Improving the accuracy of the MFIE with the choice of basis functions

Improving the

Accuracy

of

the

MFIE

with the Choice of

Basis Functions'

Ozgiir Ergiil' and Levent Giirel

Deparhnent of Electrical and Electronics Engineering Bilkent University

Ankara,

Turkey

(ergul@e.bilkent.edu.tr, Igurel@bilkent.edu.t)

1. Introduction

In the method-of-moments (MOM)

[I]

and the fast-multiple-method (FMM) [21 solutions of the electromagnetic scattering problems modeled by arbitray planar triangulations, the magnetic-field inlegal equation (MFIE) can be observed to give less accurate results compared to the electric-field integral equation (EFIE), if the current is expanded with the bo-Wilton-Glisson (RWG) [3] basis functions. The inaccuracy is more evident for problem geometries with sharp edges or tips 141. This paper shows that the accuracy of

the MFIE depends strongly on the quality of the current modeling and that the accuracy can be significantly improved by the choice of the basis functions. Campansons are performed for four different basis functions.

2. The

Use

of

Basis Functions in the MFIE

Application of the MOM an the MFIE requires the evaluation of the impedance matrix elements

2 - = - p -

r t , ( F ) . r i x

j & ' C f l ( ~ ' ) x ~ k ( ~ , ~ ' ) ,

(1)

S" S"

where

.

i

and inrepresent the testing and basis functions, respectively. Different from the interaction expression for the EFIE [3], calculation of Equation (1) does not put any resmction on the choice of the basis and testing functions. Therefore, bath divcrgence- conforming and curl-conforming functions can be used lo expand the current, if the interactiore will be calculated by this equation. On the other hand, replacement of the V operator on the basis leads to a new expression as

'

This work was supponed by the Turkish Academy of Sciences in the framework of the Young

SCientiSt Award Program (LGITUBA-GEBIP~002-1-12), and by the Scientific and Technical Research Council of Turkey (TUBITAK) wdcr Rcscarch Grant l03E008.

,

(a) (h)

Figure I . (a) Cube with I A edge, (h) triangulation with NI0 mesh sire

2. Results

Figure I(a) shows a scattering problem that involves a cube with Ih edge. The cube is

hiangulaled with N I 0 mesh size (Figure l(b)) and an incident field propagates in the -x direction with a y-polanzed eleclric field. Figure 2(a) represents the magnitude of the y (dominant) component of the induced c w e n t on the x = 0.5 (front) iiurface found by the FMM implementations using the MFIE formulation and different basis functions. It can

be observed that the current i s not modeled sufficiently with the RWG functions. The curl-conforming n x RWG gives better representation: but the TL and D x T L functions

have even better performances. The singularity of the current near the edges can be observed to be significantly better modeled with the latter two functions.

Figure 3 shows the total radar cross section (RCS) values on the x-y plane far the cube

geomehy in Figure I(a). For all types of the basis functions, the mesh size is changed

from

N5

to N20. It can be observed that the RCS curves are converging faster for the TL or n x TL hasis functions. The RCS values obtained by the M I 0 mesh size is more

reliable for these functions as compared to the RWG function that leads to very slow convergence.

Figure 2. Magnitude of the y component of the induced current on the kont surface of the cube in F i y R ](a) represented by (a1 RWG, (b) n x RWG, ( e ) TL and (d) n x TL functions.

4.

Conclusion

The accuracy of the MFIE for the MOM and FMM ~alutions of the electromagnetic scattering problems modeled by arbitrary planar triangulations can be significantly improved by the choice of basis functions. The cunent distribution and the RCS results

can be observed to be more accurate for the TL and n x TL functions. With the usual

choice of the mesh size as NlO, the values obtained by the MFIE with RWG hasis

functions are not very reliable if the geometry of the problem has sharp edges and tips. Far these problems, TL and n x TL functions can be used to improve the efficiency in spite of the increased computational cost.

c

8

U w 1s l W

c

(*“I

(3

~ ~ pl*Ifl.pr”)

(4

Figure 3. RCS valucs OD the x-y plane for the cube geometry with different mesh Sizes obtained by the use of (a) RWG, (b) n X RWG, (c) TL and (d) n x TL functions in MFlE formulation.

References

[ I ] R F. Harringlon, Field Compumlion by Momenf Mefhodr. IEEE Press, 1993.

[2] R. Coihnan, V. RoWllin, and S. Wandzwa, “The fast multipole method (FMM) for the wave

equation: apedestrian prsscriptios”lEEEAn1. Propag. Mag., vol. 35, no. 3, pp. 7-12, June 1993. [3] S. M. b o , D. R. Wilton, and A. W. Glissan, “Eleetramagnctic Scattering by surfaces Of

arbitrary shape,”lEEE Tram. Antenna Propogol., vol. AP-30, pp. 409-418, May 1982. [4]

0.

E& and L. GUrel, “Investigation of the Inaccuracy of the MFlE Discretized with the

RWG Basis Functions,” 2004 IEEE AP-S Inrernolionol Symporium and URSl Radio Science

Meefing, Montsrey, CA, June 2004.

[SI L. C. Trintinalia and H. Ling, “An improved hiangular patch basis for the method of

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[6] R. E. Hadgcs and Y. Rahmat-Samii, “The evaluation of MFIE integrals with the use of vector triangle basis functions,” Micro. Opf. Tech. L d l . , vol. 14, no. I , pp. 9-14, Jan. 1997.