Progress In Electromagnetics Research, PIER 80, 29–44, 2008

FAST ANALYSIS OF ANTENNA MOUNTED ONELECTRICALLY LARGE COMPOSITE OBJECTS

J. Yuan † and Y. Qiu

EMC Laboratory of school of electromechanical engineeringXidian UniversityXi’an, Shaanxi 710071, P. R. China

J. L. Guo, Y. L. Zou, and Q. Z. Liu

National Key Laboratory of Antennas and Microwave TechnologyXidian UniversityXi’an, Shaanxi 710071, P. R. China

Abstract—Several characteristics of the wire antenna on electricallylarge composite body are analyzed by an adaptive multilevel fastmultipole algorithm (MLFMA). Adaptive MLFMA is applied to theboundary integration of the analysis model. With the basis functionsand testing functions expanded with Dirac functions on differentposition, the calculation of impedance integration can be simplifiedand all the translation process can be calculated by fast Fouriertransformation (FFT). Good agreement between the computed andmeasured results of antenna characters is obtained.

1. INTRODUCTION

In engineering the antenna characteristics are usually obtained inlaboratory by experiment and then the antenna is installed oncarrier directly. In fact the antenna characteristics such as radiationpattern, input impedance and gain are always affected by the carrierbody, which will influence the work performance and electromagneticcompatibility of antenna system, that is to say, the carrier bodybecomes one part of the antenna system and the new system has newcharacteristics. For the profile complexity, large size and material† Also with National Key Laboratory of Antennas and Microwave Technology, XidianUniversity, Xi’an 710071, P. R. China

30 Yuan et al.

characteristics of the carrier, it is always very difficult to obtainantenna characteristics by experiment after installation. So, thesimulation is usually an effective method.

In this paper the characteristics of antennae on electrically largecomposite object is analyzed by an adaptive MLFMA. The boundary-integral equations of composite object are established first. However,the use of the conventional boundary-integral method for integralequations always results in full matrices. MLFMA [1, 2] is alwaysapplied to the integral equation to significantly reduce the memoryrequirement and computational time. But some targets are veryelectrically large and the electromagnetic simulation of these targetscan not be solved by traditional method on single computer inengineering, so the performance of traditional MLFMA should beimproved for electrically larger problems.

In the adaptive MLFMA presented, by expanding the basisfunctions and testing functions with Dirac functions on differentposition, the calculation of impedance integration can be simplifiedand all the translation operation can be calculated by FFT and theCPU time and computer memory are greatly reduced.

2. ANALYSIS MODEL AND FORMULATIONS

We assume there is an antenna on a composite body illustrated inFig. 1. The composite body consists of metallic and dielectric medium.In this paper, antenna is also seen as metal. The integral equations onthe surface of composite object should be established first. In Fig. 1,the metallic medium is in volume V1 and the surface of V1 is S1. The

antenna

S1V1

V2

S2

S1 Scd

Sce

Sd

V0 S0

Figure 1. An antenna on a composite object.

Progress In Electromagnetics Research, PIER 80, 2008 31

dielectric medium is in volume V2 and the surface of V2 is S2. VolumeV0 is free space and S0 is the boundary surface of free space and thecomposite object. Scd denotes the boundary surface of V1 and V2,Sce denotes the boundary surface of V0 and V2 and Sd denotes theboundary surface of V0 and V1, so S0 = Sce + Sd and S1 = Scd + Sd.

The equivalent electric current and magnetic current on Si(i =0, 1, 2) are

Ji = ni × Hi (i = 0, 1, 2) (on Si) (1)Mi = Ei × ni (i = 0, 1, 2) (on Si) (2)

where E i and H i are the total electromagnetic field in volume Vi. Involume V0 the total electromagnetic field is

θ(r)E0(r) = Ei − L0J0(r′) + K0M0(r′) (3)

θ(r)H0(r) = H i −K0J0(r′) − 1η20

L0M0(r′) (4)

where

θ(r) =

{ 1, r ∈ Vi, i = 0, 11/2, r ∈ Si, i = 0, 10, r /∈ Vi, r /∈ Si, i = 0, 1

(5)

Li and Ki are integration operators and Gi is the Green’s function infree space

LiZ(r′) = jωµi

∫S

[Z(r′) − 1

ω2µiεi∇∇′ · Z(r′)

]·Gids′ (6)

KiZ(r′) = −∫S

Z(r′) ×∇Gids (7)

Gi =e−jki|r−r′|

4π |r − r′| (8)

where ki = ω√µiεi and ηi =

õi

εi. We applied impedance boundary

condition (IBC) to surface Scd and Sce

E1 − (n1 · E1)n1 = ηcdη0(n1 × H1) (9)E0 − (n0 · E0)n0 = ηceη0(n0 × H0) (10)

where ηcd and ηce are the normalized surface impedance of Scd and Sce.On surface Sd the tangential components of electromagnetic field are

32 Yuan et al.

continuous

E1|tan = E0|tan (on Sd) (11)n1 × H1 = −n0 × H0 (on Sd) (12)

The combined electric field integration and magnetic field integration(CFIE) on Scd and Sce can be obtained by (9) and (10),

α

η0{E1 − (n1 · E1)n1} = β {ηcdn1 × H1} (on Scd) (13)

α

η0{E0 − (n0 · E0)n0} = β {ηcen0 × H0} (on Sce) (14)

with (3), (4), (13) and (14), we can obtain the CFIE on Scd, Sce andSd .

α

η0

{L1cdJcd − ηcdη0

[K1cd(Jcd × n1) +

12Jcd

]− L1dJd + K1dMd

}tan

−βηcd

{12Jcd + n1 ×

[K1cdJcd + L1cdηcdη0(Jcd × n1)/η2

1 −K1dJd

−L1dMd/η21

]}= 0 (15)

α

η0

{L0ceJce − ηceη0

[K0ce(Jce × n0) +

12Jce

]− L0dJd + K0dMd

}tan

−βηce

{12Jce + n0 ×

[K0ceJce + L0ceηceη0(Jce × n0)/η2

0 −K0dJd

−L0dMd/η20

]}=

1η0

{αEi

tan − βηceη0n0 × Hi

}(16)

1η0

{−L1cdJcd + K1cdηcdη0(Jcd × n1) + L0ceJce −K0ceηceη0(Jce × n0)

+(L1d + L0d)Jd − (K1d + K0d)Md}tan =1η0

Eitan (17)

n0 × {−K1cdJcd−L1cdηcdη0(Jcd×n1)+K0ceJce + L0ceηceη0(Jce × n0)

+(K1d + K0d)Jd +(

1η21

L1d +1η20

L0d

)Md

}= n0 × Hi (18)

Solving Equations (15)–(18), we can obtain the electric currentsand magnetic currents J cd, J ce, J d and M d. MLFMA is very suitable

Progress In Electromagnetics Research, PIER 80, 2008 33

for solving integral equations, in MLFMA Green’s function Gi can bewritten as

Gi =e−jki|r−r′|

4π |r − r′| ≈−jk

16π2

∫d2ke−jk·(rjm−rim′ )TL(k · rmm′) (19)

where

TL =L∑

l=0

(−j)l(2l + 1)h(2)l (krm′)pl

(k · rmm′

)(20)

is the translation operator of MLFMA. The gradient expression of Gi

is

∇Gi = −jk

[ −jk

16π2

∫d2ke−jk·(rjm−rim′ )TL(k · rmm′)

](21)

Applying the vector basis function f i and combining (6), (7), (19) and(21), we can obtain the multipole expansion expressions of L and Koperators

LZ(r′) =ωµk

16π2

∫ds

∫ds′e−jk·(rjm−rim′ )TL(k·rmm′)

(I−kk

)· fi(r′)ai

(22)

KZ(r′) =k2

16π2

∫ds

∫ds′e−jk·(rjm−rim′ )TL(k·rmm′)

(fi(r′)×k

)ai (23)

where ai is the expansion coefficients of electric current and magneticcurrent. Substituting (22) and (23) into (15)–(18), the aggregationoperator and disaggregation operator of MLFMA can be written as

Vsm′i(k) =∫

fi

(r′)e−jk·rim′ds′ (24)

Vfmj =M∑

m=1

αm

∫dse−jk·rjm

(I − kk

)· fj(r)

+N∑

n=1

βn

∫dse−jk·rjm

(fj(r) × k

)(25)

in (25), αm and βn are the coefficients of Li and Ki operators , M andN are the number of Li and Ki operators in Equations (15)–(18).

In traditional MLFMA there are three computation processes thatconsume CPU time and memory: 1) the computation of impedance

34 Yuan et al.

integration; 2) the computation of translation process; and 3) thecomputation of translation factor TL.

While computing the integral formulation of the near-fieldinteraction in MLFMA, the basis function f i and testing function f j

can be approximated as linear combinations of Dirac delta functionson the Gaussian integration points

g ≈ g =IG∑s=1

psr′δ3s(r − r′) g ∈ {fx, fy, fz} (26)

We let∫(x− xj0)(y − yj0)(z − zj0) ×

[g −

IG∑s=1

psr′δ3s(r − r′)

]ds = 0 (27)

the point (xj0, yj0, zj0) is at the center of f i or f j . According to (27),a matrix equation of the expansion coefficient psr′ can be obtained andthen psr′ can be derived.∑

WIGr′psr′ = OsIG(28)

where

OsIG=

∫(x− xj0)(y − yj0)(z − zj0)gjds (29)

WIGr′ =(x′ − xj0

) (y′ − yj0)(z′ − zj0

)(30)

Substituting (26) into the near-field interaction integral formula-tions, we can see that the calculation speed and precision of integralformulations are controlled by the Gaussian integral coefficient ws andthe expansion coefficient psr′ of basis and testing functions.∫

fi(r′) · fj(r) ·G(r, r′)ds =IG∑

si=1

IG′∑sj=1

wsipsiwsjpsjG (31)

For computing V fmj , testing function f j can be approximated inthe same way as in the near-field integral.

Through the expansion, the integral computation of near-fieldinteraction and V fmj can be simplified. The speed and precision canbe controlled also. For an arbitrarily shaped body, if the followingrules are used, the calculation error of integral computation can becontrolled within 1%.

Progress In Electromagnetics Research, PIER 80, 2008 35

1) For the surface whose curvature is discontinuous, if the distancebetween the triangle elements center and curvature center lessthan 0.5λ, where λ is wave length, then the normal number ofGaussian integral points should be employed.

2) For the surface whose curvature is continuous and finite, suchas sphere surface and cylinder surface, the number of Gaussianintegral points employed is less than 0.7 times of normal number.

3) For the surface whose curvature is continuous and infinite, forexample plane surface, the number of Gaussian integral pointsemployed is less than 0.6 times of normal number.While computing V sm′i, basis function f i can be approximated

as linear combinations of Dirac delta functions on the centers of gridsgrouped uniformly in MLFMA (Fig. 2).

basis function fi

Dirac function

Figure 2. Expansion of basis function fi.

gi ≈ gi =∑

r′∈Ci

βir′δ3(r − r′) gi ∈ {fix, fiy, fiz} (32)

where Ci is the number of grid centers around basis f i.We let∫(x− xi0)(y − yi0)(z − zi0) ×

[gi −

IG∑s=1

βir′δ3(r − r′)

]ds = 0 (33)

the point (xi0, yi0, zi0) is at the center of f i. According to (33), amatrix equation of the the expansion coefficient βir′ can be obtain.edand then βir′ can be derived.∑

WCir′βir′ = OiCi (34)

36 Yuan et al.

where

OiCi =∫

(x− xi0)(y − yi0)(z − zi0)gids (35)

WCir′ = (x′ − xi0)(y′ − yi0)(z′ − zi0) (36)

Through this expansion operation, the integral computation ofV sm′i can be simplified like V fmj and near-field interaction that wehave discussed above and there is another advantage by using thisexpansion. We notice that basis function f i is mapped to the gridcenters around f i through this expansion and the grids around f i arefilled by Dirac functions, then the translation formulation of MLFMAis a discrete circle convolution and can be calculated by FFT.∑

m′∈Gm′

TLSm′ = FFT−1{FFT ([Sm′ ]) × FFT ([TL])} (37)

where

Sm′

(k)

=∑

i

Vsm′i

(k)

ai, (38)

In practice we can enclose every parts of the target by multiplecubes (Fig. 3). Each cube is divided by grids in uniform size onconnection boundary and at same level of MLFMA, thus the lessempty grids are generated and higher efficiency of FFT can beobtained. While expanding basis function f i, expansion coefficient canbe calculated at the coarsest level of MLFMA only and the expansioncoefficient of other levels can be obtained by interpolation method.

cube 1

cube 3

cube 2

target

Figure 3. Target enclosed by multiple cubes.

While computing translation operator TL, the method given by [3]is used on the base of RPFMA [4], adaptive-RPMLFMA [5] and

Progress In Electromagnetics Research, PIER 80, 2008 37

FAFFA-MLFMA [6], thus the TL can be calculated on 1/8 spheresurface only and the computation complexity of MLFMA can bereduced significantly.

Using all of the improvement measures we have introduced above,the MLFMA have adaptive computation performances as follow: 1)adaptive computation of the integration of near-field interaction, V fmj

and V sm′i by the expansion of basis function and testing function;2) the empty grid around basis function f i are filled adaptively andthe computation of translation process meets the requirement of FFTadaptively; and 3) the precision and CPU time of the computation ofTL can be controlled adaptively. So the algorithm we presented is anadaptive algorithm.

To discrete the integral equations, the RWG basis function [7] isused for the surface of the object, and the basis function illustrated inFig. 4(a) is used for antenna (wire), the expression of which is shownas follow.

fWn (r) =

ρ+

n

l+n, r ∈ W+

ρ−n

l−n, r ∈ W−

0, etl

(39)

The basis function illustrated in Fig. 4(b) is used to wire-surfacejunctions, and the expression of this kind of basis function is [8]

fCn (r) =

fCP

n (r), r ∈ T+ln

fCWn (r), r ∈ Wire

0, etl

(40)

(a) (b)

Figure 4. Basis function used to wire and wire-surface joint.

38 Yuan et al.

where

fCPn (r) =

α+ln

αtn

1l+ln

[1 − (h+l

n )2

(h+ln · ρ+l

n )2

]ρ+l

n

h+ln

(41)

fCPn (r) = −ρ−

n

l−n(42)

3. RESULTS AND DISCUSSION

In this section, the characteristics of two HF communication antennaeon a ship are analyzed. The positions of these two antennae and thegrid model of ship are shown in Fig. 5. The coordinates of these twoantennae on the shipboard are (112.0,7.5,8.5) and (48.5,8.5,6.5). Thegrid model is used to obtain the geometrical data of the elementsonly. When we analyze this model, all the buildings on shipboardare seen as dielectric objects (εr = 9.0, µr = 1.1) and the shipbody is seen as perfect conductor (PEC). The ship dimension is about170×18×6.5 m3 and 116796 unknowns are obtained. All the simulationprocess is finished on microcomputer with PIV-2.4 GHz processor and4 GB memory.

antenna1

antenna2

X

Y

Z

Figure 5. Model of antenna on a ship.

The radiation pattern of these two antenna in horizontal plane andvertical plane are illustrated in Fig. 6–Fig. 9. For the radiation patternin horizontal plane, the experiment result in free space is presented firstand then the experimental result on shipboard. Experiment results are

Progress In Electromagnetics Research, PIER 80, 2008 39

0.0

0.3

0.6

0.9

1.20

30

60

90

120

150

180

210

240

270

300

330

0.0

0.3

0.6

0.9

1.2

Antenna1 in free space(Experimental) Antenna1 on shipboard(Numerical) Antenna1 on shipboard(Experimental)

Figure 6. Radiation pattern of antenna1 (25 MHz H-H).

0.0

0.3

0.6

0.9

1.20

30

60

90

120

150

180

210

240

270

300

330

0.0

0.3

0.6

0.9

1.2

Antenna2 in free space(Experimental)Antenna2 on shipboard(Numerical)Antenna2 on shipboard(Experimental)

Figure 7. Radiation pattern of antenna2 (25 MHz H-H).

obtained by using a scaled ship model with the scaled ratio 1:80. Wecan see that radiation pattern has been modified by the effect of theship body. The numerical result of radiation pattern in horizontal planeon shipboard is also shown in Fig. 6 and Fig. 7. The good agreement ofnumerical and experimental results show that the algorithm presentedin this paper has very high precision. For the radiation pattern invertical plane, only the experimental result in free space and the

40 Yuan et al.

0.0

0.2

0.4

0.6

0.8

1.0

0

30

60

90

120

150

180

210

240

270

300

330

0.0

0.2

0.4

0.6

0.8

1.0

Antenna1 in free space(Experimental) Antenna1 on shipboard(Numerical)

Figure 8. Radiation pattern of antenna1 (25 MHz V-V).

0.0

0.2

0.4

0.6

0.8

1.0

0

30

60

90

120

150

180

210

240

270

300

330

0.0

0.2

0.4

0.6

0.8

1.0

Antenna2 in free space(Experimental) Antenna2 on shipboard(Numerical)

Figure 9. Radiation pattern of antenna2 (25 MHz V-V).

numerical result on shipboard are presented. The experimental resulton shipboard is not given because it is very difficult to be obtained.

In Fig. 10–Fig. 12, the numerical results of the input resistance,input reactance and gain of these two antennae on shipboard areshown, where the experiment results in free space is also presented.Through both the experiment and numerical results we can see thatthe antenna characteristics have also been modified by the effect of theship body.

Progress In Electromagnetics Research, PIER 80, 2008 41

0 5 10 15 20 25 30

0

100

200

300

400

500

Rin/(o

hm)

f(MHz)

In free space(Experimental) Antenna1 on shipboard(Numerrical) Antenna2 on shipboard(Numerrical)

Figure 10. Input resistance of antennae.

0 5 10 15 20 25 30-800

-700

-600

-500

-400

-300

-200

-100

0

100

Xin/(o

hm)

f(MHz)

In free space(Experimental) Antenna1 on shipboard(Numerical) Antenna2 on shipboard(Numerical)

Figure 11. Input reactance of antennae.

To show the effectiveness of the adaptive MLFMA presented inthis paper, we also simulated all antennae characteristics by traditionalMLFMA, and the performances of them are shown in Table 1. Wecan see that the adaptive MLFMA has better performance than thetraditional one.

Table 1. Performance of Adaptive MLFMA and MLFMA.

Unknowns CPU time (s) Memory (MB)

Adaptive MLFMA 116796 139828 2659

MLFMA 116796 207669 3798

42 Yuan et al.

0 5 10 15 20 25 30

-6

-4

-2

0

2

4

Gai

n/(d

Bi)

f(MHz)

In free space(Experimental) Antenna1 on shipboard(Numerical) Antenna2 on shipboard(Numerical)

Figure 12. Gain of antennae in free space and on shipboard.

4. CONCLUSION

The computational complexity of the adaptive MLFMA presented inthis paper is only about 0.7×O(Ns lgNs) and its memory requirementis 0.7 × O(Ns), where Ns denotes the number of surface unknowns.The numerical results demonstrate that the method presented also hashigh precision, and the characteristics of antenna on electrically largercomposite targets can be generally, accurately, efficiently computed bythis technique.

REFERENCES

1. Song, J., et al., “Multilevel fast mutipole algorithm forelectromagnetic scattering by large complex objects,” IEEETrans. Antennas Propagat., Vol. 145, No. 10, 1488–1498, 1998.

2. Song, J. M. and W. C. Chew, “Multilevel fast multipole algorithmfor solving combined field integral equations of electromagneticscattering,” Microwave Opt. Tech. Lett., Vol. 10, No. 1, 14–19,1995.

3. Wang, H.-G., Z.-P. Nie, and J. Wang, “Optimization of theinvariant terms’ calculation in three dimensional MLFMA,” ActaElectronica Sinica, Vol. 28, No. 9, 105–107, 2000.

4. Wagner, R. L. and W. C. Chew, “A ray propagation fast multipolealgorithm,” Microwave Opt. Tech. Lett., Vol. 7, No. 10, 435–438,1994.

Progress In Electromagnetics Research, PIER 80, 2008 43

5. Hu, J., J. Hu, and Z.-P. Nie, “An adaptive ray-propagationmultilevel fast multipole algorithm,” Chinese Journal of RadioScience, Vol. 19, No. 6, 669–672, 2004.

6. Chew, W. C., T. J. Cui, and J. M. Song, “A FAFFA-MLFMAalgorithm for electromagnetic scattering,” IEEE Trans. AntennasPropagat., Vol. 50, No. 11, 1641–1649, 2002.

7. Rao, S. M., D. R. Wilton, and A. W. Glisson, “Electromagneticscattering by surfaces of arbitrary shape,” IEEE Trans. on AP,Vol. 30, No. 3, 409–417, 1982.

8. Hwu, S. U., D. R. Wilton, and S. M. Rao, “Electromagneticscattering and radiation by arbitrary conducting wire/surfaceconfigurations,” AP-S Digest, Vol. 6, No. 2, 890–893, 1988.

9. Guo, J.-L., J.-Y. Li, and Q.-Z. Liu, “Analysis of arbitrarilyshaped dielectric radomes using adaptive integral method basedon volume integral equation,” IEEE Trans. Antennas Propagat.,Vol. 54, No. 7, 1910–1916, 2006.

10. Zhou, B., Q. Liu, J. Guo, and Y. Ji, “Research on a sleeve antennamounted on metallic cylinder,” Journal of Eletromagnetic Wavesand Applications, Vol. 19, No. 6, 769–777, 2005.

11. Guo, J. L., J. Y. Li, and Q. Z. Liu, “Electromagnetic analysis ofcoupled conducting and dielectric targets using mom with a pre-conditioner,” Journal of Electromagnetic Waves and Applications,Vol. 19, No. 9, 1223–1236, 2005.

12. Zou, Y., Q. Liu, and J. Guo, “Fast analysis of body-of-revolutionradomes with method of moments,” Journal of ElectromagneticWaves and Applications, Vol. 21, No. 13, 1803–1817, 2007.

13. Zou, Y. L., J. Y. Li, and Q. Z. Liu, “Modified mode decompositionfor analyzing antennas with body of revolution radome,” Journalof Electromagnetic Waves and Applications, Vol. 21, No. 10, 1403–1410, 2007.

14. Guo, J. L., J. Y. Li, and Q. Z. Liu, “Analysis of antennaarray with arbitrarily shaped radomes using fast algorithm basedon VSIE,” Journal of Electromagnetic Waves and Applications,Vol. 20, No. 10, 1399–1410, 2006.

15. Zhou, H.-J., Q.-Z. Liu, Y.-Z. Yin, and W. B. Wei, “Studyof the band-notch function for swallow-tailed planar monopoleantennas,” Progress In Electromagnetics Research, PIER 77, 55–65, 2007.

16. Liu, S.-F. and S.-D. Liu, “Study on the impedance-matching tech-nique for high-temperature superconducting microstrip antennas,”Progress In Electromagnetics Research, PIER 77, 281–284, 2007.

44 Yuan et al.

17. Gupta, R. C. and S. P. Singh, “Analysis of radiation patternsof compound box-horn antenna,” Progress In ElectromagneticsResearch, PIER 76, 31–44, 2007.

18. He, Q.-Q. and B.-Z. Wang, “Radiation patterns synthesis for aconformal dipole antenna array,” Progress In ElectromagneticsResearch, PIER 76, 327–340, 2007.

19. Wu, Y.-J., B.-H. Sun, J.-F. Li, and Q.-Z. Liu, “Triple-bandomni-directional antenna for WLAN application,” Progress InElectromagnetics Research, PIER 76, 477–484, 2007.

20. Zhang, Y. J. and E. P. Li, “Fast multipole accelerated scatteringmatrix method for multiple scattering of a large number ofcylinders,” Progress In Electromagnetics Research, PIER 72, 105–126, 2007.

21. Lee, K.-C., J.-S. Ou, and C.-W. Huang, “Angular-diversityradar recognition of ships by transformation based approaches —including noise effects,” Progress In Electromagnetics Research,PIER 72, 145–158, 2007.