NASA/TM-2003-2 12 165 

Electromagnetic Scattering From a 

Polygonal Thin Metallic Plate Using 

Quadrilateral Meshing 

Manohar D. Deshpande 

Langley Research Center, Hampton, Virginia 

May 2003 

https://ntrs.nasa.gov/search.jsp?R=20030055676 2018-05-15T16:37:42+00:00Z
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NASA/TM-2003-2 12 165 

Electromagnetic Scattering From a 

Polygonal Thin Metallic Plate Using 

Quadrilateral Meshing 

Manohar D. Deshpande 

Langley Research Center, Hampton, Virginia 

National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton, Virginia 2368 1-2 199 

May 2003 
Contents 

List of Symbols 

List of Figures 

Abstract 

1 .O INTRODUCTION 

2.0 THEORY 

Scattered Far Field 

3.0 NUMERICAL RESULTS 

RCS of Hexagonal Plate 

RCS of Equilateral Triangular Plate 

RCS of Equilateral Triangular Plate with 

Concentric Equilateral Hole 

RCS of Diamond Shaped Plate 

4.0 CONCLUSION 

APPENDIX A 

REFERENCES 

iv 

V 

1 

i 

3 

9 

10 

10 

15 

19 

24 

28 

28 

30 

... 

111 
List of Symbols 

AI;'" 

A, 74 

A,,, 

A Magnetic vector potential. 

Bill 

dsrr 'dS"1 

det[J] 

Elf1 Incident electric field 

ij" element of matrix due to mrh and nrh quadrilateral patches. 

x- and y- components of magnetic vector potential 

Magnetic vector potential due to i'" edge on 12''' quadrilateral patch. 

Basis function for i"' edge of n'" quadrilateral patch. 

Elementary area on n"' and mrh patches respectively. 

Determinant of Jacobian for coordinate transformation 

- 

- 

- 

- 

Theta component of incident electric field 

Phi component of incident electric field. 

x-,y-, and z-components of incident electric field. 

Electric and magnetic scattered field. 

Theta and phi components of electric far field. 

E@, 

E, 

E,, , E,, , E,, 

E, 9 H, 

E@ 9 E, 

EM Electromagnetics 

EFIE Electric Field Integral Equation 

ESP Electromagnetic Surface Patch Code 

HI,' Incident magnetic field 

J Surface electric current density. 

- 

- 

- 

ki Vector wavenumber in direction on incidence angle 

ko 

li , i = 1,2,3,4 

MOM 

N 

PoMePl 

RWG 

r 

r 

N; (5, 7) 

- 

-/ 

Tri 

Free space wave number 

Length of four sides of quadrilateral patch 

Method of Moments 

Total number of quadrilateral patches. 

Linear transformation function. 

Polygonal Metallic Plate Code 

Rao Wilton Glisson triangular basis function 

Position vector for observation point 

Position vector for source point 

Complex number representing amplitudes of edge currents 

[TI 

[VI 

2, jj, 2 

xi 9 Y; i'" node coordinates 

Column matrix representing amplitudes of edge currents 

Column matrix representing incident wave values. 

Unit vectors along x-, y-, and z-axes 

iv 
[ZI Impedance square matrix 

a0 Angle = 90 (degrees) for E-polarization and equal to 0 for H-polarization 

€0 3 Po 

170 

U Radian frequency 2nf 

Permitivity and permeability of free space medium 

Free space impedance equal to 120~ 

44 Unit vectors along theta and phi directions 

4' Scalar Potential 

V Gradient operator 

oEE , oHH , oEH , oHE RCS in dB-m2 

List of Figures 

Figure 1 

Figure 2 

Figure 3 

Figure 4 

Figure 5 

Figure 6 

Figure 7 

Figure 8 

Figure 9 

Figure 10 

Geometry of arbitrarily shaped thin metallic plate in x-y plane and 

illuminated by a plane wave. 

Arbitarily shaped thin metallic plate discretized into quadrilaterals, m"' 

and n'h are typical elements with their four nodes. 

(a) nfh quadrilateral in x-y plane , (b) nfh quadrilateral in 4,~ plane 

(a) n'" quadrilateral with four edges, (b) and its adjacent quadrilateral. 

Thin metallic flat plate of hexagonal shape with a = 2.074 cm and 

descritized into quadrilateral patches. 

Monostatic RCS of hexagonal plate (shown in Figure 5) when illuminated 

by E-polarized plane wave for various discretizations, frequency = 

1 1.8 1 1 GHz. 

Monostatic RCS of hexagonal plate ( shown in Figure 5) when illuminated 

by H-polarized plane wave for various discretizations, frequency = 

1 1.8 1 1 GHz. 

Monostatic RCS of hexagonal plate (shown in Figure 5) when illuminated 

by E-polarized plane wave a, = 90" at 4; = -90" and 4; = 90°, solid 

line- present method, dashed line RWG functions, solid square symbolmeasurement, frequency = 11.81 1GHz. 

Monostatic RCS of hexagonal plate (shown in Figure 5) when illuminated 

by H-polarized plane wave a, = 0" at 4; = -90" and 4; = 90°, solid linepresent method, dashed dot line ESP-code, solid square symbolmeasurement, and dashed line RWG function, frequency = 11.81 1GHz. 

Monostatic RCS of hexagonal plate (shown in Figure 5) when illuminated 

by H-polarized plane wave a. = 0" at 4; = 0" and 4; = 180°, solid lineV 
present method, dashed dot line ESP-code, solid square symbolmeasurement, and dashed line RWG function, frequency = 11.81 1GHz. 

Figure 11 Monostatic RCS of hexagonal plate (shown in Figure 5) when illuminated 

by E-polarized plane wave a. = 90" at 4; = 0" and 4; = 180°, solid linepresent method, dashed line RWG functions, solid square symbolmeasurement, frequency = 11.81 1GHz. 

Figure 12 Equilateral triangular thin metallic flat plate with a = 5.08 cm and 

descritized into quadrilateral patches. 

Figure 13 Monostatic RCS of equilateral triangular plate (shown in Figure 12) when 

illuminated by E-polarized plane wave a, = 90" at 4; = 0" and 

4; = 180°, solid line- present method, short dashed line- RWG functions, 

long dashed line ESP-code, solid square symbol- measurement, frequency 

= 11.811GHz. 

Figure 14 Monostatic RCS of equilateral triangular plate (shown in Figure 12) when 

illuminated by H-polarized plane wave a. = 0" at Qi = 0" and Qj = 180°, 

solid line- present method, short dashed line- RWG functions, dashed dot 

line ESP-code, solid square symbol- measurement, frequency = 

11.81 lGHz 

Figure 15 Monostatic RCS of equilateral triangular plate (shown in Figure 12) when 

illuminated by E-polarized plane wave a. = 90" at q3j = -90" and 

qj = 90°, solid line- present method, short dashed line- RWG functions, 

dashed dot line ESP-code, frequency = 1 1.8 1 1 GHz 

Figure 16 Monostatic RCS of equilateral triangular plate (shown in Figure 12) when 

illuminated by H-polarized plane wave a. = 0" at 4; = -90" and 

qi = 90°, solid line- present method, short dashed line- RWG functions, 

dashed dot line ESP-code, solid square symbol- measurement, frequency 

= 11.81 1GHz. 

Figure 17 Monostatic RCS of equilateral triangular plate (shown in Figure 12) when 

illuminated by E-polarized plane wave a. = 90" as a function of 4; for 

frequency = 1 1.8 1 1 GHz, 8; = 90" . solid line- present method, short 

dashed line- RWG functions, dashed dot line ESP-code, frequency = 

11.81 1GHz. 

vi 
Figure 18 

Figure 19 

Figure 20 

Figure 21 

Figure 22 

Figure 23 

Figure 24 

Figure 25 

Monostatic RCS of equilateral triangular plate (shown in Figure 12) when 

illuminated by E-polarized plane wave a. = 90" as a function of 

frequency for 4,. = 90 and 0; = 90". solid line- present method, short 

dashed line- RWG functions, dashed dot line ESP-code, solid square 

symbols- measurement. 

Equilateral triangular thin metallic flat plate with a concentric equilateral 

triangular hole, a = 5.08 cm, b = 2.54 cm and descritized into 

quadrilateral patches. 

Monostatic RCS of equilateral triangular plate with equilateral triangular 

hole (shown in Figure 19) when illuminated by E-polarized plane wave 

a, = 90" at 4; = -90" and 4; = 90°, solid line- present method, short 

dashed line- RWG functions, dashed dot line ESP-code, solid square 

symbols- measurement, frequency = 1 1.8 1 1GHz. 

Monostatic RCS of equilateral triangular plate with equilateral triangular 

hole (shown in Figure 19) when illuminated by H-polarized plane wave 

a. = 0" at 4; = -90" and 4; = 90°, solid line- present method, short 

dashed line- RWG functions, dashed dot line ESP-code, solid square 

symbols- measurement, frequency = 1 1.81 1GHz. 

Monostatic RCS of equilateral triangular plate with equilateral triangular 

hole (shown in Figure 19) when illuminated by E-polarized plane wave 

a. = 90" at 4; = 0" and 4; = 180°, solid line- present method, short 

dashed line- RWG functions, dashed dot line ESP-code, solid square 

symbols- measurement, frequency = 1 1.81 1GHz. 

Monostatic RCS of equilateral triangular plate with equilateral triangular 

hole (shown in Figure 19) when illuminated by H-polarized plane wave 

a. = 0" at 4; = 0" and 4; = 180°, solid line- present method, short dashed 

line- RWG functions, dashed dot line ESP-code, solid square symbolsmeasurement, frequency = 1 1.8 1 1GHz. 

Monostatic RCS of equilateral triangular plate with equilateral triangular 

hole (shown in Figure 19) when illuminated by E-polarized plane wave 

a. = 90" as a function of 4; for frequency = 1 1.8 1 1 GHz, 0; = 90" . solid 

line- present method, short dashed line- RWG functions, dashed dot line 

ESP-code, solid square symbols- measurement, frequency = 1 1.8 1 1 GHz. 

Monostatic RCS of equilateral triangular plate with equilateral triangular 

hole (shown in Figure 19) when illuminated by E-polarized plane wave 

a, = 90" as a function of frequency for 4; = -90 and Oi = 90" (grazing 

incidence at edge). solid line- present method, short dashed line- RWG 

functions, dashed dot line ESP-code, solid square symbols- measurement. 

vii 
Figure 26 Monostatic RCS of equilateral triangular plate with equilateral triangular 

hole (shown in Figure 19) when illuminated by E-polarized plane wave 

a, = 90" as a function of frequency for 4; = 90 and 8, = 90" (grazing 

incidence at tip). solid line- present method, short dashed line- RWG 

functions, dashed dot line ESP-code, solid square symbols- measurement. 

Thin, metallic diamond-shaped flat plate with a = 3.592 cm and 

descritized into quadrilateral patches. 

Figure 27 

Figure 28 Monostatic RCS of diamond-shaped plate (shown in Figure 27) when 

illuminated by H-polarized plane wave a, = 0" at 4; = 90" and 

4; = -90 , solid line- present method, short dashed line- RWG functions, 

dashed dot line ESP-code, solid square symbols- measurement, frequency 

= 11.81 1GHz. 

0 

Figure 29 Monostatic RCS of diamond-shaped plate (shown in Figure 27) when 

illuminated by E-polarized plane wave a, = 90" at 4; = 90" and 

#; = -90°, solid line- present method, short dashed line- RWG functions, 

dashed dot line ESP-code, solid square symbols- measurement, frequency 

= 11.811GHz. 

Figure 30 Monostatic RCS of diamond-shaped plate (shown in Figure 27) when 

illuminated by H-polarized plane wave a, = 0" at 4; = 0" and 4; = 180°, 

solid line- present method, short dashed line- RWG functions, dashed dot 

line ESP-code, frequency = 11.81 1GHz. 

Figure 3 1 Monostatic RCS of diamond-shaped plate (shown in Figure 27) when 

illuminated by E-polarized plane wave a, = 90" at 4; = 0" and 

4; = 180°, solid line- present method, short dashed line- RWG functions, 

dashed dot line ESP-code, solid square symbols- measurement, frequency 

= 11.811GHz. 

Figure 32 Monostatic RCS of diamond-shaped plate (shown in Figure 27) when 

illuminated by E-polarized plane wave a. = 90" as a function of 4; for 

frequency = 1 1.81 1 GHz, 8, = 90" . solid line- present method, short 

dashed line- RWG functions, dashed dot line ESP-code, solid square 

symbols- measurement, frequency = 1 1.8 1 1GHz. 

... 

VI11 
ABSTRACT 

The problem of electromagnetic (EM) scattering from irregularly shaped, thin, metallic flat 

plates in free space is solved using the electric field integral equation (EFIE) approach in 

conjunction with the method of moments (MOM) with quadrilateral meshing. An irregularly 

shaped thin plate is discretized into quadrilateral patches and the unknown electric surface 

current over the plate is expressed in terms of proper basis functions over these patches. The 

basis functions for the electric surface current density that satisfy the proper boundary conditions 

on these quadrilateral patches are derived. The unknown surface current density on these 

quadrilateral patches is determined by setting up and solving the electric field integral equation 

by the application of the MOM. From the knowledge of the surface current density, the EM 

scattering from various irregularly shaped plates is determined and compared with the earlier 

published results. The novelty in the present approach is the use of quadrilateral patches instead 

of well known and often used triangular patches. The numerical results obtained using the 

quadrilateral patches compare favorably with measured results. 

1. INTRODUCTION 

The EM scattering from a complex-shaped metallic object is of practical interest to 

electromagnetic analysts as well as to engineers. The EM scattering from polygonal shaped 

plates is of special interest because a complex-shaped object can always be modeled as an 

interconnection of these polygonal flat plates. 

metallic plates can be accomplished by using various numerical techniques [l-41. One of the 

widely used techniques is the MOM in which a polygonal plate is first discretized into number of 

triangular patches called sub domains. The unknown surface current density on these sub 

An EM scattering analysis of polygonal shaped 

1 
domains is then expressed in terms of well known Rao-Wilton-Glisson (RWG) [3] basis 

functions. The unknown surface current density on these sub domains is then determined using 

the EFIE in conjunction with the MOM. Although the triangular meshing and use of RWG basis 

functions is a very popular and mature concept, the non-vanishing normal component of surface 

current on one of the inclined edges of sub-domain triangles attached to the open /boundary edge 

of polygonal plate will give a non-zero normal component of the surface current. However, the 

non-zero normal component of the surface current on the boundary edge will approach zero 

when very fine discretization is used in the region close to the openboundary edges. This 

problem will not arise if one uses a sub domain element (near the boundary) whose edges will be 

either along the boundary or normal to the boundary (such as quadrilateral element). Another 

approach as implemented in [4] divides a polygonal plate into small rectangles. The surface 

current density over each sub domain is expressed in terms of overlapping triangular functions 

(roof-top functions) in the direction of current flow and a pulse function in the orthogonal 

direction. The EFIE in conjunction with the MOM is then used to determine the surface current 

density. An important advantage of this procedure is that the impedance matrix encountered in 

this procedure has a block Toeplitz nature, a property that is useful in reducing the matrix filling 

time. However, the major drawback of this scheme is that the curve boundaries are 

approximated by stair cases and hence for accurate results requires very fine discretization. 

In this report the problem of electromagnetic scattering of plane waves by an arbitrarily 

shaped, thin metallic plate is studied using the EFlE method in conjunction with quadrilateral 

meshing. Use of the equivalence principle and the free space Green's function in deriving the 

EFIE with the surface current density as an unknown is described in detail. With proper choice 

of the expansion and testing functions over the quadrilateral mesh, the MOM is described to 
convert the EFIE into a matrix equation. The procedure to obtain the scattered far field by the 

plate in terms of the electric surface current is also described. 

The remainder of report is organized as follows. The formulation of the problem in 

terms of the EFIE using the surface equivalence principle is developed in section 2. Numerical 

results on the mono-static radar cross section of various polygonal shaped metallic plates are 

presented in section 3 and compared with earlier published data. The advantages and limitations 

of the present formulation are discussed in section 4. 

2. THEORY 

Consider a time harmonic electromagnetic plane wave incident on an irregularly shaped, 

infinitesimally thin metallic plate as shown in Figure 1. The incident field with a time 

dependence eir* may be written as 

Arbitrarily Shaped 

Metallic Plate 

Incident 

Field Point Wave ,, 

Figure 1 : Geometry of arbitrarily shaped thin metallic plate in x-y plane 

and illuminated by a plane wave 

3 
- 

E,, = (&Eoi + 4i E@, 

where /c, = -k,[ico~($~)sin(O~) + jsin($;)sin(B,) + icos(B,)], 7 = 2x+ jjy + ?z , 

E6, = (E,(cos(ao), and E@, = If?;/ sin(ao) , k, being the free- space wave number. With 

reference to Figure 1, a. = 0 corresponds to H-polarization and a, = 90' corresponds to Epolarization. From equation (1), the x-, y-, and z-components of the incident field may be 

written, respectively, as 

Exl = E, COS(@, 1 COS($, ) - E, sin($! ) 

= E&?, c0s(8~ ) - E@, ) 

E- -1 = -E, sin(@!) 

The corresponding magnetic field components are obtained through 

H;,, - =- k; x E;,* korlo 

where qo is the free-space impedance. The incident field with E, f Oand E, = 0 is called the 

H-polarized wave and E, = 0 and E, # 0 is called the E-polarized wave. 

To facilitate the solution of the problem using the equivalence principle, the thin 

metallic plate is replaced by the equivalent electric surface current density j(x, y) . The 

scattered field due to J' may be obtained through the vector magnetic potentials as 

Es <.?> = - jmi - v 4 

4 
1 

PO 

Hs (J) = -v x ;i 

where the magnetic vector and scalar potentials are given by 

(7) 

where E, and p, are the permitivity and permeability, respectively, of free-space, r‘ and 7’ are 

respectively the coordinates of field and source points. From the knowledge of the scattered 

field and incident field, the EFE is set up by equating the total tangential electric field over the 

surface of the polygonal plate to zero. Hence 

To determine the approximate solution of the integral equation in (10) using the MOM, 

the polygonal surface is approximated by a union of quadrilaterals as shown in Figure 2. On the 

nth quadrilateral the electric surface current density can be represented by a superposition of 

vector functions Bni (x, y) as 

i=l 

where Tli is the amplitude of electric current normal to the if” edge of the izf” element 
Node #I 

Node #2 

11 

Figure 2: Arbitrarily shaped thin metallic plate discretized into 

quadrilaterals. mth and nth are typical elements with their four nodes 

Node #3 

:L ' 

Edge #3 

(as shown in Figure 4). The vector basis function fill; (x, y) representing the surface current 

Edge #1 

j=-I 

density on the n"' quadrilateral is required to satisfy the appropriate boundary conditions; such 

as the normal component of B,,; (x, y) to the i"' edge must be unity, and must have a value of 

zero on the other remaining edges. Construction of such a vector basis function with help of 

Figure 3 is explained in detail in the Appendix A. 

*5 I 

$= 1 

y4 

(x39 Y3) 

Node #3 

Node #I 

IX 

Node#1 Edge# Node #4 

q =-I 

Figure 3: (a) nih quadrilateral in x-y plane, (b) nt'* quadrilateral in 5.71 plane 

6 
Node #2 

Node #5 

Figure 4: (a) nfh quadrilateral with four edges, (b) nfh and its adjacent quadrilateral 

where and 4,,; are obtained from equations (8) and (9), respectively, by replacing a by 

Ell; . Selecting Bmj with j = 1,2,3,4 and rn = 1,2,3, ... N as a testing function and application of 

the MOM to equation (12) yields 

The left side of equation (13) over the n‘” quadrilateral (Figure 4(a) ) can be written as 

7 
where 

Now consider union of two adjacent quadrilaterals with a common edge as shown in Figure 

4(b). The left side of equation (13) for two adjacent quadrilaterals can be written as 

In the above expression, the first letter of superscript represents the test quadrilateral and the 

second letter represents the source quadrilateral element. To ensure continuity of current at the 

common edge it is assumed that T11+1.6 = -q13 . With an assembled mesh of all quadrilaterals 

over the surface of polygonal plate, the left side of equation (13) can be written as 

8 
where the elements of the matrix are obtained from Alrjlnr and the column matrix [TI represents 

the unknown amplitude of current at the edges of quadrilateral elements. Using (16) the equation 

(1 3) can be written in a matrix form 

[ZITI = [VI 

where the elements of column vector matrix are obtained from 

Vj = - IB,, [I?,,] ds,, 

The edge currents obtained after solving the matrix equation (17) are then used to determine the 

EM scattering from polygonal thin metallic plates. It is noted that the T, on the boundary edges 

are forced to zero before solving the matrix equation (17). 

Scattered Far Field: 

The scattered far field due to the polygonal plate is estimated from 

EB = - jkoqO(A, cos($)cos(8) + A, sin($)cos(8)) 

E@ = -jkoqO(-A, sin($) + A, cos(@)) 

where A, and A, are obtained from 

From the far fields, the radar cross sections, for co and cross polarizations are obtained as 

9 
3. NUMERICAL RESULTS 

In this section the Radar Cross Section (RCS) of polygonal, thin metallic plates of various 

shapes is calculated using the quadrilateral meshing scheme. To validate the present analysis 

and the computer code developed, the numerical results obtained by the present method are 

compared with (1) the RCS computed using the ESP code [4], (2) the RCS computed using 

triangular meshing', and (3) the RCS measured in the Langley Experimental Test Range Facility 

c41. 

RCS of Hexagonal Plate: 

As a first example, a hexagonal plate with a = 2.074 cm, as shown in Figure 5, is 

considered. Using the hexa.SES file (listing given on the attached disk), the hexagonal plate is 

modeled and meshed (also shown in Figure 5 using the Geostar/COSMOS commercial 

software). The hexa.MOD file generated by Geostar is then is imported into PoMePl (Polygonal 

Metallic Plate, listing given on the attached disk ) electromagnetic code written in FORTRAN to 

estimate the RCS of polygonal, thin metallic plates. The mono-static RCS of the hexagonal plate 

' MOM code developed at the Electromagnetic Research Branch of NASA Langley Research Center by the author. 

10 
is calculated for both E- and H-polarized incident waves for various discretization levels and is 

shown in Figures 6-7. It is clear from the Figures 6-7 that the discretization level with the 

number of edges more than 232 yields fairly consistent numerical results. Hence for the RCS 

computation of hexagonal plate shown in Figures 8-1 1 the number of edges used was 232. The 

Figures 8-1 1 also give a comparison of the results obtained using the quadrilateral meshing with 

the results obtained using triangular meshing, the ESP code [4], and the measured data [4 1. 

From Figures 8-1 1 it can be observed that the results obtained using the quadrilateral meshing 

agrees well with the other numerical data as well as with the measured results. However, for EFigure 5: Thin, metallic flat plate of hexagonal shape with side a = 2.074 

cm and descritized into quadrilateral sub-domains. 

11 
a,dB - m2 

ei - 

Figure 6: Monostatic RCS of hexagonal plate shown in Figure 5 when 

illuminated by E-polarized plane wave as a function of number of edges 

ei - 

Figure 7: Monostatic RCS of hexagonal plate shown in Figure 5 when 

illuminated by H-polarized plane wave as a function of number of edges 

12 
Figure 8 Monostatic RCS of hexagonal plate (shown in Figure 5) when illuminated by E-polarized plane wave 

a0 = 90" at qi = -90" and qi = 90°, solid line- present method, dashed line RWG functions, solid square 

symbol- measurement, frequency = 1 1.8 1 1GHz. 

oHH ,dB-m' 

-IQ f 

A 

e,, deg . -b 

Figure 9 Monostatic RCS of hexagonal plate (shown in Figure 5) when illuminated by H-polarized plane 

wave cl0 = 0" at qj = -90" and qi = 90°, solid line- present method, dashed dot line ESP-code, solid 

square symbol- measurement, and dashed line RWG function, frequency = 11.81 IGHz. 

13 
aHH ,dB - m' 

-20 . 

-30 , 

A 

- e, , deg. + 

Figure 10 Monostatic RCS of hexagonal plate (shown in Figure 5) when illuminated by H-polarized 

plane wave a, = 0" at qi = 0" and qi = 180°, solid line- present method, dashed dot line ESPcode, solid square symbol- measurement, and das ed line RWG function, frequency = 1 1.8 1 1GHz. 

dB-m' 

f -1 0 

-5 5 

e, , deg. + 

Figure 11 Monostatic RCS of hexagonal plate (shown in Figure 5) when illuminated by E-polarized 

plane wave a, = 90" at @i = 0" and @j = 180°, solid line- present method, dashed line RWG 

functions, solid square symbol- measurement, frequency = 1 1.8 1 1GHz 

14 
polarized incidence close to grazing angles, the computed results do not agree well with the 

measured data. This is due to the fact that in the computation the finite thickness of the plate is 

assumed to be zero. 

RCS of Equilateral Triangular Plate: 

A equilateral triangular plate as shown in Figure 12 is considered as a second example 

for the validation of the present code based on the quadrilateral meshing. Using the 

COSMOS/GEOSTAR the equilateral triangle is discretized into quadrilateral patches as shown 

in Figure 12. The *.MOD file, in conjunction with PoMePl code is used to estimate the 

monostatic RCS of the equilateral triangular plate. The results of the computations are shown in 

Figures 13-18 along with the experimental and other numerical data. The monostatic RCS 

a 

Figure 12 Equilateral triangular, thin, metallic flat plate with a = 5.08 cm 

calculated using the quadrilateral meshing compares well with the results obtained using the 

triangular meshing and the ESP - code. The experimental data also compares well with the 

numerical data as long as the angle of incidence is not close to grazing. The discrepancy 

15 
between the measured and numerical data at angles close to grazing is because of the 

assumption of zero thickness in the numerical simulation. The discrepancy between measured 

and numerical data as a function of frequency for the grazing incident angles, as shown in Figure 

17, illustrate the necessity of taking into account the thickness of the plate in the numerical 

simulation techniques for more accurate results. 

oEE ,dB - nz2 

-1 5 

-28 

-2 5 

-3 0 

-3 5 

435 

__+ 0, , deg . 

Figure 13 Monostatic RCS of equilateral triangular plate (shown in Figure 12) when 

illuminated by E-polarized plane wave a. = 90" at 4, = 0" and 4i = 180°, solid line- present 

method, short dashed line- RWG functions, long dashed line ESP-code, solid square symbolmeasurement, frequency = 1 1.81 1GHz. 

16 
-1 0 

-1 5 

-2 0 

-2 5 

-3 w 

-3 5 

40 

45 

-5 0 

-5 5 

40 

8, , deg . __+ 

Figure 14 Monostatic RCS of equilateral triangular plate (shown in Figure 12) when 

illuminated by H-polarized plane wave a,, = 0" at qbj = 0" and 4; = lSOo, solid line- present 

method, short dashed line- RWG functions, dashed dot line ESP-code, solid square symbolmeasurement, frequency = 1 1 .S 1 1 GHz 

oEE, dB - m2 

-lo € 

-1 5 

-20 

-25 

-30 

-35 

45 

-50 

-55 

e;, deg - 

Figure 15 Monostatic RCS of equilateral triangular plate (shown in Figure 12) when 

illuminated by E-polarized plane wave ab = 90" at 4; = -90" and 4; = 90°, solid linepresent method, short dashed line- RWG functions, dashed dot line ESP-code, frequency = 

11.81 lGHz 

17 
~HH 

Figure 16 Monostatic RCS of equilateral triangular plate (shown in Figure 12) when 

illuminated by H-polarized plane wave a. = 0" at qi = -90" and qi = 90°, solid linepresent method, short dashed line- RWG functions, dashed dot line ESP-code, solid square 

symbol- measurement , frequency = 1 1.8 1 1GHz. 

a,,dB-m2 

Figure 17 Monostatic RCS of equilateral triangular plate (shown in Figure 12) when 

illuminated by E-polarized plane wave ab = 90" as a function of qi for frequency = 

1 1.8 1 1 GHz, 6, = 90" . solid line- present method, short dashed line- RWG functions, 

dashed dot line ESP-code, frequency = 1 I .811GHz. 

18 
oEE, dB - m 

-20 I 

LLu.l - 17 18 

Frequency, GHz 

Figure 18 Monostatic RCS of equilateral triangular plate (shown in Figure 12) when 

illuminated by E-polarized plane wave a,, = 90" as a function of frequency for 

4; = 90 and Oi = 90". solid line- present method, short dashed line- RWG functions, 

dashed dot line ESP-code, solid square symbols- measurement. 

RCS of Equilateral Triangular Plate With Concentric, Equilateral Triangular Hole. 

An equilateral triangular plate with a concentric equilateral triangular hole as shown in 

Figure 18 is considered as an another example for the validation of the present code based on the 

quadrilateral meshing. Using the COSMOS/GEOSTAR the equilateral triangle is discretized 

into quadrilateral patches as shown in Figure 18. The *.MOD file, in conjunction with PoMePl 

code is used to estimate the mono-static RCS of the equilateral triangular plate with an 

equilateral triangular hole. The results of the computations are shown in Figures 19-24 along 

with the experimental and other numerical data. The mono-static RCS calculated using the 

19 
quadrilateral meshing compares well with the other numerical data as well as with the 

experimental results. 

Figure 19 Equilateral triangular thin metallic flat plate with a concentric equilateral 

triangular hole, a = 5.08 cm, b = 2.54 cm and descritized into quadrilateral patches. 

For this example, it is also observed that the numerical simulation data do not match well 

with the experimental results for grazing incidence. 

20 
o,,dB-m2 

-'"E1 5 

-2 0 

-25 

-30 

-3s 

40 

4' 

*O%i*%*%** - 8, ,deg. 

Figure 20 Monostatic RCS of equilateral triangular plate with equilateral triangular hole (shown in 

Figure 19) when illuminated by E-polarized plane wave a, = 90" at 4, = -90" and 4, = 90°, solid 

line- present method, short dashed line- RWG functions, dashed dot line ESP-code, solid square 

symbols- measurement, frequency = 1 1.8 1 1GHz. 

o,,dB-m2 

-20 

-2 5 

-30 

-35 

-90 

-45 

-50 

65 

so -Bo -eo -40 -20 o a0 40 eo 80 

Figure 2 1 Monostatic RCS of equilateral triangular plate with equilateral triangular hole (shown in 

Figure 19) when illuminated by H-polarized plane wave a, = 0" at 4; = -90" and 4; = 90°, solid 

line- present method, short dashed line- RWG functions, dashed dot line ESP-code, solid square 

symbols- measurement, frequency = 11.81 1GHz. 

21 
oEE,dB-m2 

-1 5 

-I0 1 

b'*8*'1* fi*.'*'*"***' 

1.d.. ..I. '..I'.. ,I.' 

-80 -BO -4 - -20 o 20 40 ea m 

Oi , deg . 

Figure 22 Monostatic RCS of equilateral triangular plate with equilateral triangular hole 

(shown in Figure 19) when illuminated by E-polarized plane wave ab = 90" at qj = 0" and 

@j = 180°, solid line- present method, short dashed line- RWG functions, dashed dot line ESPcode, solid square symbols- measurement, frequency = 1 1.81 1GHz. 

o, , dB - m2 

-1 

-1 

-2 

-3 

-3 

4 

4 

a 

-5 

a 

Oi , deg . --+ 

Figure 23 Monostatic RCS of equilateral triangular plate with equilateral triangular hole (shown 

in Figure 19) when illuminated by H-polarized plane wave a, = 0" at qj = 0" and 4; = 180°, 

solid line- present method, short dashed line- RWG functions, dashed dot line ESP-code, solid 

square symbols- measurement, frequency = 1 1.81 1GHz. 

22 
1 Figure 25 Monostatic RCS of equilateral triangular plate with equilateral triangular hole (shown in Figure 19) 

when illuminated by E-polarized plane wave a. = 90" as a function of frequency for 4,. = -90 and 

Si = 90" (grazing incidence at edge). solid line- present method, short dashed line- RWG functions, dashed 

dot line ESP-code, solid square symbols- measurement 

23 

oEE,dB-m2 

::$ 

-25 

-30 

-35 

-40 

-45 

-50 

-55 

-80 

o 30 eo 90 120 190 180 210 290 mo a00 330 980 

Figure 24 Monostatic RCS of equilateral triangular plate 

4j 7 deg . - 

with equilateral triangular hole (shown in Figure 19) 

when illuminated by E-polarized plane wave cto = 90" as a function of qj for frequency = 1 1.8 1 1 GHz, 

6; = 90" . solid line- present method, short dashed line- RWG functions, dashed dot line ESP-code, solid 

square symbols- measurement, frequency = 11.81 1GHz. 

6, , dB - m - QUadflleWW UMhlng 

= = - = Trlenguler Ywhlng 

-20 D1..81911 

~ ueaouremnt 

-25 

-30 

-35 

-40 

-45 

-50 

-55 

6 7 8 9 10 11 12 13 14 15 18 17 18 -60 

Frequency, GHz - 
oEE ,dB - m2 

-25 

-30 

-4 Frequency, GHz - 

Figure 26 Monostatic RCS of equilateral triangular plate with equilateral triangular hole (shown 

in Figure 19) when illuminated by E-polarized plane wave a,, = 90" as a function of frequency 

for qji = 90 and 0, = 90" (grazing incidence at tip). solid line- present method, short dashed lineRWG functions, dashed dot line ESP-code, solid square symbols- measurement 

RCS of Diamond Shaped Plate 

A diamond shaped polygonal plate as shown in Figure 25 is considered as a last example 

for validation of the present code. Using the COSMOS/GEOSTAR the diamond shaped plate is 

discretized into quadrilateral patches as shown in Figure 25. The *.MOD file, in conjunction 

with PoMePl code is used to estimate the mono-static RCS of the diamond shaped plate. The 

results of the computations are shown in Figures 26-30 along with the experimental and other 

numerical data. The mono-static RCS calculated using the quadrilateral meshing compares well 

with the other numerical data as well as with the experimental results. For this example, it is 

also observed that the numerical simulation data do not match well with the experimental results 

for grazing incidence. 

24 
Figure 27 Thin, metallic diamond-shaped flat plate with a = 3.592 cm 

and descritized into quadrilateral patches 

OHH, dB - m2 

-1 0 

-1 5 

-20 

-2 5 

-30 

-35 

-40 

45 

40 

65 

6, , deg . ___, 

Figure 28 Monostatic RCS of diamond-shaped plate (shown in Figure 27) when 

illuminated by H-polarized plane wave ab = 0" at $i = 90" and #i = -90°, 

solid line- present method, short dashed line- RWG functions, dashed dot line 

ESP-code, solid square symbols- measurement, frequency = 11.81 1GHz. 

25 
oEE ,dB - m2 

-lo Ir 

e,, deg . -+ 

Figure 29 Monostatic RCS of diamond-shaped plate (shown in Figure 27) when illuminated 

by E-polarized plane wave a, = 90" at 4; = 90" and 4; = -90°, solid linepresent method, short dashed line- RWG functions, dashed dot line ESP-code, 

solid square symbols- measurement, frequency = 1 1.8 1 1 GHz. 

~HH ,dB - m2 

-lo F 

-I5 E 

-20 f- 

* Yeesurernent 

- B,,deg. ___, 

Figure 30 Monostatic RCS of diamond-shaped plate (shown in Figure 27) when 

illuminated by H-polarized plane wave ab = 0" at 4; = 0" and @i = 180°, solid linepresent method, short dashed line- RWG functions, dashed dot line ESP-code, 

frequency = 11.81 1GHz. 

26 
oEE ,dB - m2 

Figure 3 1 Monostatic RCS of diamond-shaped plate (shown in Figure 27) when 

illuminated by E-polarized plane wave a, = 90" at q4i = 0" and q4; = 180°, solid 

line- present method, short dashed line- RWG functions, dashed dot line ESPcode, solid square symbols- measurement, frequency = 1 1.81 1 GHz. 

oEE,dB-m2 

-30 

-35 

-40 

- 45 

-50 

-55 

-00 I 

0 30 BO 90 - 120 150 180 210 240 270 300 330 380 

&,deg. 

Figur 32 Monostatic RCS of diamond-shaped plate (shown in Figure 27) when illuminated 

by E-polarized plane wave a, = 90" as a function of 4,. for frequency = 11.8 1 1 

GHz, 0; = 90". solid line- present method, short dashed line- RWG functions, 

dashed dot line ESP-code, solid square symbols- measurement, frequency = 

1 1.8 1 1 GHz. 

27 
4 CONCLUSION 

The electric field integral equation approach in conjunction with the method of moments 

with quadrilateral meshing has been described to estimate the electromagnetic scattering from 

irregularly shaped thin metallic plates. Discretizing an irregularly shaped thin plate into 

quadrilateral patches and using proper expansion and testing functions over the quadrilateral subdomains, the integral equation has been reduced to a matrix equation. From the knowledge of 

the surface currents density, obtained after solving the matrix equation, the EM scattering from 

various irregularly shaped plates has been determined and compared with earlier published 

results. The novelty in the present approach is the use of quadrilateral patches instead of well 

known and often used triangular patches. It has been observed that the use of quadrilateral 

patches instead of triangular patches reduces the number of unknowns and hence it requires 

handling of reduced size matrices. Also, the quadrilateral patches are more natural to satisfy the 

boundary edge currents condition, Le., zero normal component compared to the triangular 

patches. This property makes it possible to achieve convergent results fewer quadrilateral 

patches. 

The numerical results obtained using quadrilateral patches have been compared with the 

numerical results obtained using the triangular patches and ESP-code. The numerical results 

obtained by the present code have also been compared with the measured data obtained in the 

NASA Langley Experimental Test Range Facility. 

APPENDIX A 

Construction of the vector basis function BIli (x, y) for an arbitrarily shaped quadrilateral 

in the x-y coordinate system is a complex problem. However, a coordinate transformation that 

28 
transforms a quadrilateral in the x-y plane into a square element in a new (5, q) coordinates 

plane simplifies the derivation as shown below. Using the following transformation 

i=l 

where 

i=l 

With the coordinate transformation given in (A1 ), the n"' quadrilateral is transformed into a 

square with sides equal to 2, as shown in Figure 3. The vector basis function along the edges 

of the quadrilateral can be shown to be [ 61 

If i? is the unit normal vector to the izfh quadrilateral, then the basis function representing the 

surface current can be written as 

(5, ?#7) = !L (1 - x v (5,q) = 5- (1 + q)i? x vq: 4 4 

where the various quantities appearing in the above expressions can be expressed as [6 ] 

Vg = (x-- y-)/det[J] , (A5) A ay A ax A ay A ax 

377 37 a5 35 Vq = (-x-+ y-)/det[J] 

With simple mathematical manipulation it can be shown that 

29 
REFERENCES 

r31 

[41 

VI 

E. H. Newman and R. J. Marhefka, “ Overview of MM and UTD methods at the Ohio 

State University,” Proc. IEEE, Vol. 77, no. 5, May 1989, pp. 700-708. 

E. H. Newman and T. Pravit, “ A surface patch model for polygonal plates, ” IEEE Trans. 

On Antennas and Propagation, Vol. AP-30, no 4, July 1982, pp.588-593 

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of 

arbitrary shape, ” EEE Trans. On Antennas and Propagation , Vol. AP-30, no. 3, pp. 

409-418, May 1982. 

M. D. Deshpande et al., “Analysis of electromagnetic scattering from irregularly shaped, 

thin, metallic plates, ” NASA Technical Paper 3361, Oct. 1993. 

E. H. Newman, “A User’s manual for the electromagnetic surface patch code: ESP: 

Version IV, Electroscience Lab, Ohio State University, 1988. 

J. Jin, “Thefinite element method in electrornagnetics, ” John Wiley & Sons, Inc., New 

York, 1993 

30 
REPORT DOCUMENTATION PAGE 

4. TITLE AND SUBTITLE 

Electromagnetic Scattering From a Polygonal Thin Metallic Plate Using 

Quadrilateral Meshing 

Form Approved I OMB No 0704-0188 

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Availability: NASA CAS1 (30 1) 62 1-0390 

6. AUTHOR(S) 

Manohar D. Deshpande 

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11. SUPPLEMENTARY NOTES 

The problem of electromagnetic (EM) scattering from irregularly shaped, thin, metallic tlat plates in free space is 

solved using the electric field integral equation (EFIE) approach in conjunction with the method of moments 

(MOM) with quadrilateral meshing. An irregularly shaped thin plate is discretized into quadrilateral patches and 

the unknown electric surface current over the plate is expressed in terms of proper basis functions over these 

patches. The basis functions for the electric surface current density that satisfy the proper boundary conditions 

on these quadrilateral patches are derived. The unknown surface current density on these quadrilateral patches 

is determined by setting up and solving the electric field integral equation by the application of the MOM. From 

the knowledge of the surface current density, the EM scattering from various irregularly shaped plates is 

determined and compared with the earlier published results. The novelty in the present approach is the use of 

quadrilateral patches instead of well known and often used triangular patches. The numerical results obtained 

using the quadrilateral patches compare favorably with measured results. 

14. SUBJECTTERMS I 15. NUMBER OF PAGES 

Electromagnetic Scattering; Moment Method; Polygonal Plates 41 

16. PRICE CODE 

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1-280-5500 standard Form 298 R ev. 2- 

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