SIMULATION OF CURVED SURFACES BY PATCH MODELING IN THE STUDY OFELECTROMAGNETIC SCATTERING BY RESONANT SIZE BODIES

Fernando Las Heras Andres, Jose Luis Fernandez Jambrina

ABSTRACT

The study of the scattering of conducting closed surfaces of resonantelectric size is accomplished using the electrical field integral equation(EFIE) with the method of moments (MM) and planar triangular patch modeling.Patch models of regular polyhedrons of different number of faces aregenerated. The backscattering radar cross sections (RCS) of these bodies arecalculated in all directions (solid angle of 4n) of incidence of the planewave. Vertical linear polarization is used. A comparison between the resultsobtained with the numerical method and the theoretical results of thecircumscribed sphere is made, which permits to obtain some conclusions ofthe limitations in the simulation of curved surfaces by planar surfacepatches.

INTRODUCTION

MM is widely used in scattering problems involving resonant size conductingbodies. It is not the purpose of this contribution to enter in newdiscussions of such well known numerical method but to emphasize theadjacent problem of the modelling techniques. The conducting bodies studiedhave been modeled by triangular patches in order to calculate their RCS. Thepatches are planar so any patch model of a curved surface implies anapproximation to real model. The accuracy of the RCS calculated results whenmodelling a curved surface will depend on the geometrical fitness betweenthe patch model and the curved surface. As it will be seen later, thegeometrical fitness can be expressed in terms of the maximum distancebetween the planar patch surface and the curved surface.

NUMERICAL CODE

The numerical code used is an adaptation for RCS calculations of the onepresented in [11, whose theoretical formulation can be seen in [2]. Thiscode uses the two-potential EFIE, which is solved by using the method ofmoments. It uses subdomain type basis functions which are defined on pairsof adjacent triangular patches, with a constant current representation inthe common edge; the testing functions are equal to the basis functions.

SCHEME OF THE PROBLEM

In order to study the effect of the maximum distance between the planarpatches and the real curved surface, the simulation of a conducting sphere[41 by means of triangular patch modeled polyhedrons is presented. Regularpolyhedrons with different number of faces are modeled: a cube, an

Grupo de Radiacion. Dpto.SSR. E.T.S.I.de Telecomunicacion. U.P.M.Ciudad Universitaria s/n. - 28040 Madrid, SPAIN

1365

octahedron, a dodecahedron, an icosahedron and an eighty faces polyhedron.All these bodies are modeled in an effective way, with optimum patch densitycriterions validated by the comparison of the calculated RCS of bodies ofsimple geometry with experimental results [3]. By this way the effect ofgeometrical approximation when modeling curved surfaces is isolated from theeffect of a non-effective patch model of a planar surface. In all cases avertical polarized plane wave is used to obtain the backscattering RCS.

RESULTS

In the figures 1 and 2, it is shown the typical representation of thebackscattering RCS of a perfectly conducting sphere of radius a, normalizingthe RCS to its maximum cross section (mra ) and as a function of its maximumelectric perimeter (27ra/A) [41.

In the figure 1-a it is also represented the backscattering RCS of aperfectly conducting cube, where ac is the radius of its circumscribedsphere. The representation is made in a similar way than in the case of theconducting sphere but with a radius ac. It can be observed that there is anoptimum dimension of the cube, in terms of the radius of its circumscribedsphere, that equates the RCS of the cube with the RCS of the sphere.

In the figure 1-b it is represented the RCS of the cube for the optimumrelationship a/ac. It can be seen that there is a limit value of theelectrical size of the cube; onward this value the RCS of the cube isdependent on the angle of incidence of the plane wave. This dependence isshown in figure 2 where the backscattering RCS of the cube is represented asa function of the angular spherical coordinates which define each directionof incidence. It can be noted how the maximum scattering passes from beingmainly due to the diagonal cross sections of the cube (0=90 and 4=45,1350, 225, 3150) for small electrical sizes to being due to the six facesof the cube (0=90 and 0=0, 900, 1800, 270 ) for bigger electrical sizes.

In the figure 3 it is represented the backscattering RCS of an eighty facespolyhedron in similar terms than for the case of the cube. For this case itresults that the optimum value of the radius a is nearer to ac than it wasfor the case of the cube, and the range of simulation (a/X)msx has becomegreater (see table 1), although not so much as it would be deduced from theexamination of the parameter Df-s/X (maximum distance between the patchesand the simulated sphere) that seems to remain constant for the precedentexamples. This is because at the value (a/A )msx the length of the edges ofthe patches is 0.311A which is approximately the limit value of the patchdensity criterion used.

CONCLUSIONS

Two main conclusions are derived from the results:

- Each polyhedron inscribed in a sphere of radius ac can represent, in termsof RCS, a sphere of radius a, where a<ac . The value of the equivalentradius a will be as nearer to the value of ac as more faces have the regularpolyhedron.

- The simulation of the sphere by each polyhedron has a range of validity interms of electrical size. The parameter that better defines the range of

1366

simulation is the distance (in wave lengths) between the planar patchsurface (one or more patches) and the curved surface to simulate (bothrelated by the optimum relationship a/ac) . It is found that the simulationis correct if the maximum distance between the patch surface and thesimulated sphere (parameter Df-s in table 1) is in the interval[x/32 X/16], where X is the wave length. For greater distances the RCS ofthe polyhedron considered becomes dependent with the angle of incidence ofthe plane wave. It can occur that it is the electric size of the patcheswhich first limits the range of simulation as in the case of the eightyfaces polyhedron presented.

The study of the simulation of the sphere can be generalized to surfaceswith different curvature radius.

REFERENCES

1 M.F. Costa, R.F. Harrington. Electromagnetic Radiation and Scatteringfrom a System of Conducting Bodies Interconnected by Wires. April 1983.Report TR-83-8, Syracuse University, Syracuse, New York 1320.2 S.M.Rao. D. R. Wilton. A. W. Glisson. Electromagnetic Scattering BySurfaces of Arbitrary Shape. May 1982. IEEE Transactions on Antennas andPropagation, vol. AP-30, no.3.3 W. Anderson. RCS Prediction Techniques: A Review and Comparison withExperimental Measurements. 24-26 June 1986. Conference Proceedings. MilitaryMicrowaves.4 J.J. Bowman, -T.B.A. Senior, P.L.E. Uslenghi. 1969. Electromagnetic andacoustic scattering by simple shapes. Radiation Laboratory, the Universityof Michigan, USA. North Holland publishing company Amsterdam.

C |EOCTAHEDRON DODECAHEDRON ICOSAIHEDRON 80-FACES

No.faces 6 8 12 20 80

No. vertex 8 6 20 12 42

No. edges 12 12 30 30 90

Ca/ac)opt. 0.75 0.72 0.86 0.86 0.95

(2ua/X)mAx 0.9 0.9 2.0 2.2 3.0

(a/X)msx 0.143 0.143 0.318 0.350 0.477

D /ac 0.423 0.423 0.205 0.205 0.066f s c_ _ _ ____-

[(D f-s )mx 0- 230 0.1l98 0. 076 0. 076 0. 017|

CD /X)msx 0.033 0.028 0.024 0.027 0.008

edge length/X 0.104 0.140 0.264 0.214 0.311|at (a/X)mhx() maximum value

Table 1: Geometrical parameters in the simulation of the RCSof a sphere by polyhedrons

1367

/

L

0.50 1.00

--- ac=aac-=1.O6aac=l. 14aac=l. 22aac=l .28aac=l. 35aac=l. 39aac=l. 43aac=1. 47a

-_Sphere (BechtelJ

(a)

1 .50 2.002.00

1-

(b)

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

2Tra/X

Fig.1. Backscattering RCS of a cube: a) Patch model and obtention ofthe optimum a/ac. b) Dependence with the angle of incidence

1368

6.00

5.50 -

5.00

4.50 9

4.00

N 3.50 -

¾3.00

X 2.50 -

2.00 9

i.50

1.00

0.50

0.0000.00

F005.50

5.00

4.50

4.00

3.50

3.00

2.50

2.00

N

I-)cs

uP34

1.50

1.00

0.50

0.00

Z

(.i.. e olfl.,---- 11-

x y

__j

1 .50

ac=o. 144x

;a t

ac=o. 202x

'-Y CD

ac=o. 259X ac=o. 317X

72, C e

A

ac=o. 374Xac=o. 432X

NU1

ac.=o. 490X

'.-

/.3 t

Fig. 2. Variation of the backscatteringof incidence of the plane wave.

RCS of a cube with the angle

1369

Z'IrXlu)4j, .

6.00 ac=

5.50

5.00 ac=1.02a4.50 ac=1. 04a-.ac=l.06a4.00 "N -- ac=l 09a

co'35 // Sphere (Bechtel3.00

2.50 I.I

2.00 t

1.50 A

0.50 2.0 2.l/ 3.z0 3-.50.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

6.00

5.50

5.00

4.50

4.00

N%-O

NU)u

Px

3.50

3.00

2.50

2.00

1.50

1.00

0.50

0.00

2ra/X

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

27ra/X

Fig.3. Backscattering RCS of an 80 faces pol.: a) Patch model and obten-tion of the optimum a/ac. b) Dependence with the angle of incidence.

1370