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A Memory-Reduced 2-D Order-Marching Time-DomainMethod for Waveguide StudiesW. Shao a , B. Z. Wang b & T. Z. Huang ca Institute of Applied Physics, University of Electronic Science and Technology of China,610054, Chengdu, P. R. Chinab Institute of Applied Physics, University of Electronic Science and Technology of China,610054, Chengdu, P. R. Chinac Institute of Applied Physics, University of Electronic Science and Technology of China,610054, Chengdu, P. R. ChinaPublished online: 03 Apr 2012.

To cite this article: W. Shao , B. Z. Wang & T. Z. Huang (2008) A Memory-Reduced 2-D Order-Marching Time-DomainMethod for Waveguide Studies, Journal of Electromagnetic Waves and Applications, 22:17-18, 2523-2531, DOI:10.1163/156939308787543822

To link to this article: http://dx.doi.org/10.1163/156939308787543822

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J. of Electromagn. Waves and Appl., Vol. 22, 2523–2531, 2008

A MEMORY-REDUCED 2-D ORDER-MARCHINGTIME-DOMAIN METHOD FOR WAVEGUIDE STUDIES

W. Shao, B. Z. Wang, and T. Z. Huang

Institute of Applied PhysicsUniversity of Electronic Science and Technology of China610054, Chengdu, P. R. China

Abstract—In this paper, a memory-reduced (MR) 2-D order-marching time-domain (OMTD) method for uniform waveguides isproposed. The OMTD method uses weighted Laguerre polynomialsand Galerkin’s testing procedure to eliminate the temporal variablesand results in an unconditionally stable scheme. Although the OMTDmethod may be more efficient than the finite-difference time-domain(FDTD) method with too many time steps to complete a simulation, ithas to deal with matrix inversion. To reduce its high computer memoryrequirements and computational power when analyzing complicatedgeometries, the divergence theorem is introduced to obtain a morememory-efficient matrix equation. Two waveguide examples arepresented to validate the accuracy and efficiency of our algorithm.

1. INTRODUCTION

The finite-difference time-domain (FDTD) method has been widelyused for electromagnetic analysis because of its accuracy and simplicity[1–11]. However, in some cases of complicated structures, the Courant-Friedrich-Lewy (CFL) stability condition imposes tiny time steps tiedto fine structure details. This main drawback often results in a longsolution time for the problems involving fine grid division. To eliminatethe CFL stability condition, a new order-marching time-domain(OMTD) algorithm was introduced [12–14]. This unconditionallystable scheme with weighted Laguerre polynomials does not haveto deal with time steps and may be computationally much moreefficient than the FDTD method with too many time steps to computethe solution. [15] applied a two-dimensional (2-D) OMTD methodand discrete Fourier transform (DFT) to uniform millimeter wavewaveguide analysis.

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2524 Shao, Wang, and Huang

Although the OMTD method solves the temporal variablesanalytically, it results in an implicit relation and has to perform thematrix inversion, which is never considered in the FDTD method.The memory storage requirements and computation time of theOMTD method is dependent on the produced sparse matrix equation.Based on divergence relationship, [16] introduced a reduced FDTD(R-FDTD) method with memory-efficient formulations to obtain amemory reduction of 33% in the storage of electromagnetic fields.

In this paper, we introduced the divergence theorem to the 2-D OMTD method. We obtain about three-fourteens memory storagereduction of nonzero unknowns by substituting a Maxwell’s divergencerelationship for one of the curl difference equations. Combined withthe DFT, the memory-reduced (MR) 2-D OMTD method is used tocalculate the cutoff frequencies of TE modes of waveguides. Thenumerical results verify the accuracy and efficiency of the proposedmethod.

2. MATHEMATICAL FORMULATION

Field analysis for TE or TM modes of a uniform waveguide only needsthree field components to find the cutoff frequencies, and the problemis in a 2-D case [17].

For simplicity, TE modes in waveguide are taken into account, andthe time-domain Maxwell’s equations with simple and lossless mediaare

∂Hz

∂t= − 1

µ

(∂Ey

∂x− ∂Ex

∂y

)(1)

∂Ex

∂t=

1ε

∂Hz

∂y(2)

∂Ey

∂t= −1

ε

∂Hz

∂x(3)

In charge-free regions, the divergence of D can be chosen to replace(3)

∇ · D =∂Ex

∂x+

∂Ey

∂y= 0 (4)

Since the Laguerre polynomials Ln(t) are orthogonal with respect tothe weighting function e−t, an orthogonal set {ϕ0, ϕ1, ϕ2, . . .} is chosenas the basis functions

ϕn(st) = e−st/2Ln(st) (5)

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A memory-reduced 2-D order-marching time-domain method 2525

where s > 0 is a time scale factor. Using these entire-domain temporalbasis functions, the electromagnetic fields Ex, Ey and Hz can beexpanded as

{Ex, Ey, Hz(x, y, t)} =NL∑n=0

{En

x , Eny , Hn

z (x, y)}

ϕn(st) (6)

The first derivative of field components U(x, y, t) with respect to timet is [18]

∂U(x, y, t)∂t

= sN∑

n=0

1

2Un(x, y) +

n−1∑k=0,n>0

Uk(x, y)

ϕn(st) (7)

Using a Galerkin’s testing procedure and a Yee’s space lattice, andeliminating magnetic fields, with reference to [12], we get

−2sµi,j−1∆yj−1

Emx |i,j−1+

(sεi,j∆yj

2+

2sµi,j∆yj

+2

sµi,j−1∆yj−1

)Em

x |i,j

− 2sµi,j∆yj

Emx |i,j+1 +

2sµi,j−1∆xi

Emy |i,j−1 −

2sµi,j−1∆xi

Emy |i+1,j−1

− 2sµi,j∆xi

Emy |i,j +

2sµi,j∆xi

Emy |i+1,j

= −sεi,j∆yj

m−1∑k=0

Ekx |i,j − 2

m−1∑k=0,m>0

(Hk

z |i,j − Hkz |i,j−1

)(8)

Emy |i,j − Em

y |i,j−1 +∆yj

∆xiEm

x |i,j −∆yj

∆xiEm

x |i−1,j = 0 (9)

where ∆xi and ∆yj are the lengths of the lattice edge where the electricfields are located, and ∆yj is the distance between the adjacent centernodes where magnetic fields are located.

Compared with the traditional OMTD method, Emy |i,j has a

relationship only with adjacent three electric field components, whichresults in a reduction of nonzero element storage by three-fourteens,and does not need to summate from order 0 to m− 1. After obtaining{E0}, we can solve (8) and (9) in an order-marching procedurerecursively.

In order to study the spectral response, the frequency resolutioncan be achieved by the DFT [19]

U(k∆f) = ∆tP−1∑p=0

U(p∆t)e−j02πk∆fp∆t (10)

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2526 Shao, Wang, and Huang

where j0 =√−1.

In the presence of conductors, the divergence of the electric fluxis not zero, but rather equals the induced charge on the conductor ρand (4) is not satisfied. The MR-OMTD formulation at a conductorboundary is discussed as follows.

Figure 1. Conductor boundaries of a rectangular waveguide.

Fig. 1 shows a rectangular waveguide filled with air. At the bottomconductor boundary, the divergence of the electric flux is not zero, andthe components Em

y |i,J1 do not satisfy (4) any more. They can becalculated by the boundary conditions and continuity equation shownas

n · D = ρs (11)n × H = Js (12)

∇ · Js = −∂ρs

∂t(13)

where n points outward from the conductor boundary, ρs is the surfacedensity of charge and Js is the surface current density. At the bottomconductor boundary, we get

εEy = ρs (14)Js = xHz (15)

∂ρs

∂t= −∂Hz

∂x(16)

Expanding ρs by using the basis functions shown in (5), we have

Emy

∣∣∣i,J1

=1

εi,J1

ρms

∣∣∣i,J1

(17)

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A memory-reduced 2-D order-marching time-domain method 2527

ρms

∣∣∣i,J1

=2

s∆x

(Hm

z

∣∣∣i−1,J1

− Hmz

∣∣∣i,J1

)− 2

m−1∑k=0

ρks

∣∣∣j,J1

(18)

Inserting (18) to (17), we obtain

Emy

∣∣∣i,J2

=2

sεi,J1∆x

(Hm

z

∣∣∣i−1,J1

− Hmz

∣∣∣i,J1

)− 2

m−1∑k=0

Eky

∣∣∣i,J1

(19)

Eliminating magnetic fields from (19), an implicit relation at thebottom conductor boundary can be written as

−2sµi−1,J1∆xi−1

Emy |i−1,J1

+

(sεi,J1∆xi

2+

2sµi,J1∆xi

+2

sµi−1,J1∆xi−1

)Em

y |i,J1

− 2sµi,J1∆xi

Emy |i+1,J1 +

2sµi,J1∆yJ1

Emx |i−1,J1+1−

2sµi−1,J1∆yJ1

Emx |i−1,J1+1

= −sεi,J1∆xi

m−1∑k=0

Eky |i,J1 + 2

m−1∑k=0,m>0

(Hk

z |i,J1 − Hkz |i−1,J1

)(20)

The formulations of the electric field components at the otherconductor boundaries can be constructed in the similar way.

3. NUMERICAL EXAMPLES

In this paper, we use the following Gaussian pulse as an incident electricfield profile:

Ei(t) = e−[(t−Tc)/Td]2 (21)

where Td = 1/(2fc) and Tc = 3Td.Firstly, we consider a rectangular waveguide with a cross-sectional

area of 1.88×3.76 mm2 to validate the proposed algorithm. The hollowwaveguide is modeled with 10 × 20 meshes using square cells. Andwe choose fc = 100 GHz, the order-marching number NL = 350 ands = 5.75×1010. At the point p1(7, 15), Ey(t) is chosen as the excitationsource. And the time-domain data at the point p2(3, 5) are recorded forDFT. We can calculate the cutoff frequencies with MR-OMTD methodfor first four TEmn (m = 0) modes. Table 1 shows the results and thepercentage changes of the results from the corresponding analyticalsolutions.

A single-ridge waveguide, shown in Fig. 2, is considered as thesecond example to verify the advantage of MR-OMTD method inefficiency. Because of the capacitive loading, fine grid spacing is takennear the ridges. Graded grid division is adopted, shown as in Fig. 3.

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2528 Shao, Wang, and Huang

Table 1. Cutoff frequencies (GHz) with MR-OMTD method for firstfour TEmn (m = 0) modes and percentage changes from the analyticalsolutions.

TE10 % TE20 % TE11 % TE21 % Analytical solutions

39.87 - 79.73 - 89.14 - 112.76 -

MR-OMTD 39.84 0.075 79.56 0.21 89.26 0.13 112.73 0.027

Figure 2. Cross section of a single-ridge waveguide (a = 3.100 mm,b = 0.775 mm, a′/a = 0.2 and b′/b = 0.35).

Figure 3. Graded grid division near the ridge.

We choose fc = 100 GHz, NL = 350 and s = 5.75× 1010. And the2-D computational space-domain is subdivided into a 21 × 46 latticealong the x- and y-directions, respectively, with the minimum gridspacing ∆xmin = ∆ymin = 1.0 µm.

From Table 2, we can see that the results of the MR-OMTD

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A memory-reduced 2-D order-marching time-domain method 2529

Table 2. Cutoff frequencies of the single-ridge waveguide.

TE10 (GHz) TE20 (GHz) CPU time (s) FDTD 32.84 105.88 186 OMTD 32.82 105.76 48

MR-OMTD 32.81 105.73 35

method are in a good agreement with that of the FDTD method andthe traditional OMTD method. The two OMTD methods consumemuch less CPU time than the FDTD method, in which the CFLstability condition imposes a tiny time step tied to the fine griddivision. In addition, compared with the traditional OMTD method,the MR-OMTD method shows improvement in the computationefficiency because of its reduction in memory requirement. Allcalculations are performed on an Intel Core2 2.1-GHZ machine.

4. CONCLUSIONS

This paper describes a memory-reduced 2-D OMTD method to studythe cutoff frequencies of TE modes of uniform waveguides. With thedivergence theorem, the memory storage of nonzero unknowns in thematrix is reduced by three-fourteens. And the field components nearthe conductors are also discussed. In the numerical examples, theproposed method yields results that show improvement in computationefficiency compared with the traditional 2-D OMTD method.

ACKNOWLEDGMENT

This work is supported by the National Natural Science Foundation ofChina (10771031) and the Youth Technology Foundation of UESTC(JX0734)

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