The HSLO(3)-FDTD With Direct-Domain and

Temporary-Domain Approaches On Infinite Space

Wave PropagationMohammad Khatim Hasan

Department of Industrial ComputingUniversiti Kebangsaan Malaysia

Bangi, Selangor, MalaysiaEmail: khatim71@hotmail.com

*Mohamed Othman, *Rozita Johari and+Zulkifly Abbas

*Department of CommunicationTechnology and Network

and +Department of PhysicsUniversiti Putra Malaysia

Serdang, Selangor, MalaysiaEmail: mothman@fsktm.upm.edu.my

Jumat SulaimanDepartment of MathematicUniversiti Malaysia Sabah

Kota Kinabalu, Sabah, MalaysiaEmail: jumat@ums.edu.my

Abstract-In this paper, some numerical simulations by anew high-speed low order (3)-finite difference time domain(HSLO (3)-FDTD) method with direct-domain (DD) andtemporary-domain (TD) approaches are conducted to simulateone dimensional free space wave propagation represented bya Gaussian pulse. The simulation is conducted on 2 meter ofsolution domain truncated by a simple absorbing boundarycondition. The efficiency of the new schemes are analyzedand compared with the standard finite difference time domain(FDTD) method in terms of processing time, amplitude andglobal error. The new schemes (HSLO (3)-FDTD by bothapproaches) solve the problem faster and produce similar resultto the output simulated by the standard FDTD.

Keywords-FDTD; HSLO-FDTD; numerical simulationof wave propagation; direct-domain approach; temporary-domainapproach.

I. INTRODUCTIONAdvancement in electronic devices, such as wireless com-

munication devices is very crucial in today technology. Clas-sical "trial and error" design paradigm for electromagneticdevices is very costly in terms of money and time taken. Thesignificant advances in computer modeling of electromagneticinteractions in the last few decades have made possible toshift from the classical method to one that heavily relies oncomputer simulation. Computational electromagnetics (CEM)has been a great technological importance and due to this,it has become a central problem in present-day computationalscience. Industrial and engineering requirements have motivateadvances in computational electromagnetic fields. Beside that,recent advances in computer technology also motivate thefields. In computational electromagnetic, numerical methodplays an important role. The method facilitates the advance-ment of research in many fields. The electromagnetics phe-nomenon can be described by Maxwell equations.Maxwell equations can be solved either in time-domain or

frequency-domain. Furthermore, the numerical method can berepresented by partial differential equation (PDE) formulation

or integral equation (IE) formulation. In frequency domain,the most popular integral solver is the Method of Moment(MoM). This method reduces the volume of the problem spa-tial dimensions by one [1]. This gives advantages to solve threedimension problems by two dimensional methods. However,MoM results in a dense linear system of equations. Solvingthis system with Gauss elimination has the complexity ofo(N3), where the size of the matrix is N x N. One wayto solve this workload is through iterative method, whichis usually based on matrix-vector multiplication, with thecomplexity of O(N2). If the Maxwell equations in PDE form,which is the Hemholtz equation, the most common method isthe Finite Element Method (FEM)[2] and the Finite DifferenceMethod (FDM)[3]. Most of the FDM method use the iterativeapproaches.

In time domain, the methods for JIE have not been usedwidely. Most of the methods are called Marching-on-in-time(MOT). The method however prone to instability [4]. Howeverthe instability can be overcome by using implicit time steppingmethod. In the PDE formulation however have several popularmethod, which are the finite-difference time-domain (FDTD)[5], finite-volume time-domain (FVTD) [6] and finite-elementtime-domain (FETD)[7]. FVTD was introduced to CEM by[6]. The method is borrowed from method that is extensivelyused in computational fluid dynamics (CFD), whereas FETDmethod [8]based on variational fonnulation of the PDE. Boththe FVTD and FETD are very suitable for unstructured grid.Most of recent research in electromagnetic solve time-

domain problem because they can solve a problem for severalfrequencies in a single calculation. The FDTD method is themost applied method used in this domain. In CEM, FDTDrefers to finite difference approximation to the Faraday's andAmpere's laws using second order accurate in time and space.The method was first proposed by Yee [9]. He used an electricfield (E) which was offset both spatially and temporallyfrom a magnetic field grid to obtain update equations that

I-4244-0000-7/05/$20.00 ©2005 IEEE. 1002

yield the present fields throughout the computational domainin term of past fields. The method was further developed[10] to solve electromagnetic scattering from a dielectriccylinder. This method is the most commonly used to solveproblem in time-domain because of its simplicity and directlyadapted to homogeneous problem. Since then, the methodhas become one of the most powerful Maxwell equationssolver of electrodynamics. It has been implemented on variousapplications such as electromagnetic penetration problem [III,electromagnetic scattering of complex object [12], radar crosssection [13], anisotropic material [14], monopole antenna([15], [16]), dipoles wire, ([17], [181), radiation pattem [19],mobile antenna ([20], [21], [22]), microstrip antenna ([23],[24]), wireless LAN ([25], [26], [27], [28]), radiation fromcable ([29], [30]) and PEC scatterer [31]). However, there aredrawbacks in the method. One of the drawbacks is it needs along processing time to simulate problems.The algorithmic simplicity, robustness, and potential for

great volumetric complexity afforded by FDTD modeling haveprompted an extraordinary level of interest in this technique.One of the primary reasons for the widespread popularity ofFDTD is it computational efficiency. FDTD is also well suitedfor parallel implementation. Beside FDTD, Transmission LineMethod (TLM)[32] is an altemative method that can beimplemented in the time-domain.

II. SOME LITERATURE ON INCREASING THE SPEED OFFDTD METHOD

To increase the speed of the method, some researchersapply higher-order scheme in FDTD method. Lan, Liu andLin [33] have developed a second order accuracy in timeand fourth order in space. This new scheme is then com-pared to FDTD by modelling plane-wave pulse propagatingthrough free-space. Result show that the higher-order methodreduces the numerical dispersion and has improved stability.Georgakapoulus, Birtcher, Balanis and Renaut [34] apply theFDTD(2,4) to a wave propagating problem. They also applythe method to solve array analysis, cavity resonance, monopoleantenna coupling and personal electronic devices ([35], [36]).In the paper, they used the same gridding concept in [37].The only difference is that they implement FDTD(2,4) at thecoarser grid. Propokidis and Tsiboukis [38] have implementFDTD(2,4) to simulate a lossy dielectric problem. All ofthe experimental result shows that FDTD(2,4) can simulateaccurately with coarser grid mesh than the FDTD(2,2). Theimplementation of higher-order truncation to Maxwell equa-tions increase the complexity of the method, however bysolving the problem in coarser grid will increase the speedof the processing time.The advancement of multiprocessor technology also influ-

ence the development of high speed FDTD algorithms. Perlik,Opsahl and Taflove [39] develop Parallel Virtual Machine(PVM) FDTD code to predict scattering of electromagneticfields on a Connection Machine (CM). The Maxwell equationsonto a massively single instruction multiple data (SIMD)parallel architecture. The machine is located at Cambridge

University. Rodohan and Saunders [40] implement a PVMparallel FDTD algorithm on network of Transputers to sim-ulate electromagnetic waves of a rectangular antenna. Theparallel machine consist of several type of heterogeneousworkstations with SIMD architechture. Jensen, Fijany andRahmat-Samii [20] proposed a new design of parallelism, inboth spatial and time. They solve circular scatterer but thedesign is merely analyzed theoretically. Nguyen, Zook andZhang [41] solve electromagnetic scattering problem usingdomain decomposition FDTD with heterogeneous network ofworkstation which composes of 4 SUN SPARCstation andfour IBM RS/6000. Schiavone, Codreanu, Palaniappan andWahid [42] implement PVM parallel FDTD code on Beowulfcluster to solve printed dipole antenna. The machine locatedat University of Central Florida. Guiffaut and Mahdjoubi[43] develop Message-Passing-Interface (MPI)FDTD code tocalculate the near and far field data. The experiment wasperformed on the CRAY T3E machine. Zhenghui, Benqing andZejie [44] describe the strategy used for parallel implementa-tion of the FDTD algorithm on cluster of workstation withtwo heterogeneous PC (Pentium-II 266/64Mb and Pentium-II 200/64Mb) and a workstation by PVM parallel softwareto solve one dimensional free space problem. Yang, Liao,Jen and Xiong [45] proposed new design of decompositionfor implementing parallel FDTD on domain decompositionusing MPI library to analyze coupling model of pulse intoslot. The researcher decompose the whole domain into severalsub-domains according to features of the problem. Moreover,each sub-domain may have its own lattices independently tosuit the special shapes. Araujo, Santos, Sobrinho and Rocha([46], [47]) analysis a monopole antenna and horn antennausing MPI library.

In this research, we increase the speed of FDTD processingtime via different approach. The new method which is calledthe High Speed Low Order (3) FDTD (HSLO(3)-FDTD)method will solve a one dimensional wave propagating in freespace problem on the same mesh size used by the standardFDTD using a single processor machine.

III. FREE SPACE MAXWELL EQUATIONS

Let's consider the Maxwell equations for free space below.

&E 1_ IV x H

At EO

aH 1=_ Iv x EAt Po

(1)

(2)

where E,H,Eo and ,a0 are the electric fields, magnetic fields,electric permittivity and magnetic permeability, respectively.For one-dimensional case using E, and Hy, the Eqs. (1) and(2) become

(9Ex 1 09Hy=E_ 1OH

Ot EO az

yHy 1 9ExAt PO Oz

(3)

(4)

1003

These are the equations of a plane wave with the electric fieldoriented in the x-direction, magnetic field in the y- direction,and travelling in the z-direction. For further details, see [481.

IV. HIGH SPEED Low ORDER (3) FINITE-DIFFERENCETIME-DOMAIN METHOD

The HSLO(3)-FDTD method was developed by borrowingthe concept implemented in Modified Explicit Group(MEG)introduced recently by Othman and Abdullah [49], which isthe extension of the concept of the half-sweep iterative methodpropose by Abdullah [50] through the Explicit DecoupledGroup (EDG) iterative method. Both of the Modified ExplicitGroup and the Explicit Decoupled Group are classified asiterative method and was used to solve eliptic type of problem.

In this research, we modify the concept used in MEGmethod and apply it to develop HSLO-FDTD for solving thefree space Maxwell equations in time-domain. The iterativeconcept in MEG is ignored because there is no matrix inHSLO-FDTD method to be solved. By taking central differ-ence approximations as below,

5F(i) Fn((i + it) - Fn(i - m2 ) O(A2) (5)6x mAX

for spatial derivatives and

6F(i) = Fn+ (i) +O(At2) (6)It At ( y6

for temporal derivatives in (3) and (4) yields

+ 2 (k)A\t

H n (k + m2 )-H&(k- m)m2e )A,I

H +(k + m) -Hyn(k + m ) EX+ (k + m)Exn (k)At miioAx

(8)By rearranging Eqs. (7) and (8) above the same way asstandard FDTD scheme, yields

2k2 2 (- k) (k+m)-Hy (k- (9)

H5'+l (k+2) = Hyn(k 2 )_t (E+x (k±m)-n-K+ (k))(10)

where= At

which m is odd number. See that (9) and (10) is a generalizedform of FDTD method where when m = 1, it is the standardFDTD, when m = 3, it is the HSLO(3)-FDTD, when m = 5,it is the HSLO(5)-FDTD, and so on. By using Eq. (9) and(10), we only calculate m of node points in the entire solutiondomain from 0 to T time steps.

In this paper, Eq. (9) and (10) with m = 3, will be usedto solve problem in solution domain given by Fig.l(b) withthe black square and circle is the magnetic and electric fields

.-... p.... u*- --u-@-u

(a)

(b)

Fig. 1. a) Solution domain for standard FDTD method and (b) Solutiondomain for HSLO(3)-FDTD method

have to be solved in the main HSLO(3)-FDTD algorithm.The uncalculated node will be solved later only at T byusing weighted average after the entire black node have beencalculated. The standard FDTD will be executed on solutiondomain given by Fig. l(a).

V. DIRECT-DOMAIN AND TEMPORARY-DOMAINALGORITHM

The HSLO(3)-FDTD can be implemented in direct-domain(DD) and temporary-domain(TD) algorithms. The al-gorithm for HSLO(3)-FDTD will be illustrated in Algo. 1 andAlgo. 2 with D* = -At = 0.5. From both algorithmsbelow (Algo. 1 and 2), we can see that both algorithms havethe same complexity which is G( NpNt), where Np is thenumber of grid points and Nt is the number of time step.

Transform Actual Solution Domain,Np toNTemporary Solution Domain,' p

Loop for T from 0 to NtLoop for i from O to N3p

Ex= -E_3 * (Hy+j - Hyi)End LoopGaussian Pulse,Ex = e-05*(Setting ABC BoundaryLoop for i from 0 to Np

Hy= H, - Dt * (Exi - Exi-1)3n *ooEnd Loop

End LoopTransform Temporary Solution Domain, "P to'3

Actual Solution Domain,NpCalculate remaining grid point in solution domain

Algo. 1. HSLO(3)-FDTD (TD) Algorithm.

Loop for T from 0 to NtLoop for i from 1,(i + 3) to Np

EX = Dt *(Yi+2 - Hyi-1)End LoopGaussian Pulse,Ex e-Setting ABC BoundaryLoop for i from O,(i + 3) to Np

Hyi = Dtu- L * (EXi+4-EXilEnd Loop

End LoopCalculate remaining grid point in solution domain

1004

Algo. 2. HSLO(3)-FDTD (DD) Algorithm.

We further analyze the complexity of both algorithms bynumber of arithmetic operation for addition/substraction(ADD/SUB) and multiplication/division (MUL/DIV). Thesummary of arithmetic calculation for FDTD, HSLO(3)-FDTD(DD) and HSLO(3)-FDTD(TD)are as Table 1.

TABLE ICOMPARISON OF FDTD, HSLO(3)-FDTD(DD) AND

HSLO(3)-FDTD(TD) ARITHMETIC COMPLEXITY

Arithmetic OperationMethod ADD/SUB MUIJDIVFDTD 4NpNt 2NpNtHSLO(3)-FDTD(DD) 4N9Nt + 2NE 2NpNt + 2Er4.3t 3 2N1 2N3HSLO(3)-FDTD(TD) 4EN + Np P2N + + I1 3 3

From Table 1, we can calculate relative gain by bothHSLO(3)-FDTD algorithms to standard FDTD. The formu-lation of gain for both method are as below.Relative gain (ADD/SUB) for HSLO(3)-FDTD(DD),

Gr(ADD/SUB) =2 1

and relative gain (ADD/SUB) for HSLO(3)-FDTD(TD),

Gr(ADD/SUB) = 2-3and taking the limit Nt -X oo, we obtain the percentage gainof 67% for both HSLO(3)-FDTD algorithms and relative gain(MUL/DIV) for HSLO(3)-FDTD(DD),

Gr(MUL'DIV) = 2 1

and relative gain (MUL/DIV) for HSLO(3)-FDTD(TD),2 1 1

Gr(MUL/DIV) =

and again taking the limit Nt -* oo, we obtain the percentagegain of 67% for both HSLO(3)-FDTD algorithms. As thecomplexity is the major contributor to processing time, we

predict that the maximum relative reduction in processing timefor both HSLO(3)-FDTD algorithms are 67%.

VI. NUMERICAL EXPERIMENT AND RESULTSThe effectiveness of HSLO(3)-FDTD method is analyzed

by executing a one dimensional free space wave propagationproblem with Gaussian pulse as the point source. We willgenerate a Gaussian pulse at the middle of the solutiondomain of 2 meter, truncated with simple absorbing boundarycondition. To ensure the accuracy of the simulated result, thesolution domain is discretize into 600 grid points with cellsize of 0.0033 meter and time slice of 5.5 x 10-12 ns. Theexperiment was run on Intel Pentium 3 of Mobile CPU 1 GHz727 MHz 256 MB of RAM with LINUX operating system.The result of simulation is given in Fig. 2 and 3. From

those figures, both HSLO(3)-FDTD algorithms results are

Fig. 2. Wave propagation from the centre of solution domain at Ins

Fig. 3. Wave propagation from the centre of solution domain at 4.4 ns

very similar with standard FDTD. These figures (Fig. 2 &3) shows the behavior of wave propagation in free space fromthe point source until being absorbed by the the absorbingboundary. However, there exist a small reduction in accuracyof approximation by HSLO(3)-FDTD method. The reductionin accuracy are shown in Fig. 3 & 4 as global error inpower density unit. From the figure, we found that HSLO(3)-FDTD(DD) has better accuracy than HSLO(3)-FDTD(TD) butHSLO(3)-FDTD(TD) method approximate amplitude similarto standard FDTD (refer Fig. 5) than HSLO(3)-FDTD(DD).The comparison of processing time is given in Fig. 6. and

we can see that both new schemes simulate the problem fasterthan the standard FDTD scheme with 49.99% to 61.77%reduction in processing time for HSLO(3)-FDTD(DD) and44.50% to 60.100% reduction in processing time for HSLO(3)-FDTD(TD). As mention before, it is expected that the max-

imum reduction in processing time gain by both HSLO(3)-FDTD approaches are 67%.

VII. CONCLUSION AND FUTURE RESEARCH

The HSLO(3)-FDTD method give us the opportunity tosolve 1 grid point of the solution domain in the main loop of

1005

0.05i -~~~~~FOTD

X HSLO3)-FDTD(DD)0 04 HSLO"3?FDTlDITDI

IA~~~~~~~~~~~~~~~~~~~~~~~~~~~~P

0.02--,As003 jA

t 0.01 A A'AlAAA60.00 A.--.

002 n [ 7 171 205 239 273 307 341 375 409 443 3

IA: I~~VA

-0.03 l A

-0.04 AA

-0.05Grid points

Fig. 4. Global error for HSLO(3)-FDTD(DD) and HSLO(3)-FDTD(TD)tostandard FDTD methods before absorbed by ABC

-.FDTDH*HSL0(3}-FDTD(DD0)

0.00 * HSLO(3)-FDTD(TD)I!

1.00

1.00

0.99 0-" ~ ~ ~ "0.99

.1. 2.2 3.3Tinmes)

Fig. 5. Amplitude for standard FDTD, HSLO(3)-FDTD(DD) and HSLO(3)-FDTD(TD)

20 -

18 - -- HSLO3)-FDTD(DD)0 16 -HSLO(3)-FDTD(TD)

r_ 16 - - =

0 4-14

10

0.6

0

2 -

1.1 2.2 3.3 44Time (ns)

HSLO(3)-FDTD and the remaining point only at the requiredtime step. This approach will increase the speed of FDTDalgorithm. The Performance of this scheme was tested forproblem in one dimensional free space propagation with ABCboundary condition. The major advantages of this schemeis that it requires less processing time and less complexityalgorithm than the existing FDTD scheme but there exista small reduction in its accuracy. It is clearly shown thatHSLO(3)-FDTD by both approaches are better than standardFDTD in one-dimension for free space wave propagatingsimulation.

In this paper, we demonstrate the effectiveness of thenew method on free space wave propagation with simpleabsorbing boundary via direct-domain and temporary-domainapproach. Both approach show tremendous result in simulatingproblems. In the near future, we will apply this method for twodimensional problem.

ACKNOWLEDGMENT

The authors would like to thank the Govemment ofMalaysia, Public Service Department of Malaysia and Uni7versiti Kebangsaan Malaysia for the first author study leavesand sponsorship.

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1 .60E-05 -HSLO(3)-FDT0(DDj

1.40E-05

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° 1.0E-05

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