iiJ.k) - -- [E*(i,j,k) - E^(i,j,k- 1)] At [E:(i,j,k)-E?(i,j-l,k)} (2a) Hj{i%j,k) = H*-\i,j,k) - -£L [E<?(i,j,k) - Enz(i - l,j,*)] +-^- [££(.,>,*) - E:(i,j,k- 1)] (2b) ff?(», j,*) = Hr\i,j,k) - -^- [££(*,**) - ^(i, j - 1,*)] A* [JEJ(i,i,Ar)-^(i-l,j,*)] (2c) /z0A:r 2e-jrAj 2A* Wj'-*> = 2^ME“”{i'j'k) 2A/ 2Aİ [H?(i,j,k)-H:(i-lJ,k)] {2e + aAt)Ax 2At + WT^Wzm''hk)-H:i''i'k-l)] (2e) In these equations h = n + 1/2 and e = £o£r- Because of the 3D-FDTD mesh structure, electric fields are calculated at the integer multiplicants of time and magnetic fields are calculated at the fractional multiplicants of time. As it is common in all numerical methods there exist unexpected reflections from the boundaries of the FDTD computation space unless the boundaries are cov ered with appropriate boundary conditions. In order to remove the effects of these reflections the methods listed below are used in FDTD applications: xi. Boundaries are covered with PEC (perfectly electric conductor) by assigning zero value to the tangential electic field components during time simulations.. Boundaries are covered with PMC (perfectly magnetic conductor) by as signing zero value to the tangential magnetic fields components during time simulations.. Field values of the cells which are adjacent to the boundary inside FDTD are assigned to field values of the cells which are adjacent to the boundary outside (symmetry condition).. Radiation condition called as absorbing boundary condition (ABC) is ap plied at the boundary planes. In this thesis, different ABCs are theoretically explained and numerically com pared with each other according to their efficiencies and accuracies in FDTD applications. First-order MUR ABC: With the help of first-order approximation of the three dimensional wave equation first-order MUR ABC [2] at x - 0 boundary is given as (dr - %ldt)Et = 0 (3a) where E% represents the tangential electric field component relative to the bound ary wall. The discretized form of Eq.(3a) will then be £,“+1(0,j,A-) = £r(U,Â0 + ^^(İ^HW,*) - £T(0,i,*)) (3b) For example, the tangential electric field components at x - 0 boundary are Ey and Ez. Eq.(3b) tells that, Ey(i,j, k) and Ez(i,j, k) for all (j, k) will be related to the current (n) and one past (n - 1) time instant values of themselves and to the nodes one inside. By doing this, first-order Mur condition is satisfied. Second- order MUR ABC: With the help of second-order approximation of the three dimensional wave equation second-order MUR ABC [2] at x = 0 boundary is given as (<b '& - c0-2^2 + \{dl + dl))\x=QEt = 0. (4) with the ”same Et definition. Higdon ABC: Higdon [3], [4] proposed a differential annihilator for a lineer combination of plane waves (U(x,y,t)) each propagating with speed c and at xiithe symmetrical incidence agles ±Qi,...,±an relative to the x- axis toward the boundary at x = 0 of a two-dimensional cartesian FDTD computation space;.J=l v U = 0. (5) This operator depending on the order of multiplication absorbs [/(.t, y, t) with no reflection at x = 0. Liao Extrapolation: First an outer grid boundary located at xmax and a sten cil of field values u located along a straight line perpendicular grid boundary are taken into consideration. In order to develop an approximation for the updated field at the boundary u(t + At,xmax) following equations [2], [10] are used. «i = u{t,xmax-acAt) (6) A1u(f, xmax - acAt) = Aruı = «i - u2 A2u{t,xmax - acAt) = A2u\ = A1uy - Alv,2 A3u(t, xmax - acAl) = A3mj = A2Mi - A2u2 (7) «o = u{t + At,xmax) S mi + A]mi + A2mi +... + AN~1u1 (8) where a, which is in (0, 2) interval, is a scaling factor needed for Yee grid. Here Mi = u(t,xmax) can be accepted as tangential electric field Et. Dispersive Boundary Condition (DBC): DBC is generally used for numer ical analysis including micros t rip components such as microstrip lines, basic mi- crostrip discontinuities and microstrip antennas. Since the major direction of the power flow is in the waveguide direction for these componenets, it is possible to use one-dimensional wave equation. Then at the boundary z = zmax DBC [6] is formulated as (d, + v-xdt){dz + v;1dt)Et = o (9) where Et is the tangential electric field component and V\ and v2 represents the velocity of propagation of the fields for two different frequencies. In order to prevent the reflections of the evanescent waves DBC is corrected with adamping factor, q = 0.1/Az (dz + v^dt + ai)(& + v?dt)Et = o (io) xiiiPerfectly Matched Layer (PML): For two-dimensional TE case the relevant field components include Ex, Ey and Hz. Maxwell's equations reduce to 8E,“ BH,,”,,0Ş, + a.ft = Ş_Ş (Uc) at ay ox where a and a* denote a possible electric conductivity and magnetic loss, respectively, assigned to free space. If the following condition is satisfied a cr“ - = - (12) co £o then the wave impedance of the lossy free-space medium equals that of lossless vacuum and no reflection occurs when a plane wave propagates normally across an interface between vacuum and the lossy free-space medium. Berenger [7] has introduced a new ABC which is called PML by splitting Hz into two part, Hzx and Hzy and then Maxwell's equations become 3EX d(Hzx + Hzy) £o-qj~ + ayE, = q- (13a).9Ey _ d(Hzr + Hzy) ”-ÖT + ****- gx (13b) 9HZX dEy llo^r + a*Hzr = ~~af (13c) 9Hzy dEx fl0~öT y zy = ~dy~ ( ^ where ax and ay denote eletric conductivities and a*x and a* denote magnetic losses. When Berenger [7] applied above equations for vacuum-PML and PML- PML interfaces respectively he got the following results for the condition that equation (12) is satisfied for both (cxx, a*) and {cry,cr*):. There is no reflection from the interface normal to x-axis between (0, 0, 0, 0) vacuum and (crx, a*, 0, 0) PML medium or between (0, 0, cry,(j*) and (ax,a*, ay,cr*) PML media. xiv. There is no reflection from the interface normal to a:-axis between (0, 0, 0, 0) vacuum and (0,0, ay, a*) PML medium or between (crx, cr*, 0, 0) and [(Tx,(T*,ay,(T*) PML media. Similar results are obtained for TM case, too. Two-dimensional FDTD with PML: PML(a..ci«v«C) pMLc^yq;,) PML <«*,<£«,«;> PML(o“.o,'.o.o) y f! X<./' /-”\ \. / / \ \ 'J jj ( ( Wave source) | f'lX '. \ i- / i I i B, Propagating waves PML(Oj.o,>i),o) PML«j“,o,;,o”.%) PML(o.ao>0'“) PML(oa.o»a”o,) Figure 1. 2D FDTD and PML structure yf PML J-JL ¦+-E, O H, H“ © H”Figure 2. Right-upper part of 2D FDTD+PML structure PML is applied to two-dimensional FDTD as shown in Figure 1. FDTD equations are applied in the inner part and PML equations are used in the PML region. For the right-upper part of the Figure 1 (see Figure 2) in PML region (/ > IL and J > JL in Fig.3) the field equations are gj(i)Ai gar(l)At Eny+i{i,j) = e~ <o E%(iJ)- (l~e *° )rrrn <7x{i)Ax + ^+1/2(«',i)-^I/a(«-i,i) K+1/2(«\i) g*(. + l/2)At H£ll^j) = t-^r-Hz;W{i>i) (l- e tr*(i+l/2)At CO ') <7;(* + l/2)A* [£T(i + l,j)-^(«.i)]- (14) (15) xvThere must be a modification for the Ey field component over the vacuum-PML interface at the right part of the Figure 2 (for I = IL and J < jL) since the magnetic field component is identified as Hz in the vacuum and as Hzx and Hzy in PML medium. Equation (14) becomes E?\u,i) = e-^q(,i,,)-(1-;~TV^(.U) * * ax(il)/\x + Hnz^l2{ilj)-H^I\U-UJ)\. (16) Three-dimensional FDTD with PML: The three-dimensional FDTD with PML structure is seen in Figure 2. FDTD Space Figure.3 3D FDTD and PML structure When both TE and TM cases are taken into consideration for three-dimensional cartesian coordinates, PML equations are found as [18] /«o- - - r <J“nxv = dt 'y”xy d(Ezx + Ezy) dy (17a) dHxz fJ-o- - + cr,H: dt zx*xz d(EyX + EyZ) Wz (17b) duy* _L“*TT yz d{Exy + Exz) Öz (17c) d(EZX+EZy) dx (17d) dt xxxzx 0\İjyX T £jyz) dx (17e) XVIPO-âf + ^ = Qy d7f) t0~W y y ~ dv e^ + a.E,,-6^**1^ (18c) 0^, r d(Hzr + Hzy) -o~~: + «^ = - v *V «y/ (18d) dt ' x w dx -dE^+(TF.ŞSi+üsi),18el :o^r + ^”- - d~x (18e) '~âr + aAf dy -. (18f) It is difficult to apply (17) and (18) equations to three dimensional FDTD compu tation space, because the parameters which are used for vacuum-PML and PML- PML interfaces differ from each other. Therefore, there exist six PML-vacuum interface layers which can be named as A regions, twelve B regions which are used for the intersection parts of A" s and eight C regions which are used for the intersection parts of £?'s. Numerical Applications: Here, a complex electromagnetic structure is chosen to show the effectiveness of FDTD algorithm and to compare different ABC simulations. Electromagnetic analysis in planar microstrip structures is a challenging work especially from nu merical computational aspects. Different wave phenomenia -such as propagating (guided) leaky and surface waves- occur in these structures. In Figure 4 a sim ple 50Q microstrip line is shown where FDTD simulation of time-domain pulse propagation is done with various ABCs are compared. The width and height of the strip are w = lmm and h = 0.5mm respectively. For the comparison quantity of various ABCs, a reflection coefficient defined as the unwanted reflected wave in frequency domain in front of the boundary plane which is expected to absorb all energy to simulate open boundary. In this study, first-order Mur, DBC which are the most efficient ABCs for microstrip structures are chosen to be compared with PML. Reflection coefficient values for second-order Mur, DBC, PML with 8 FDTD (PML-8) cells and PML with 16 xviiFDTD cells (PML-16) are plotted with respect to frequency in Figure 5. As it is seen, PML-16 is the most efficient method because when the thickness of the PML layer increases the propagating waves are absorbed better. Figure 4. Microstrip line -100- -125 DBC PML-8 PML-16 First-order Mur I I I I | I I I I | I I I I | I I II | I II I | M I I | I I I I | I I I I | I I I I | I I 10 35 60 85 110 135 160 165 210 235 Frequency [GHz] Figure 5. Reflection coefficient In order to show the qualification of FDTD method some complex microstrip structures such as a quarter wavelength transformer, a microstrip coupler and a low-pass filter are analyzed in Section 5.3. » This study gives a brief outline of FDTD method and the important numeri cal aspects in its implementation. Also, absorbing boundary simulation, which is the most important point in FDTD, is investigated. Various ABCs are sum marized and coded in FDTD algorithm. Their comparisons are given and finally, PML is taken into account in this study. PML, since introduced by Berenger in 1994, has been applied mostly in 2D structures. Only a few numerical imple mentations are found in literature for 3D structures [17], [18], [19], [22]. At the end of this study, a compact-powerfull 3D algorithm is built with PML. Many complex electromagnetic structures can be analyzed with the help of this code as illustrated here for planar microstrip structures. xvi

Benzer Tezler

  1. Tez No

    101364

    FDTD yöntemiyle insan kafası modelinde elektromagnetik etkilerin belirlenmesi

    FDTD analysis of electromagnetic field effects induced on a human head

    SEVİNÇ İNCE

    Yüksek Lisans

    Türkçe

    Türkçe

    2001

    Elektrik ve Elektronik Mühendisliğiİstanbul Teknik Üniversitesi

    Y.DOÇ.DR. SELÇUK PAKER

  2. Tez No

    76434

    On the stability of the shallow water wave equation solutions: Formulation, analysis and application

    Sığ su dalga denklemlerinin nümerik çözümlerinin kararlılığı üzerine: Formulasyon, analiz uygulama

    EGEMEN KURTOĞLU

    Yüksek Lisans

    İngilizce

    İngilizce

    1998

    Makine MühendisliğiBoğaziçi Üniversitesi

    Mekanik Ana Bilim Dalı

    DOÇ. DR. HALUK ÖRS

  3. Tez No

    101199

    Foto ve MOM sayısal yöntemleriyle bazı dizi antenlerin tasarımı

    The Design of some antenna arrays with foto and moment numerical methods

    HALİL ERDEM BAŞEĞMEZ

    Yüksek Lisans

    Türkçe

    Türkçe

    2000

    Elektrik ve Elektronik Mühendisliğiİstanbul Teknik Üniversitesi

    PROF.DR. ERCAN TOPUZ

  4. Tez No

    95254

    Bilgisayar destekli tasarım ve şekil optimizasyonu

    Computer aided design and shape optimization

    AYTEKİN TOLGA PERÇİN

    Yüksek Lisans

    Türkçe

    Türkçe

    2000

    Makine MühendisliğiUludağ Üniversitesi

    Makine Mühendisliği Ana Bilim Dalı

    PROF. DR. FERRUH ÖZTÜRK

  5. Tez No

    39417

    Meteorolijide nokta semivariogram ile objektif analiz

    Objective analysis with point cumulative semivariogram in meteorology

    ZEYAD Z. HABİB

    Yüksek Lisans

    Türkçe

    Türkçe

    1993

    Meteorolojiİstanbul Teknik Üniversitesi

    PROF.DR. ZEKAİ ŞEN

Geri Dön