a memory-efficient fdtd algorithm for solving maxwell equations in cylindrical grids

3
1382 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 5, MAY 2004 [4] R. S. Elliott and W. R. O’Loughlin, “The design of slot arrays including internal mutual coupling,” IEEE Trans. Antennas Propagat., vol. 34, pp. 1149–1154, Sept. 1986. [5] G. J. Stern and R. S. Elliott, “Resonant length of longitudinal slots and validity of circuit representation: Theory and experiment,” IEEE Trans. Antennas Propagat., vol. AP-33, pp. 1264–1271, Nov. 1985. [6] S. R. Rengarajan, “Compound radiating slots in a broad wall of a rectangular waveguide,” IEEE Trans. Antennas Propagat., vol. 36, pp. 1116–1123, Sept. 1989. [7] H. Y. Yee, “The design of large waveguide arrays of shunt slots,” IEEE Trans. Antennas Propagat., vol. 40, pp. 775–781, July 1992. [8] J. Y. Li and C. H. Liang, “Moment method analysis of rectangular waveguide longitudinal slot arrays,” Chinese J. Radio Sci., vol. 4, pp. 428–432, 1998. [9] A. Enneking, R. Beyer, and F. Arndt, “Rigorous analysis of large fi- nite waveguide-fed slot arrays including the mutual internal and external higher-order mode coupling,” in Proc. IEEE AP Symp. Digest, vol. 38, July 2000, pp. 74–77. [10] Y. Zhang, Y. J. Xie, and C. H. Liang, “Application of the SN-precondi- tioning method to the integral equations for a slot-array antenna,” Mi- crowave Opt. Technol Lett., vol. 37, pp. 302–305, 2003. [11] F. X. Canning and J. F. Scholl, “Diagonal preconditioners for the EFIE using a wavelet basis,” IEEE Trans. Antennas Propagat., vol. 44, pp. 1239–1246, Sept. 1996. [12] K. Chen, “On a class of preconditioning methods for dense linear systems from boundary elements,” SIAM J Sci. Comput., vol. 20, pp. 684–698. A Memory-Efficient FDTD Algorithm for Solving Maxwell Equations in Cylindrical Grids Tongbin Yu and Bihua Zhou Abstract—A memory-efficient finite-difference time-domain (FDTD) al- gorithm for solving Maxwell equations in cylindrical grids is introduced. In the proposed algorithm, two equations that spatially link the three electric- field and three magnetic-field components respectively are used to eliminate one component each of E and H. This enables us to get a direct memory re- duction of 33% in the storage of the fields by using a spacial field updating procedure. A numerical example considering a two dimensional case is pre- sented to validate the efficiency of the proposed algorithm. Index Terms—Cylindrical coordinates, finite-difference time-domain (FDTD) methods, memory-efficient. I. INTRODUCTION The finite-difference time-domain (FDTD) method [1] has been widely used for solving electromagnetic problems [2], [3]. However, This method has a drawback of high computational requirements. An attractive advance to overcome this drawback has been the introduction of the reduced FDTD(R-FDTD) algorithm[4], which can reduce com- puter memory requirements in the storage of the fields by up to 33%. The original R-FDTD concept is applied only to Cartesian coordinates. This paper extends R-FDTD concept to cylindrical coordinates, and presents a memory-efficient FDTD algorithm for solving Maxwell equations in cylindrical grids. The proposed algorithm allows a 50% Manuscript received January 12, 2003; revised July 8, 2003. This work is supported by the National Science Foundation of China under Grant 60172002. The authors are with the EMP Laboratory, Nanjing Engineering Institute, Nanjing Jiangsu 210007, China. Digital Object Identifier 10.1109/TAP.2004.827508 increase in the computational volume for a given computer memory size, with a little increase in computation and code complexity. II. THEORY In a homogeneous isotropic media, Maxwell’s curl vector equations can be represented by a system of six scalar equations in cylindrical coordinates. Using time step and space step , , , we have the FDTD updating formula for (1) Where (2) (3) Assuming that initially at time , all the field components are zero over the whole computational domain, and using recurrence relation of (1), we have (4) where (5) (6) 0018-926X/04$20.00 © 2004 IEEE

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1382 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 5, MAY 2004

[4] R. S. Elliott and W. R. O’Loughlin, “The design of slot arrays includinginternal mutual coupling,” IEEE Trans. Antennas Propagat., vol. 34, pp.1149–1154, Sept. 1986.

[5] G. J. Stern and R. S. Elliott, “Resonant length of longitudinal slots andvalidity of circuit representation: Theory and experiment,” IEEE Trans.Antennas Propagat., vol. AP-33, pp. 1264–1271, Nov. 1985.

[6] S. R. Rengarajan, “Compound radiating slots in a broad wall of arectangular waveguide,” IEEE Trans. Antennas Propagat., vol. 36, pp.1116–1123, Sept. 1989.

[7] H. Y. Yee, “The design of large waveguide arrays of shunt slots,” IEEETrans. Antennas Propagat., vol. 40, pp. 775–781, July 1992.

[8] J. Y. Li and C. H. Liang, “Moment method analysis of rectangularwaveguide longitudinal slot arrays,” Chinese J. Radio Sci., vol. 4, pp.428–432, 1998.

[9] A. Enneking, R. Beyer, and F. Arndt, “Rigorous analysis of large fi-nite waveguide-fed slot arrays including the mutual internal and externalhigher-order mode coupling,” in Proc. IEEE AP Symp. Digest, vol. 38,July 2000, pp. 74–77.

[10] Y. Zhang, Y. J. Xie, and C. H. Liang, “Application of the SN-precondi-tioning method to the integral equations for a slot-array antenna,” Mi-crowave Opt. Technol Lett., vol. 37, pp. 302–305, 2003.

[11] F. X. Canning and J. F. Scholl, “Diagonal preconditioners for the EFIEusing a wavelet basis,” IEEE Trans. Antennas Propagat., vol. 44, pp.1239–1246, Sept. 1996.

[12] K. Chen, “On a class of preconditioning methods for dense linearsystems from boundary elements,” SIAM J Sci. Comput., vol. 20, pp.684–698.

A Memory-Efficient FDTD Algorithm for Solving MaxwellEquations in Cylindrical Grids

Tongbin Yu and Bihua Zhou

Abstract—A memory-efficient finite-difference time-domain (FDTD) al-gorithm for solving Maxwell equations in cylindrical grids is introduced. Inthe proposed algorithm, two equations that spatially link the three electric-field and threemagnetic-field components respectively are used to eliminateone component each of E and H. This enables us to get a direct memory re-duction of 33% in the storage of the fields by using a spacial field updatingprocedure. A numerical example considering a two dimensional case is pre-sented to validate the efficiency of the proposed algorithm.

Index Terms—Cylindrical coordinates, finite-difference time-domain(FDTD) methods, memory-efficient.

I. INTRODUCTION

The finite-difference time-domain (FDTD) method [1] has beenwidely used for solving electromagnetic problems [2], [3]. However,This method has a drawback of high computational requirements. Anattractive advance to overcome this drawback has been the introductionof the reduced FDTD(R-FDTD) algorithm[4], which can reduce com-puter memory requirements in the storage of the fields by up to 33%.The original R-FDTD concept is applied only to Cartesian coordinates.This paper extends R-FDTD concept to cylindrical coordinates, andpresents a memory-efficient FDTD algorithm for solving Maxwellequations in cylindrical grids. The proposed algorithm allows a 50%

Manuscript received January 12, 2003; revised July 8, 2003. This work issupported by the National Science Foundation of China under Grant 60172002.

The authors are with the EMP Laboratory, Nanjing Engineering Institute,Nanjing Jiangsu 210007, China.

Digital Object Identifier 10.1109/TAP.2004.827508

increase in the computational volume for a given computer memorysize, with a little increase in computation and code complexity.

II. THEORY

In a homogeneous isotropic media, Maxwell’s curl vector equationscan be represented by a system of six scalar equations in cylindricalcoordinates. Using time step�t and space step�r,�',�z, we havethe FDTD updating formula for Er

En+1r i+

1

2; j; k

= XA � Enr i+

1

2; j; k +XB

Hn+

z i+ 1

2; j+ 1

2; k �H

n+

z i+ 1

2; j� 1

2; k

i+ 1

2�r�'

�XB

Hn+

' i+ 1

2; j; k+ 1

2�H

n+

' i+ 1

2; j; k� 1

2

�z(1)

Where

XA =1� ��t

2"

1 + ��t

2"

(2)

XB =�t

"

1 + ��t

2"

(3)

Assuming that initially at time t = 0(n = 0), all the field componentsare zero over the whole computational domain, and using recurrencerelation of (1), we have

En+1r i+

1

2; j; k

= XB �

SUMHn+

z i+ 1

2; j + 1

2; k

i+ 1

2�r�'

SUMHn+

z i+ 1

2; j � 1

2; k

i+ 1

2�r�'

SUMHn+

' i+ 1

2; j; k + 1

2

�z

+SUMH

n+

' i+ 1

2; j; k � 1

2

�z(4)

where

SUMHn+

z i+1

2; j +

1

2; k

= XA � SUMHn�

z i+1

2; j +

1

2; k

+Hn+

z i+1

2; j +

1

2; k (5)

SUMHn+

' i+1

2; j; k +

1

2

= XA � SUMHn�

' i+1

2; j; k +

1

2

+Hn+

' i+1

2; j; k +

1

2: (6)

0018-926X/04$20.00 © 2004 IEEE

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 5, MAY 2004 1383

Similar treatment of the two updating formulas for E' and Ez yields

En+1' i; j +

1

2; k

= XB �

SUMHn+r i; j+ 1

2; k+ 1

2

�z

SUMHn+r i; j+ 1

2; k� 1

2

�z

SUMHn+z i+ 1

2; j+ 1

2; k

�r

+SUMH

n+z i� 1

2; j+ 1

2; k

�r(7)

En+1z i; j; k +

1

2

= XB � SUMHn+' i+

1

2; j; k +

1

2

i+ 1

2

i�r

� SUMHn+' i�

1

2; j; k +

1

2

i+ 1

2

i�r

� SUMHn+r i; j +

1

2; k +

1

2

1

i�r�'

+SUMHn+r i; j�

1

2; k+

1

2

1

i�r�'(8)

where

SUMHn+r i; j +

1

2; k +

1

2= XA � SUMH

n�r

� i; j +1

2; k +

1

2+H

n+r i; j +

1

2; k +

1

2: (9)

From (4), (7) ,and (8) we can get three equations for En+1r (i �

1=2; j; k), En+1' (i; j � 1=2; k), and En+1

z (i; j; k� 1=2). Combiningthese three equations with (4), (7), and (8), we can easily find thefollowing:

i+ 1

2

iEn+1r i+

1

2; j; k �

i� 1

2

iEn+1r i�

1

2; j; k

+1

i�'En+1' i; j+

1

2; k �

1

i�'En+1' i; j�

1

2; k

+�r

�zEn+1z i; j; k+

1

2�

�r

�zEn+1z i; j; k+

1

2:=0 (10)

Similarly, the equation that spatially links the three magnetic-fieldcomponents is yielded

i+ 1

i+ 1

2

Hn+r i+ 1; j +

1

2; k +

1

2

i

i+ 1

2

Hn+r i; j +

1

2; k +

1

2

+1

i+ 1

2�'

Hn+' i+

1

2; j + 1; k +

1

2

1

i+ 1

2�'

Hn+' i+

1

2; j; k +

1

2

+�r

�zH

n+z i+

1

2; j +

1

2; k + 1

�r

�zH

n+z i+

1

2; j +

1

2; k : = 0 (11)

Fig. 1. Geometry for metallic monopole fed through an image plane by acoaxial transmission line.

Fig. 2. Reflected voltage in the coaxial line for a cylindrical metallicmonopole antenna excited by a 1 V Gaussian pulse, calculated by the twoFDTD algorithms.

Equations (10) and (11) can be used to eliminate one component eachof E andH. So following the field updating procedure similar to thatof [4], a memory-efficient FDTD algorithm for solving Maxwell equa-tions in cylindrical grids is achieved. In the presence of sources andconductors, some field components cannot be calculated by (10) and(11). A simple way to solve this problem is calculating these field com-ponents via original FDTD formulas.

III. NUMERICAL VALIDATION

In order to validate numerically the derived memory-efficient FDTDalgorithm, a metallic cylindrical monopole driven through an imageplane by a coaxial transmission line [5] has been considered, as shownin Fig. 1. The conductors of the coaxial line, the monopole and theimage plane are assumed to be perfect. The parameters in Fig. 1 isselected as follows: a = 0:0035 m, b = 0:008 m, h=a = 230, c=a =200, d=a = 100. On the plane A � A0 in the coaxial line, the time-varying, incident field is

~Ei(t) =U i(t)

r ln b

a

(12)

where

U i(t) = exp�t2

2� 2p(13)

1384 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 5, MAY 2004

The monopole antenna is characterized by the time �a = h=c. Theratio of the characteristic time for the Gaussian pulse, �p, to the char-acteristic time for the antenna, �a, is taken to be �p=�a = 1:61�10�1.Using 2-D original FDTD algorithm and the proposed FDTD algorithmfor cylindrical grids, the reflected voltage on planeB�B0 in the coaxialline is calculated, as shown in Fig. 2. From Fig. 2, we can see that theresults calculated by the two algorithms agree exactly.

IV. CONCLUSION

Amemory-efficient FDTD algorithm is derived for solvingMaxwellequations in cylindrical coordinates. This algorithm is based on therelationship that spatially links the three electric-field and three mag-netic-field components respectively. The efficiency of the proposed al-gorithm is validated by a 2-D numerical example.

REFERENCES

[1] K. S. Yee, “Numerical solution of initial boundary value problems in-volvingMaxwell’s equations in isotropic media,” IEEE Trans. AntennasPropagat., vol. AP-17, pp. 585–589, 1966.

[2] A. Taflove, “Review of the formulation and applications of the finite-dif-ference time-domainmethod for numerical modeling of electromagneticwave interactions with arbitrary structures,” Wave Motion, vol. 10, pp.547–582, Dec. 1988.

[3] K. Kunz and R. Luebbers, The Finite Difference Time Domain Methodfor Electromagnetics. Boca Raton, FL: CRC, 1993.

[4] G. D. Kondylis, F. D. Flaviis, G. J. Pottie, and T. Itoh, “A memory-efficient formulation of the finite-difference time-domainmethod for thesolution of Maxwell equations,” IEEE Trans. Microwave Theory Tech.,vol. 49, pp. 1310–1320, July 2001.

[5] J. G. Maloney and G. S. Smith, “A study of transient radiation fromtheWu–King resistive monopole-FDTD analysis and experimental mea-surements,” IEEE Trans. Antennas Propagat., vol. 41, pp. 668–676,May1993.