the finite-difference–time-domain method and its application to eigenvalue problems

7/22/2019 The Finite-DifferenceTime-Domain Method and its Application to Eigenvalue Problems

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1464 IEEETRANSACTIONSON MICROWAVETHEORYAND TECHNIQUES, VOL. MTT-34, NO. 12, DECEMBER1986

The Finite-DifferenceTime-DomainMethod and its Applicationto Eigenvalue Problems

DOK HEE CHOI AND WOLFGANG J. R. HOEFER, SENIOR MEMBER, IEEE

Abstract This paper describes the application of the finite-differencemethod in the time domain to the solution of three-dimensional (3-D)eigenvahre problems. Maxwells equations are discretized in space andtime, and steady-state solutions are then obtained via Fourier transform.While achieving the same accuracy and versatility as the TLM method, thefinite-difference-time-domain (FD-TD) method requires less than halfthe CPU time and memory under identical simulation conditions. Otheradvantages over the TLM method include the absence of dielectricboundary errors in the treatment of 3-D inhomogeneous planar strnctnres,such as microstrip. Some numerical results, including dispersion curves ofa microstrip on anisotropic substrate, are presented.

I. INTRODUCTIONT HE TLM METHOD has been successfully applied tovarious microwave circuit problems for more than tenyears. The special advantages of the TLM technique overother numerical methods are well illustrated by Johns andBeurle [1] in their original paper on the method. Sincethen, several improvements have been made to this tech-nique by various authors in order to enhance the accuracyof the solution and economize CPU time and memoryspace [2][5]. Mariki [7] has extended the TLM method toanalyze anisotropic media, and Saguet and Pic [4] as wellas A1-Mukhtar and Sitch [5] have employed a graded meshto make the algorithm faster and more efficient.Although the graded mesh algorithm reduces memory

space requirements, it demands far more iterations thanthe original method for equal frequency resolution [5]. Thisis especially obvious in the case of three-dimensional (3-D)simulations where the grade ratio N requires an additionalN 2 1 iterations. These requirements of large computerresources may critically limit the applicability of the TLMtechnique.Thus, Saguet [16] has proposed a simplified node which

reduces the number of variables to be processed and storedat each node by one third. However, this modificationincreases the velocity error. A further reduction in compu-tational expenditure has been proposed by the authors [8];instead of the original vector solution, we obtain a scalarpotential solution using a scalar 3-D network. However,Manuscript received March 22, 1986; revised July 8, 1986. This workwas supported in part by the National Science and Engineering ResearchCouncil of Canada.The authors are with the Department of Electrical Engineering, Uni -versity of Ot tawa, Ottawa, Ontario, Canada KIN 6N5.IEEE Log number 8610560.

the scalar approach is limited to problems which lead touncoupled modal solutions, i.e., TE and TM or LSE andLSM fields.The major reason for the large CPU memory demand of

this technique resides in the basic 3-D TLM concept. Inorder to represent each electromagnetic component, each3-D unit cell requires 26 real memory spaces, 12 for pulsestorage and 14 for additional network parameters. There-fore, each operation on each node involves a large numberof variables, requiring considerable computer CPU spaceand time. Furthermore, experience has shown that thenumber of iterations increases with the complexity of thestructure under study. For example, the accurate analysisof a finline requires easily over 1000 iterations. Given thesemassive requirements, we have searched for an alternativenumerical technique that possesses the advantages of theTLM approach but needs fewer computer resources. As aresult, a new algorithm is proposed based on both thefinite-difference-time-domain (FD-TD) and TLM meth-ods.The FD-TD method was first formulated by Yee [6],

and has been applied extensively to scattering and cou-pling problems with open boundaries [9][15], i.e., to thesolution of deterministic problems. We noted the similaritybetween this method and the TLM method, which hasbeen widely used in the numerical solution of the electro-magnetic eigenvalue problems in the time domain. Sincethe TLM method is based on the computation of theimpulse response of a large mesh of transmission lines,much unwanted information is usually generated.We have therefore developed a novel procedure which

increases the numerical efficiency of the time-domain ap-proach without sacrificing its advantages. The methoddiffers from the classical FDTD method in the assign-ment of initial field values and the application of theFourier transform to the time-domain solution. In thefoIlowing, we will describe this method and its applicationto some typical microwave problems.

II. YEES ALGORITHMMaxwells equations have been expressed in finite-dif-

ference form by Yee [6] to solve two-dimensional (2-D)wave scattering problems. Subsequently, 3-D scatteringproblems have been solved by Taflove and other workers

0018-9480/86/1200-1464$01.00 01986 IEEE


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CHOI AND HOEFER: FDTD METHOD AND EIGENVALUE PROBLEMS 1465

[9]-[15]. We will adopt Yees original algorithm for thethree-dimensional Maxwells equations. Also, we will ex-tend the concept further to include anisotropic media.In a rectangular coordinate system, the source-free

Maxwells equations can be written as first-order hyper-bolic equations

d~/dt = IAXI d~/dx + IAYI d~/dy + IA,l dH/dzdH/dt = IBXI d~/dx + IBYI d~/dy + IB,I d~/dz (1)

where ~= (EX, EY, E=)f, ~= (HX, HY, H=)f, and

10 6;: 01 1-,;. o 0][1


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1466 IEEETRANSACTIONSON MICROWAVETHEORYAND TECHNIQUES, VOL. MTF34, NO, 12, DECEMBER 1986

main in which the field must be computed is unbounded.Since no computer can store an unlimited amount of data,a special technique must be used to limit the domain inwhich the numerical computation is made, by introducingso-called absorbing or soft boundary conditions. Theseconditions have been described by Taylor et al. [9], whouse a simple extrapolation method, and by Taflove andBrodwin [10], who simulate the outgoing waves and use anaveraging process in an attempt to account for all possibleangles of propagation of the outgoing waves. Kunz andLee [12] use the radiation condition at a large distancefrom the center of the scatterer to obtain an absorbingboundary condition. Mur [13] employs a second-orderradiation condition to improve the accuracy of the results.Although these schemes have been used in scattering prob-lems in the past, no ideal reflection-free boundary condi-tion has been proposed so far.

However, in the formulation of eigenvalue problems,only hard boundaries usually represented by conduct-ing wallsoccur. At these boundaries, the tangential elec-tric and the normal magnetic field components are main-tained at zero. For example, on a perfectly conducting wallin the plane i = 1 (see Fig. 1)

{

E;(l, j+l/2, k) =0for all n E;(l, j,k+l/2) =0

H;(l, j+l/2, k+l/2)=o(the third condition is implicit in the previous two, but itsimplementation reduces numerical errors).

IV. INITIAL VALUESIn most scattering problems, an impulsive or sinusoidal

plane wave is injected at the beginning of the computation.However, in eigenvalue problems, the direction of thepropagation vector is usually not known and depends onthe space coordinates and on the eigenvalue that is to befound. In these cases, the logical choice is an isotropicpulse that propagates in the radial direction. The spatialpulse envelope should be wide enough with respect to themesh size not to accumulate numerical errors due toovershoot and ringifig as it propagates through the spacelattice.

A better way to start the computation is to estimate thefield distribution of the desired mode in the structure firstand then choose the initial value accordingly. The expe-rienced researcher usually has a good idea of the ap-proximate modal field distributions in a structw-e and istherefore able to make an educated guess of the steady-statefield pattern for a particular eigenmode. This procedure isequivalent to the excitation of a TLM mesh with a weightedimpulse distribution, and is somewhat similar to the way inwhich one chooses appropriate basis functions in the spec-tral-domain approach.

V. OUTLINE OF THE NUMERICAL PROCEDUREThe application of the FDTD method will be discussed

using a rectangular resonator as an example. A continuous

I

(a)JE,I

G05 067 AI!IA0.11 0.15(b)

Fig. 2. (a) Field distribution of an empty rectangular resonator ob-tained with the FDTD method. (b) Output spectrum obtained withboth the FDTD and TLM methods under identicaf conditions

medium is replaced with a 3-D uniform mesh. To solve thesystem of equations (2) in this mesh, initial values must beassigned first as described in the previous section. For arectangular-type resonator, a simple sinusoidal function isan appropriate choice for the dominant mode eigenvalue.As n increases, the discrete time functions for ~ and ~fields evolve towards the steady state which is characteris-tic of the desired mode in the geometry. In this way, theevolution of all six field components is obtained simulta-neously at discrete time points n At. The final steady-statefield distribution may be calculated by taking the timeaverage of the time-domain solution at each mesh point.Thus, the steady-state solution is given by

F(iO, jO, kO) = ~lF(iO, jO, kO) l/N. (4)n

This simple procedure to obtain the final field distributionis another advantage over the TLM method, which re-quires two simulations for finding the fields of a givenmode.

In eigenvalue problems, the steady-state solution is atime-harmonic function, from which the eigenvalues can


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CHO1 AND HOEFER: FDTDMETHOD AND EIGENVALUE PROBLEMS 1467

(a)

a=l 2dlb= 6dIc= 8dlS= 2dlE,=3.75

(b) a=20dlb= 6dlC= 8dl

ha----i a =5d lb =4d lC =3dlc c,= 16

(c)i

Fig. 3. Three-d imensional inhomogeneous resonators ahalyzed in th isstudy.

TABLE ICOMPARISON OF RESULTS FOR THE NORMALIZED RESONANTFREQUENCIES A1/A OBTAINED WITH THE TLM AND FD-TD

METHODS UNDER IDENTICAL SIMULATION CONDITIONS FOR THEGEOMETRIESN FIG. 3

Fig.3 Mode s/a TRM TLM (CPU time) FD-TD (CPU time).- -.1 . . .. ...14E...H==:lIIl{~::til-oo{?~?!5)d)d/6 0.05220 0.0516 (144) I 0.0517 (51)b) Hybrid ! 0.0278 (357) 0.0278 (1 17)c) Hybrid 0. 0405 (357) 0.0405 (1 17)

be extracted by discrete Fourier transform, as in the TLMmethod

S(~) =~F(iO, jO, kO)exp(j2rsnj). (5)nBoth the stability factor s and the number of iterations nstrongly affect the spectral response.In order to test this algorithm for validity, it has been

applied to a simple rectangular cavity with sides 12 Al x6 Al x 8 Al. We have assumed a dominant TEIOI mode inthe initial value assignment. The time-domain solution isgiven in Fig. 2(a). Discontinuous field figures are due tothe numerical error caused by the finite-difference form of(2). Fig. 2(b) compares the frequency responses obtainedwith the FDTD and TLM methods under identical con-ditions. The responses are not distinguishable. Fivehundred iterations have been used. The peak of the solu-tion is located at A1/X = 0.0750 in both methods. Theexact analytical solution is 0.07511. Even though a smallnumb& of meshes is used in this algorithm when com-pared with the scattering problems in [9]-[13], it is notedthat the accuracy in the solution of the eigenvalue prob-lems is better than that of the scattering problems by oneorder of magnitude.

VI. NUMERICAL RESULTSWe have applied this technique to most of the examples

described in the TLM literature and obtained practicallyidentical results. The method requires less than one-half of

l------=Fig. 4. Finline cavity

TABLE IIRESONANT FREQUENCIES OF THE UNILATERALFINLINE CAVITY IN FIG. 4, OBTAINED WITH

VARIOUS METHODS

a=20mmb.10mc=15mmd=4mmS=lmnE = 2.22r

GHz

~p0.0 0.1 0.2 0.3 0.4

Fig. 5. Dispersion diagram of unilateral finline with the cross-sectionalgeometry given in Fig. 4.

the CPU time spent by the equivalent TLM programunder identical conditions, including the initial excitationdistribution. Furthermore, while the TLM procedure re-quires 22 real memory stores per 3-D node in an isotropicdielectric, the FDTD method requires only seven realmemcmy stores per node. Fig. 3(a), (b), and (c) showsstructures for which solutions have been computed withthis method. The dominant resonant frequencies of thesestructures are given and compared with the TLM solutionin Talble I. The inhomogeneous rectangular cavity of Fig. 3(b) and (c) illustrates the capability of this algorithm tosolve hybrid field problems. The number of nodes chosenin each problem is the same as that employed in the TLMsolution.Furthermore, we have computed the resonant frequency

of a finline resonator (Fig. 4.) treated previously by Saguet[16]. Results are compared in Table II, which includes avalue obtained with the spectral-domain method by Saguet.Fig. 5 shows the dispersion characteristics of a finline withthe same cross section, as obtained with our method.Resulks calculated with our spectral-domain program arealso shown in the same figure. In order to compare conver-gence of both time-domain methods, solutions obtainedafter every fifth iteration are drawn in Fig. 6. The resultsshow virtually identical convergence.To show the versatility of this method, the characteris-

tics c~f a microstrip resonator on anisotropic substrate are


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1468 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-34, NO 12, DECEMBER 1986

F!GHz)11.5 +

t----- TLM FD-TD

los~mo2CQ 400 6(X) 8~Number o f i te ra tionsFig. 6. Stability and convergence of the TLM and FDTD methods asa function of the number of Iterations. The solution of the finlineproblem in Fig. 4 N represented.

t-~+ h/Al = 3w/Al = 3Al = 0.5 mlxx = 22 = 9.4

YY = 11.6Fig. 7. Microstnp cavi ty on anisotropic subst rate,

TABLE IIIDOMINANT RESONANT FREQUENCIES OBTAINEDBY BOTH THE TLM AND THE FDTD METHODS

computed in the last example, shown in Fig. 7. Severaldifferent resonant frequencies obtained by changing thelength c are tabulated in Table III. It is well known [17]that the TLM simulation of 3-D inhomogeneous planarstructures involves dielectric interface ambiguity. The bestway to resolve this error is to employ two dielectric sub-strate thicknesses differing by one Al. In our case, 3Al and4 Al are used. The final result is obtained by taking theaverage of the solutions obtained for these two values. In

order to illustrate thiswith the TLM method

process, frequency spectra obtainedfor the two cases where c is equal to

10 Al are shown in Fig. 8. The solution obtained with theFDTD method is also drawn in the same figure. Asexpected, the latter solution is located exactly between thetwo values obtained with the TLM method. This clearlyillustrates the accuracy and convenience of the FDTDmethod in such situations. Fig. 9 shows the dispersioncharacteristics of the microstrip which has the same crosssection as that in Fig. 7. Again, both methods give verysimilar results except at higher frequencies, where thediscretization errors associated with both methods becomemore pronounced, and their differences are more visibIe.

VII. CONCLUSIONSThe proposed new application of the FD-TD method to

3-D eigenvalue problems gives practically the same results


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CHOI AND HOEFER: FDTD METHOD AND EIGENVALUE PROBLEMS 1469

GHz

16

12

8

Fig. 8. Frequency spectra for the microstrip problem in Fig. 7, showingthe effect of dielectric interface error in the TI.M solution.

v,* . . . : iso trop ic substra te E,=9.4,, --: TLM \, anisotrop ic medium. : FDTD

[4]

[5]

[6]

[7]

[8]

[9]

/ em= e,? 9.4, EV=l I. 6P

[10]o. 0.2 0.4 0.6 0.8 1.0

Fig. 9. Dispersion diagram of the anisotropic microstrip in Fig. 7.[11]

as the TLM method under identical simulation conditions.However, the overall CPU time and storage requirements 1121are typically less than half those needed in the TLMsolution. Other advantages reside in the ease with whichfield distributions can be computed, and in the elimination ~1~1of dielectric boundary error in planar structures. Furtherefforts are being made to implement losses, variable meshsize, and nonlinearity in the FDTD procedure. [14]

[1]

[2]

[3]

REFERENCESP. B. Johns and R. L. Beurle, Numerical solution of two-dimen- [15]sionaf scatter ing problems using a transmission l ine matrix, Proc.Inst. Elec. Eng., vol. 118, no. 9, pp. 12031208, Sept. 1971.S. Akhtarzad and P. B. Johns, Solution of Maxwelfs equations in [16]three space dimensions and time by the TLM method of numericalanafysis~ Proc. Inst. Elec. Eng., vol. 122, no. 12, pp. 1344-1348,Dec. 1975. [17]Y. C. Shih and W. J. R. Hoefer, The accuracy of TLM analysis offinned rectangular waveguides~ IEEE Trans. Microwave Theory

Tech., vol. MTT-28, pp. 743-746, July 1980.P. ISaguet and E. Pie, Le maillage rectangulaire et le changementde maille clans la methode TLM en deux dimensions, Electron.Lett., vol. 17, no. 7, pp. 277-279, Apr. 1981.D. A. A1-Mukhtar and J. E. Sitch, Transmission line matrixmethod with irregular ly graded space, Proc. Inst. Elec. Eng., vol.128, no. 6, pp. 299-305, Dec. 1981.K. S. Yee, Numerical solution of initial boundary value problemsinvolving Maxwells equations in isotropic media, IEEE Trans.Antennas Propagat., vol . AP-14, pp. 302-307, May 1966.G. E. Mariki, Analysis of microstnp lines on inhomogeneousanisotropic substra tes by the TLM numerical technique) Ph.D.thesis, Univ. Cali fornia, Los Angeles, CA, June 1978.D. Choi and W. J.R. Hoefer, The simulation of three dimensionalwave propagation by a scalar TLM model, IEEE MTT-S Synrp.Di&, 1984, pp. 70-71,C. D. Taylor, D. H. Lam, and T. H. Shumpert, Electromagneticpuke scattering in time varying inhomogeneous media, IEEETrizns. Antennas Propagat., vol. AP-17, pp. 586-589, Sept. 1969.A. Taflove and M. E. Brodwin, Numerical solution of steady stateelectromagnetic scattering problems using the time dependentMaxwells equations, IEEE Trans. Microwave Theory Tech., vol.Ml_T-23, pp. 623-630, Aug. 1975.R. Holland, Threde: A free-field EMP coupling and scatteringcode, IEEE Trans. Nucl. Sci., vol. NS-24, pp. 24162421, Dec.1977.K. S. Kunz and K. M. Lee, A three-dimensional finite-differencesolution of the extemaf response of an aircraft to a complextransient EM environment: Part 1The method and its implemen-tation, IEEE Trurrs . Electromagn. Compat ., vol. EMC-20, pp.328-33, May 1978.G. Mur, Absorbing conditions for the f inite-difference approxima-tion of the time-domain electromagnetic-field equations, IEEETrans. E[ectromagn. Compat ., vol. EMC-23, Nov. 1981.A. Taflove and K. R. Umashankar, A hybrid momentmethod/finite difference time domain approach to electromagneticcoupling and aperture penetration into complex geometries, IEEETrans. Antennas Propagat., vol. AP,30, pp. 617-627, July 1982.K. R. Umashankar and A. Taflove, A novel method to analyzeelectromagnetic scattering of complex objects, IEEE Trans. Elec-tromagn. Compat., vol. EMC-24, pp. 397405, Nov. 1982.P. Saguet, Le maillage paralle14pip4dique et le changement demaille clans la mdthode TLM en trois dimensions, Electron. .Mt .vol. 20, no. 5, pp. 22224, Mar. 15, 1984.W. J. R. Hoefer, The transmission-line matrix methodTheoryand applications, IEEE Trans. Microwave Theory Tech., VOLMT1 33, no. 10, pp. 882-893, Oct. 1985.


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1470 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-34, NO. 12, DECEMBER1986

Dok Hee Choi was born in Seoul, Korea, in 1949and graduated from Sogang University, Seoul, in1973. From 1973 to 1977, he served as a radarofficer in the Korean air force. He received theM.SC.degree in electrical engineering from Con-cordia University, Montreaf, in 1981 and thePh.D. f rom the Univers ity of Ottawa, Ottawa, in1986.From 1977 to 1978, he worked for the Korean

Institute of Science and Technology as a micro-wave engineer involved in quality control in themanufacture of radar systems. F~om 1981 to 1984, he was employed by

Canadian Astronautics Ltd., as a microwave engineer specializing inelectronic warfare system design and millimeter-wave VCO development.He joined the Spacecraft Division of Telesat Canada, Ottawa, in 1986,and is current ly developing numerical models to simulate communica-tions satellite antennae. He is also involved in satellite transponderfront-end design. His special interests include mil limeter-wave compo-nent design and the application of numericaf techniques in appliedelectromagnetic.

Wolfgang J. R. Hoefer (M71-SM78) was bornin Urmitz/Rhein, Germany, on February 6, 1941.He received the diploma in electrical engineeringfrom the Technische Hochschule Aachen, WestGermany, in 1965, and the D.-Ing. degree fromthe Universi ty of Grenoble, France, in 1968.After one year of teaching and research at theInsti tut National Polytechrrique de Grenoble, hejoined the Department of Electrical Engineering,the University of Ottawaj Canada, where he iscurrent ly a professor. HIS sabbatical act ivit iesincluded six months with the space division of the AEG-Telefunken in

Backnang, West Germany, in 1976, six months with the Insti tut NationalPolytechnique de Grenoble, France, in 1977, and one year with the SpaceElectronics Directorate, the Communications Research Centre in Ottawa,Canada, during 1984/85. His research in terests include microwave andmil limeter-wave circuit design, numerical techniques for solving electro-magnetic field problems, and microwave measurement techniques.Dr. Hoefer is a Registered Professional Engineer in the Province ofOntario.

the finite-difference–time-domain method and its application to eigenvalue problems

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