-Ales 479 AN EFFICIENT NUMERICAL ALGORITHM FOR SOLVING SCATTERING in1AND INVERSE SCRTT (U) NUMERICAL COMPUTATION CORP STONYBROOK NV V M CHEN 21 SEP 87 N814-B6-C-6189

UNCLASSIFIED F/G 28/14 NL

EEEEEEEEEIhhhIEEE...IIII

lot..h

U I 1'. -li

,WSPY I.TWT

VI-C FILE cop. ,.-

FINAL REPORT F

AN EFFICIENT NUMERICAL ALGORITHM FORI

SOLVING SCATTERING AND INVERSE SCATTERINGPROBLEMS OF ELECTROMAGNETIC WAVES

00 PREPARED BY Y. M. CHEN00 NUMERICAL COMPUTATION CORP.

57 QUAKER PATH< STONY BROOK, NEW YORK 11790-1309 :

PHONE: 516-751-9518

SEPTEMBER 21, 1987

APPROVED FOR PUBLIC RELEASE

DISTRIBUTION UNLIMITED

DTICSLECTEI

SBIR PHASE I CONTRACT No. N00014-86-C-0109 .OFFICE OF NAVAL RESEARCHDEPARTMENT OF THE NAVY

800 N. QUINCY STREETARLINGTON, VA 22217-5000

SCIENTIFIC MONITOR:DEFENSE ADVANCED RESEARCH PROJECTS AGENCY - TTO1400 WILSON BLVD.ARLINGTON, VA 22209

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11. TITLE (Include Security Classification)An Efficient Numerical Algorithm for Solving Scattering and Inverse Scattering

Problems of Electromagnetic Waves - Unclassified

12. PERSONAL AUTHOR(S)Chen, Yung M.

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16. SUPPLEMENTARY NOTATION

17. COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary'and identify by block number)FIELD I GROUP I SUB-GROUP Numerical, algorithm, scattering, electromagnetic,

wave

19. ABSTRACT (Continue on reverse if necessary and Identify by block numbe rhe development of an efficient numerica]algorithm of determining the unknown material composition and shape of an arbitrary target(from the measured electromagnetic waves in the far field region will enhance the capabilit#of the defense radar system to defeat known evasive schemes. The first step in this researceffort is the development of an efficient and versatile numerical algorithm for calculatingthe scattered electromagnetic waves/radar cross section by a target with known complexgeometry and material property. Hence the pu ose of this Phase I research is to develop anefficient numerical algorithm for solving two imensional scattering problems. This isachieved by using a special finite difference method based upon a natural spatial discreti-zation of the integral form of Maxwell's equations on a non-orthogonal grid-system and theleap-frog finite differencing in the time domain. It has the advantages of being (a) moreefficient than any other known numerical methods, (b) highly accurate due to the body-fittedgrid system, and (c) the easiest numerical method to implement boundary conditions. Thecapability and feasibility of this two-dimensional computer code are tested by performing,--

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Block 19:

numerical simulations on few realistic examples, e.g., cylindrical objects withcross sections of a metallic jet and a composite airfoil. In these processes,the radar cross sections as functions of both the incident angle and the scatteredangle are calculated and they seem to be quite good. Hence this Phase I researchis completed successfully. 7r c A .

I 1-'

SBIR PHASE I CONTRACT NO. N00014-86-C-0109

OFFICE OF NAVAL RESEARCH

DEPARTMENT OF THE NAVY

800 N. Quincy Street

Arlington, Virginia 22217-5000

TITLE: AN EFFICIENT NUMERICAL ALGORITHM FOR SOLVING SCATTERING

AND INVERSE SCATTERING PROBLEMS OF ELECTROMAGNETIC WAVES

FINAL REPORT: Prepared by Y. M. Chen

NUMERICAL COMPUTATION CORP.

57 Quaker Path Ph. 516-751-9518

Stony Brook, N.Y. 11790-1309

DATE: September 21, 1987

TABLE OF CONTENTS

SUMMARY i

DEGREE TO WHICH PHASE I OBJECTIVES HAVE BEEN MET ii

ANTICIPATED BENEFITS ii

INTRODUCTION sow 1

FINITE DIFFERENCE METHOD 4

NUMERICAL SIMULATION V 8

TECHNICAL DISCUSSION £eessin Fw 11

REFERENCES s out 19

APPENDIX - COMPUTER CODE Unsoneww" 20Justifleati-

Availability CodesA*' vail wm/or,

Dist

SUMMARY

The development of an efficient numerical algorithm capable of determining

the unknown material composition and shape of an arbitrary target from the

measured electromagnetic waves in the far field region will enhance the capabi-

lity of the defense radar system to defeat known evasive schemes. The first

step in this research effort is the development of an efficient and versatile

numerical algorithm for calculating the scattered electromagnetic waves/radar

cross section by a target with known complex geometry and material property.

Hence the purpose of this Phase I research is to develop an efficient numerical

algorithm for solving two-dimensional scattering problems. This is achieved

by using a special finite difference method based upon a natural spatial

discretization of the integral form of Maxwell's equations on a non-orthogonal

grid-system and the leap-frog finite differencing in the time domain. It has

the advantages of being (a) more efficient than any other known numerical

methods, (b) highly accurate due to the body-fitted grid system, and (c) the

easiest numerical method to implement boundary conditions. The capability

and feasibility of this two-dimensional computer code are tested by performing

numerical simulations on few realistic examples, e.g., cylindrical objects

with cross sections of a metallic jet and a composite airfoil. In these

processes, the radar cross sections as functions of both the incident angle

and the scattered angle are calculated and they seem to be quite good.

Hence this Phase I research is completed successfully.

DEGREE TO WHICH PHASE I OBJECTIVES HAVE BEEN MET

The purpose of the Phase I research is to develop an efficient numerical

algorithm for solving two-dimensional scattering of electromagnetic waves by

a target with complex geometry and material property. Now, we are happy to

announce that the objectives of the Phase I research have been achieved

completely and successfully. The complete 2-D computer code is given in the

Appendix in this final report.

ANTICIPATED BENEFITS

As it stands by itself, the usefulness of the 2-D computer code developed

here in Phase I research is rather limited, because in the real life there is

no truly two-dimensional target. However, it has demonstrated beyond any

doubt that the above developed numerical algorithm can be extended to compute

efficiently and accurately the scattered E-M waves or radar cross section

for the three-dimensional target with complex geometry and material property.

The 3-D code will be an important subroutine of the Generalized Pulse-

Spectrum Technique for solving the 3-D inverse scattering problems; moreover,

it will be a very useful tool in advanced vulnerability analysis for any

postulated target.

__ii

INTRODUCTION

The numerical methods for solving the electromagnetic waves scattering

problems have a recent history of only two decades. Although the true and

original radar scattering problem is a problem of solving the initial-boundary

value problem of Maxwell's equations in the space-time domain, in earlier days

most numerical methods are based upon first Fourier transforming the original

problem in the space-time domain into a corresponding problem in the space-

frequency domain and then solving it numerically for several chosen frequencies.

In this way, some useful but incomplete scattering information can be extracted

from these results without performing inverse Fourier transformation. In

particular, the antique definition of the radar cross section was originally

defined for a single-frequency source; its proportionality to the square of

the electric field and the invalidity of the principle of superposition for

this nonlinear situation will introduce further error into the calculation of

radar cross section by this approach as compared to the true situation.

Moreover, all numerical methods for solving the E-M scattering problem in the

space-frequency domain can be reduced to the problem of solving a very large

matrix equation (in particular the three-dimensional scattering problems)

which requires an extra ordinarily large amount of CPU time and memory storage;

thus it renders these methods inefficient.

An efficient finite difference method for solving E-M scattering problems

in the space-time domain was first introduced by Yee [1] for the two-dimensional

case and later applied to the three-dimensional case by Taflove and Brodwin [21,

Holland [3], and Kunz and Lee [4]. This finite difference method requires

a uniformly rectangular/cubical grid system and it is the most efficient method

(see the review by Chen [5]). Unfortunately, for scatterers with curved

boundaries, one needs extra ordinarily large amount of uniform rectangular/

cubical grid zones to approximate the curved boundaries and minimize the

undesirable "staircasing phenomenon", and thus it renders this numerical method

inefficient in general.

Later, Mei, Cangellaris, Angelakos and Lin [61,[7] have presented the

"Point-matched Time Domain Finite Element Method", a combination of the

essential features of the standard Yee's finite difference method and the

finite element method. Its efficiency is an improvement over the standard

Yee's finite difference method due to its ability to compute on a body-fitted

non-orthogonal grid system, but it is still not efficient enough due to the

additional work in implementing any boundary conditions, e.g., extra inter-

polation and extrapolation are needed at the boundary grid zones.

Recently, Yee [8] has made a dramatic improvement of his method by applying

his finite difference discretization in the most natural way to the integral

form of Maxwell's equations on a general non-orthogonal grid system which

makes the implementation of boundary conditions extremely easy. One can

show that it is the most efficient (for the same accuracy) method available

by performing the standard computational complexity analysis, i.e., to count

the total floating point arithmetic operations needed in a typical calculation

[5]. This algorithm can be vectorized and parallelized with great ease and

hence it will be ideal for the vector and multi-processor computers.

Here in the Phase I research, Yee's

improved method is generalized for

solving the two-dimensional scattering

problems of E-M waves by targets with

complex geometry and material property.

First, the whole space domain Q is

divided into three connected but non- r

overlapping sub-domains, the interior

region QI representing the target

and possessing a non-orthogonal

cylindrical grid system centered in

itself, the intermediate region 12

representing the free space Just

outside of the target and possessing

the same grid system, and the exterior

region Q3 representing the far-field free 3

space but truncated at a large distance away

from the target and possessing the standard Fig. I

orthogonal cylindrical grid system (Fig. 1).

Mathematically, an initial-boundary value problem of the integral form of

Maxwell's equations must be solved numerically,

-2-

L- - JiH/t *ds,(

- "~ d_ = (CE1 + EaE /t) - ds,

E 2 - 0aH2 /3t *ds, (2)

H H2"dR = C 0E2/Dt *ds,

-K3 LZ d fi- 0VH3/at • ds, xC3 3

!!3 " = (J+ C 0E3 /t) *ds,

with boundary conditions (assuming no surface charges and currents),

xnx2 = nx , n xx 2 x H ()x C DQ 12, (4,)

E 2n = c)E *n

and the asymptotic terminating condition,

x E3 = (I0/CO) (n x H3), X Eail3 (5)

where n is the unit outer normal vector at the interfaces, aI12 is the

interface between 11 and Q2' a3 is the outer boundary of Q3 J(x) is the

source distribution, e0 and i0 are the free space permittivity and permeabi-

lity respectively, E(x) is the 3 x 3 real positive symmetric permittivity

matrix of the target, p(x) is the 3 x 3 real positive symmetric permeability

..patrix of the target, and _(x) is the 3 x 3 real positive symmetric conduct-

ivity matrix of the target.

Since there exists no realistic two-dimensional E-M scattering problem,

for the Phase I research the scattering of normal inc..dent TEM electromagnetic

wave by a cylindrical target with its axis along i, e.g., E - E i + E iS x-x y-yand H - Hzi, where i is the unit vector in the a-direction.

-3-

FINITE DIFFERENCE METHOD

To discretize (l)-(3), the rectangle rule is used to approximate both

the line and area integrals, the values of E and H fields are calculated on

two different but staggered grid systems, and the leap-frog finite difference

scheme is used to approximate the first order derivative in time.

Let each grid point of the basic non-orthogonal cylindrical grid system

be denoted by (ri.j, aj) = (i,j), where "i" and "j" denote the i-th closed

cylindrical grid line and the j-th radial grid line respectively. Let the

center of the quadrilateral defined by (i,j), (i+l,j), (i,j+l) and (i+l,j+l)

be denoted by (i4+ ,j+ -) 0 { (rij+ri+, 3 i,j+ +ri+ ), g I j+l+8a),

where all these centers form another non--orthogonal cylindrical grid system

staggered on the basic grid system. Let the E fields be evaluated at the

mid-points of the four edges of the quadrilaterals (Fig. 2) and at the integer

time increments t n= nAt, n = 1,2,3 ......,; let the H fields be evaluated atn

the centers of the quadrilaterals (Fig. 2) and at the half time increments

Sn+ = (n+I)At, n = 0,1,2,3 ...... For simplicity, let

00x 10 0 0'

C 0 , = ( 0 ] 0= O(a scalar), and J be a point

Z' PZZ

source located at the grid point

(a,$). Let i = 1,2,3,...,1

and j =1,2,3,... ,J. E(i+ ,J+l ~lIA

y i .

J-1

x .- Fig. 2

-4-

The discretization of (l)-(3) in SlIare,

H( ,j+ ,n+k) = H( ,J+ ,n- ) - 2At j+1 r~jl~~ (6)

p z (k'J)1,j+1

{A to(0I,j)62 01,j) + c 0(11j)IE( ,j n+l) 27

=Atu( J)fr '~l(-0Jl 8 - r j- i(1~- jo ) 2/8)169.( j,j)J l E(k,j,n)

+ ~( 2, j) E(1-, j ,n) + At6 ,* ( ,j ) (H (k,j+ , n+ ) - H ( ,j - , n+) j= 1,2,3,.... ,J,n = ,1,2,3,.....,.

H(i+ j,j+k,n+ ) = H(i+ll,j+ ',n- ) (8)

+ E(i+1,j+ j,n)9R(i+1,J+ j) - E(i+!I,J+1,n)(r +,+-rj)

- E(i,j+ll,n)SR.(i,j+k)},

(AtO~~i~j~~jj)(r i~~-rij)I(jlei)2/)tca(i,J ) JE(i+41,j n+l)

=6Y,(ij+l )E (i,J+ )E(i,j+ ,n) - c a(i,J+ )E(i+ ,j n)

tt6.*jj (H(i+ ,j+ g,n+ ) -H(i- l,J+,n+ )),r,+ -r,I, j+ i- j+

(Ao(+',j) A A(- ~ -E) ) 2 /8) -. r j~ij (1-0 (J-..GJ12 8) IE A 0(0

6 9.( i , j ) E (ij A, n+l)

+ {Ata(i-0jj)61 2 (i+ ,IJ) + x (i+ ,J) E(i+ ,j n+l)

=- 6,t(i+k,j)E c(i+ j)E(ijA,n) + '£x (i+,J)E(i+ ,j n)

+ At69t*(i+lj,j) (H(iA+ j,n+l ) -H(i+ ,J- ,n+ )),

j - 1,2,3,... ,J, n - 0$1,2,3......

The discretizatlon of (1)-(3) in Sl2 are,

H(i+ ,J+ ,n+ ) - H(i+ ,j+ ,n-k) (11)

At Eikj ( Eilj+,6?(+,A

11 0A~ihJ115i~-5-i

{(AtcT +E e) 6 9(i,j+ )D}E(ij+,n+l) (12)0+ 0

R ~(ilj+ )£ 0E(i,j+ ,n) - E 0 (r i.j~ 1-r .. M(-(0 1-e .) 2/ 8) E(i+ , j ,n)

At 6Z* (i ~j+)'ikj ~+-2 H(i- ,j+k,n+ )),

Atc E: ){ri ,+ (1-(0 -e ) 2/8) (1-(0 -e. 2 /8)}6Z(i+ ,j)E(i,j+ ,n+l)0 ~'j+1 j+1 j :ij- j- -

(At +E0 )6k 2 (i-,)~~~ ~)(13)

= .0 {r i+-2 j 2k(- ~-0i)/8) r (--,j- 1O0) j1/8) }6Y-(i+ ,j)E(i,j+ ,n)

+ E 0 R 2 (i+ ,j)E(i+ ,j,n) + At6t*(i+ .',j)(H(i4-1,j+ l,n+ ) -H(i+ ,j-k,n+k))

j = 102,3,... ,J, n = 0,1,2,3 ......

The discretization of (1)-(3) in 03are,

H(i+ ',j+ g,n+ -) = H(i+ ,j+ ,n-k) (14)

-2At r . . . .~~)E~~~~ln)r1 0(%+iej6 )(r - r )(r i +r i ~

+ (r i E(i+1,j+ j,n) - r.iEi,j+ ,ro)XO 0 -0 j}

E(i,j+ ,,n+1) =(Atc + )'(1£{EE(i,J+ ,n) -2At ) (H (i+ j+A, n+ ) -H (i-11, 4--1,n+f) )

- 6 ia6 1 1,0At ( -J (a,8, n+k) sink(6 J +0 ) (15)

+ J (ct,S,n+ )cos (O 0 +6 ))}

-146t(H (i+k, JA, n+k)-H (i--, j-k, n+h))E(i+h,j,n+l) = (&o0b+F0) {E 0E(i+,j n)+ (r i~1 +r i)(0 J 1 - i1

i'ct A+, aAtE- x (a, 0,n~k) (sink (e 0 +0 1)-sin (O 0J1+J y aOnk(ok6~~ )-cosk(e i+e6-l))J}, (16)

J= 1,2,3,... ,J, n = 0,1,2,3 ...

where cA(i+ ,j+ -) - k(e J+1-0 1){(r1~ i~j -r i )r jl+(r+,ji ij ) lj

6t*(,JA)- k(r ij +r )(i e l 61)

r .J - ;4(ruij.l +ruj~i+r i.j +r i+lj)

-6-

= {2 + r(2

{(ri,j+l-r i,j ri,jr,j+1 j+1 *j2 2-(j~-j

6R(i+ ,j) = {2(ri+, j+ -ri+, j_ ) + ri+ ,j+ ri+ ,(6 } +,0 1Ca (ij+ ) = £ (ij+ )sin 2 (Oj+i+O) + E (i,j+ )cos 2 (ej+l+j )

y j+1+) - y'

_E (ij+ ) = x(ij+ )(r ij+isin 0j+1 - ri Jsin 0 )sin ( j++e )

+ (i,j+ )(r Cos + - r Cos 0.)cos (0 +8.),y (i,j+I° 8j+1 ri,jc~ j cs j+l+Sj)

2 2 2 2C XUAi+, = Ex (i+ J) ir+ ,j+ sin (j+l+j ) + r i+1,j-in j +j-)

- 2 ri+ ,j+ ri+ ,j_sin (6j+l+0j)sin (Sj+Oj_ l )}

+ E (i+ ,j)r 2 CosC2 iCO +1 0 + 2y i+15, i + j+1 + r2.kj o 1,J 1 (6i+

-2ri+ , + ri+ ,j_..cos (j+l+8 )cos (ej +8 J )},

(i+ J) =E (i+ ,j)sin 8j (ri+ ,j+ sin (6jl+ ) - ri+ ,j_ sin (j+j_ ))

+ Y (i+ ,j)cos J (ri ,A+ j cos (6j++j) - ri+ ,,j_ cos (j+OjJ)) ,

ex(ij+ ) = (ex(i+ ,j+ ) + Cx(i- ,j+ )),

y (i,j+) = (c y(i+ ,j+) + E (i- ,J+))Iyyy'

C (i+ A,j) = (Ex(i+1 ,j+) + CxUi j- ))

C (i+ ,j) = (e y(i +,j+ ) + e (i+ ,J- )).

Theoretically, the boundary conditions (4) must be imposed at the inter-

face of two different materials. But here, there is no need to impose the

boundary conditions explicitly in programming this numerical algorithm,

because the boundary condition for the tangential component of E is satisfied

automatically, and the other three boundary conditions are also automatically

satisfied in the approximate sense if the differences of the material properties

spread linearly across a complete grid zone instead of just across the inter-

face. In this way, there is no cumbersome programming instruction at the

interface to slow down the calculation and the application of boundary

conditions is replaced by the process of assigning the material parameters into

the grid zones. In particular, the programming instruction is extremely

simple if the material interface is located either at a constant i-line or at

a constant J-line.

-7-

At the exterior boundary a3 (i = I), the following simple but effectivediscretization of the asymptotic terminating of non-reflecting condition (5)

is used,

E(I,j+l),n+) /n+ ), (17)

j = 1,2,3,...,J, n = 0,1,2,3 .....

Finally, to achieve a stable computation for this explicit finite

difference scheme, one must impose the following well known stability condition,

At < Min. R 'i Min. x'y'z(i+ ,j+ ) • Min. 1xiyz(i'J), (18)i'j i,j,x,y,z i,j ,x,y,z

where 6. ij is the typical dimension of the i-jth grid zone.

NU MRICAL SIMULATION

From Eqs. (6)-(16), the discrete values of the E-M fields in the space-

time domain scattered by a two-dimensional target with complex geometry and

material property can be calculated as accurate as one desires. However,

the main interest for the radar scattering application is not the detail

description of the E-M fields everywhere; rather it is a certain average

quantity to characterize the scattering and absorbing properties of a

scatterer in the incident E-M fields. The quantity mostly common used for

this purpose is the radar cross section which initially was defined for the

time-harmonic E-M fields [9] as a normalized scattered power intensity averaged

over a cycle or a normalized scattered energy intensity per cycle,Iz,2

r ) r 0' 2 , (19)r- C 14n12

where r - 2wr for two-dimensional problems,

- 4ir 2 for three-dimensional problems,

r - distance between the transmitter and the scatterer,.

- angle between the transmitter and the zero angle axis,

E = incident E-field at the target,in

and E -scattered far field E-field.

-8-

Since the antique definition of the radar cross section (19) was defined

only for a single-frequency source, errors will be introduced when it is

applied to the scattering problems with general time-varying sources. In

order to be able to deal with more realistic situations, the definition of the

radar cross section (19) is generalized to cases with C-W pulse as source as

the normalized scattered energy intensity,f Ts+ATSJE 12 dt

EM = lim r T _2 (20)r-ao ,T.n+ATno2

in EinIEI' dt

Tin -i

where AT. is the duration of the incident E-M pulse and AT is the durationin , °50frt n n

of the scattered E-M pulse, i.e., 0 for t < T and t > T. +.i. , andnin in in

E m 0 for t < T and t > T +AT-s S S SIt is clear that the definition (19) is a special case of the definition (20).

Now, there are two approaches to calculate the radar cross section of a

given target. One approach is first to approximate the time-harmonic source

by a C-W pulse with very long duration, next one runs the above developed

computer code long enough for all of the transients to died off, and finally

the definition (19) is used to compute the radar cross section. The other

approach is to use a very short C-W pulse as the source, next the computer

code is run long enough for the scattered C-W pulse passing through the loca-

tion of the receiver completely, and finally the definition (20) is used to

compute the radar cross section. It is obvious that the cost for the first

approach to calculate the radar cross section is prohibitively high and the

second approach is the preferred one.

To achieve our goal, a simple and almost costless sub-routine is added to

the computer code for calculating f0 E(X,T)I dTin the center of every grid0

zone at the outer boundary M39 i.e.,-Sn2

Y IE(I-1,j+ ,m) 12At, j = 1,2,3,...,J, n = 1,2,3 .......mr-1

are calculated in the code; in general, it is a monotonic non-decreasing

function of t.

In the region where the durations of E and E pulses are distinctively:-in -

non-overlapping ( < 1000), the value of the energy integral rises to a

-9-

T +AT.n 2constant value f 1 nE(x,T)I dT after the passing through of the the E.

0 Ts+AT --inpulse and later it will rise again to another constant value f0 IE(x, T)1 2dT

after the passing through of the E pulse. Then the value offTs+AT s - .Ts+ATs 2 in+ATin 2

TS IEI1 2dT can be obtained simply from f0 IE 2dT - f lET2td.Ts -s 0 0

A schematic diagram of this is given in Fig. 3.T +ATT.n+ATin

As for the value of fin+T in, E0 2dT, it can be obtained as f in in 2 d"Tin -in' 3 -!in'

at the target location with the target replaced by the free space in a separate

calculation.

f IE(xT)l dT0

I1 d

0 TnTnA~ T +AT

f I s 1 d

Fig. 3

As Example one, a jet consisting of perfectly conducting material with a

characteristic dimension ". 10 m is used as the target. The computational

grid system is shown in Fig. 4 with the radial grid size Ar - 1 m in 13"

The computational time increment At is chosen to be 1.25 ns. A dipole point

source with Gaussian distribution in time and a duration of 1Oat is placed in Q1

short C-W pulse of E-M fields with frequency 1 40OMz (wavelength 7. 5 a).

The numerical results of radar cross sections E(6 0), E(16°0), E(35°), E(62.5°),

-10-

I S

and E(88.5 ) as functions of 6 are plotted in Figs. 4, 5, 6, 7, and 8,

respectively. Moreover, the radar back-scattering cross section as a function

of e(0 < 6 < 90 ) is plotted in Fig. 9; since it is symmetric with respect to

O = 0, its values in the range of -900 < 0 < 0 are omitted in Fig. 9.

As a matter of fact, the newly developed computer code in this project is

used to calculate the values of the radar back-scattering cross section in the

range of negative 0 and it is found that the maximum deviation between its

corresponding values at either side of 0 = 0 is less than 3.5%.

As Example two, a composite airfoil with the leading edge consisting of

anisotropic lossy dielectric, ex = 22.989x102 farad/m, s = 44.210x102 farad/m2'x y

and a = 5xl0 mho/m, and the trailing edge consisting of perfectly conducting

material is used as the target. The characteristic length of the airfoil is

I0 m. The computational grid system is shown in Fig. 10 and the computational

time increment At is chosen to be 0.15 ns. A dipole point source with

Gaussian distribution in time and a duration of 2.1 ns is placed at (14m, ).

This current source generates a short C-W pulse of E-M fields with frequency

It, 238MHz (wavelength "" 1.26m). The numerical results of radar cross sections

Z(32.50), E(22.50), E(12.50) and E(2.5 ) as functions of e are plotted inFigs. 11, 12, 13, and 14 respectively. Similarly, the radar back-scattering

cross section as a function of positive 0 is plotted in Fig. 15; again due to

the symmetry, the radar back-scattering cross sections for the negative values

of 0 are omitted here.

TECHNICAL DISCUSSION

Obviously, the numerical results for the radar cross section obtained from

the calculation in the previous section are correct only in a very approximate

sense. The main reason is that the supposedly scattered far field E is--scomputed at the location about few wavelengths away from the target surface

(actually in the intermediate field region), an error in physical representation.

The simplest way to improve the accuracy is to add many radial zones in 13 so

that the boundary M3 will be in the truly far field region. Unfortunately,

this approach will not really solve the accuracy problem, because the ratio of

the maximum dimension to the minimum dimension of the grid zone in the far

field region will be so enormous that it will introduce large numerical errors.

-11-

x -Location of

E(6 0

m 1 6m 26.4

Fig. 4

x -Location of

I E(160)

17.im 26. 41;

Fig. 5

-12-

x -Location of

E~(350)

17. m26.4m

Fig. 6

x -Location of

E(62.5*

17. 26.4m2

Fig. 7

-13-

x -Location of

17.6 26.

Fig. a

-14-

m 17. m 26.

Fig. 9 Radar Back-Scattering Cross Section

To overcome this problem, a multi-grid system should be implemented in 13 to

reduce the huge ratio of the grid dimensions. For this, an interpolation

scheme must be devised to transmit the information of E-H fields across the

boundary separating the finer grid system from the coarser grid system.

There is another computational efficiency problem existing here. For a

stable calculation, the smallness of the dimension of the triangular grid zones

surrounding the origin of the coordinates makes At extremely small, and hence

it takes too many At to reach a pre-determined time. To overcome this

difficulty, one can replace the cluster of small triangular grid zones by a

single circular zone and modify the finite difference equations in this

neighborhood accordingly.

-15-

20m

Fig. 10 The shaded area represents the composite airfoil

-16-

Directionof Source

00

loin 20m 32m

Fig. 11

Direction of Source

E(22.5)

loin 20m 30m

Fig. 12

Direction

I rho (12.5 0

20. 30.

Fig. 13

-17-

Fig. 14

0loin 20m 30m

Fig. 15 Radar Back-Scattering Cross Section

REFERENCES

[1l K. S. Yee, "Numerical solution of initial boundary value problems

involving Maxwell's equation in isotropic media," IEEE Trans. Ant. Prop.,AP-14, pp.302-307, 1966.

[21 A. Taflove and M. E. Brodwin, "Numerical solution of steady-state

electromagnetic scattering problems using the time dependent Maxwell's

equations," IEEE Trans. Microwave Theo. Tech., MTT-23, pp.623-630, 1975.

L[3 R. Holland, "THEDE: A free-field EMP coupling and scattering code,"

IEEE Trans. Nucl. Sci., NS-24, pp.2416-2421, 1977.

[41 K. S. Kunz and K. M. Lee, "A three-dimensional finite-difference solution

of the external response of an aircraft to a complex transient EM

environment: Part I - The method and its implementation," IEEE Trans.

E-M Comp., EMC-20, pp.328-333, 1978.

[5] Y. M. Chen, "Efficiency of numerical methods for solving Maxwell's

equations in space-time domain," SGEMP Note #8, Lawrence Livermore

National Lab., Livermore, California, July 1986.

[6] K. K. Mei, A. Cangellaris and D. J. Angelakos, "Conformal time domain

finite difference method," Radio Science, 19, pp.1145-1147, 1984.

[7] A. C. Cangellaris, C. C. Lin and K. K. Mei, "Point-matched time domain

finite element methods for electromagnetic radiation and scattering," Mem.

#UCB/ERL M85/25, Electronic Research Lab., University of California,

Berkeley, April, 1985.

[8] K. S. Yee, "Numerical solution to Maxwell's equations with non-orthogonal

grids," SGEMP Note #4, Lawrence Livermore National Lab., Livermore,

California, January 1985.

[9] R. W. P. King and T. T. Wu,,The Scattering and Diffraction of Waves,

Harvard University Press, Cambridge, Mass., 1959.

-19-

APPENDIX:N>

0* OPt -.

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* ft t C -iC'-

t 4 ft C Z,3

.. ftC .

151 + 1 '*r Q) .. -4 . -4 I1

j C-4 0. a.' -4~ -ft~'Caf ft -4 1-I

av1 y .11 -lC v a '* I C4 I- 4 j- p9C4

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404 c4N 441 " :.* 1 %4 m+-mCN Q) W y0 x.. x x~H 'x YX-0)

AI I-fE 4J444141-44

-a -

0 EmU

ft ft ft 7. * a 0 0.30 43

N. ~~44 04

C - N. a '.-.N.'.'--~4 - . - N '- *-4 __ _ _ _ _ _ _ _ _ _

A[2 . N

* A a f

f; C, ao ft ft

ft1 ft4 aa4

ft44M.- ~Mf t a$

a0 (0 4- 4

* 1.4 oft.20-

.'22

41N X

V. CVft f

04* Xt',ft

04 aX"I -MIS 4 1

ft ft 04; lt4

C14 We 1:Ca 4

>1 e: 4.1 -Mcla m a ?~~*- . a : = , , 4

I1; r- - ' Wt 0r o-Vt -ft 44

0 0

4.- 41 C4

fn 44 ~OftAi4CF I>j ft *

ftftft - ~ ft4.4f 4.44.4.

-4 co-en4 - a4 a I~f ft .0t.-

If - If t4 .- -- C.4J )1 00, F6c 04 i '

A tt4 IQ n mv>>~ -4'H

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ft~t~f 44V44 'U ftt4-ft -21- f

N N01

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N N. N N N .0

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NA N44 - 5

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1,: 1 o

4~~~ ~ ~ -4041000 a

01 ~ -4& - -n ;. -i0

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N00 40 N 4 N 4 CO

NN

Np N,

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a . --

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Z; a) 00 U'. -41:

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00

o- 'o +00 .'0" UW

a, 04 -' + '4 4 W NIlu cV*.- -N + 14 XVY.XV9 X1

*U 0 O4 )

0~ 41 -4 - 1)4J0-

'0'4l'flCJ s4vfluw

a, 44 4, N-4 IN -- -4X " 4444. ~ * U

01 '.

~~v41~~~' U) 1 4- -". - W

04-4-444 ~~~~'2 0 N -.. 4I '' - ~ 4'.-

$4 I > 44 ~ 0X 4NN-4nI4' ~ 44 - 0U0

C 4-) 0 - 40.10 ! Il hl -I 0 A 41

0~0 .4.~ N''4'6l0-.4 4- X.. " '-4 I) ol 440U41

m 41 0 IC-' 14)'-d en + ~ ~ n

04 'd4U, 0 4C 0* N :4J W4(D ,4 14 09 1 op1N.

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C4C

0~" '4- "AU

cc 41 : 41 41

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U-23- l

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* I 4. a .4a .

- aI or-0' - a a~u1~ CC %4 1 -61 S4 C

*0 + - -0 .0 4Q)4- I a-l a . -1 .. 4. ~ 4- X

a~~~' - - ;-..

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77 + +.4 -,- -4 a *

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0 x- "I 1.0

8It *I.44 -9 0 + 1 (D .01-**l4I H . lN- f 1-. -1) S<a s I '.141'.- - W U-'- a"o '0a *- -,a *- OZ: - n- n a -C'C

1.4- C4 4.

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a-v a m-: m.0 0 I

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4J-. ar qj1 44 C1 1'

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3< *~V# -4 -4--I .4-~-1-3

A' ~ - t 04 E-'*-~ ~~ ~ -. < 0 00 - *.4 a V - - WCW). ' 0I I<

00 414 4j.1Oi - a0* o~- *3

0~~~~' -4<<333 (4~ -40*--~ 141 3<33<3 .

Na~~. a :0011 Na 4P24 -H 4 a4 . : a (.-N4O

-NNN, .0.'--.4 - .4%N 04~4'

o aa 41

10,S 0 a

+ 0H

0 ft .4 A 0 41' 0 ft4*, j 01 ft ft- ... r-

*~L .. 41 0tUf~4. f

:3 : 0 . A' cI :ft, -t f.,4* .- 41 0 W4 I

0 11104

f - - 44, -4J mm~ .0t-

ft mt It f - - - -1 X r,*.-. f

fo f t ft * * * 1t t ~ f

Ct ftftfic0 t f t

ft~~-- ft= xt4tf 4f q f

ftV ft to 0f 0 ftC4

ft Iftft ft ~4.4. N : t. L4clf *. 0 f ~ C f

ft 0 a f - ft2O f

ft~~~~Wft~~A *--xt.* I t .r .

0". f -0 .4 -01 -. *4.- Kfo4..

ft~~ ~ 1'~f 1.- "D t 4 -r-4

0' 4. 41 ,. -.n ., . C - 4

f, ft C13 0+ +t44 +. 4.0- 4-*.- +4 a a-- A4. ft

A 0 0-4ftft*. ft. E K. N . ft t CK-- C--I f.4.

4-. t V tC -4. ~ .- 4K N-I t.4-25C- N~ I S40'f.4f i4f

Nc 41 '

C, a1 1* a C-1

* a a+

C. C IC

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4a to-a* N a I- 1 C1

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0- $4 -4 C C

tr CN I C4 4J NC fl0 . .

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CC 0-4.J-44j4 4j C -1 41

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C a-.Na ---- n O5'a-- CU-~-'-' 4.4 -C!

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>4.i .. , 1 I *1..04- a -

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a 041-4 444-.

C Mlu f CCa : :N

C. aaaa- O-I 4.a raa .4.

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a, I p NN z

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to 4j 3-4 -a4 j

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0a 4J aJ a 4j0.14aj a rI 6- n I r-e46-

0 0- 4j40400tt -- ~142.-.. -~ 6-.-*4

eq4 a4~ ~ V V ~ X a a - '' -aN-a -X X *~~ ~ a a 6K K . - .

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