390573-3-T ON THE SOLUTION OF THE E-FIELD INTEGRAL EQUATION: PART II J. L. Volakis J.-M. Jin L. C. Kempel D. C. Ross Northrop Corporation 8900 E. Washington Blvd. Pico Rivera CA 90660-3737 October 1991 390573-3-T = RL-2578

390573-3-T ON THE SOLUTION OF THE E-FIELD INTEGRAL EQUATION: PART II J. L. Volakis J.-M. Jin L. C. Kempel D. C. Ross Northrop Corporation 8900 E. Washington Blvd. Pico Rivera CA 90660-3737 October 1991

On the Solution of the E-field Integral Equation - Part II J.L. Volakis, J.-M. Jin, L.C. Kempel, and D.C. Ross October 1991 Abstract In a previous report [1], we derived an E-field integral equation for the scattering from resistive cards. Current and charge integral equations were developed using linear and doublet basis functions, respectively, and both formulations were shown to yield identical results for flat strips. In this follow-up report, we extend these formulations to arbitrary curved strips and reformulate the charge integral equation for solution using pulse basis. Of the presented current and charge integral equations, both have a field matrix time of 0(N2). In addition, new results are presented for the S-shaped surface which illustrate that a small surface blemish can cause significant scattering returns at near grazing incidences.

1 Introduction In a previous report [1] we derived a version of the E-field integral equation for scattering by resistive cards (see Figs. 1 and 2) in terms of the charge density rather than the current density as is usually done. The resulting charge integral equation was then solved for a flat strip via Galerkin's method using doublet basis functions, and it was shown that the resulting system was identical to that obtained from the current integral equation in conjunction with linear basis and Galerkin's testing. In this follow-up report the same integral equations are implemented via Galerkin's method for a curved resistive strip. Linear basis are employed for the current integral equation (see Fig. 3) and the resulting matrix elements are developed in detail. The charge integral equation is implemented using pulse basis to yield the same accuracy as the current integral equation with linear basis but because of the simplicity of pulse basis, the charge integral equation leads to a simpler implementation. In this case, however, the standard Galerkin's method (where the test pulse is the same as the expansion pulse) is not applicable because it leads to vanishing self cell elements for one of the two integrals in the equations. To avoid this, the test pulses are shifted one-half segment width but since the integrands are slowly varying over the test pulse, it was found sufficient to sample at two symmetric location over the test pulse. This makes the implementation of the charge integral equation rather simple and provided the CPU time for generating the matrix is maintained at O(N2), where N denotes the number subdivisions, the charge integral equation is more attractive. A way to achieve this is discussed and results are presented which demonstrate that the simpler charge equation is at least as efficient as the standard current integral equation. Further new results are presented for the scattering by the S-shaped surface which illustrates that a small blemish placed at the lower knee can cause substantial backscattering near edge-on incidence. 1

2 The integral equations Referring to [1] and assuming the plane wave Hi = ejko (x cos ko +ysin o) (1) to be incident upon the resistive strip shown in Figure 1, the pertinent current integral equations are YoE'(s) = RJ(s) + - J (s) {s s'H2)(kor)} ds' R. dJ,) )+0 (2) 4ko Jc dsi 9s H (kor)ds' (2) where r (x - x')2 + (y - y')2, Yo = l/Zo is the free space admittance and J,(s) denotes the current density on the strip of resistivity R. Setting the charge density p(s) as p()- () (3a) jW where dJ, (s) = ds' (3b) we can alternatively write (2) as YoE'(s) RJ(s) - J (s')Gi(x, y; x ) ds' + 4ko c(s')-s Ho(kor) (4) which is the charge integral equation. In this GI(x, y; x',y')= J s- s^)H (k ) ds", (5) = /(x-x ) + (y - y")2, (6) 2

and it can be shown (see Appendix) that J) 2 c L L ) [(s + 6)- (s - )] d -1 { 2+s [ 2(s' - s)L - [ + 2(-)] (') d'} (7) L/2+s L with L denoting the length of the strip and s' is the cumulative distance along the strip up to the point (x', '). Substitution of (7) into (4) then leads to an integral equation only in terms of the charge density 4(s'). 3 Galerkin's Solution of the Current Integral Equation for an Arbitrary Surface Using Linear Basis Consider now an arbitrarily curved strip or a closed cylinder, shown in Fig. 4, whose surface satisfies the resistive sheet condition. The excitation is a plane wave and we are interested in the solution of the surface current density Js(s), where s is a measure of the distance along the cylinder's or strip's contour. Hereon, s will denote the unit vector tangent to the resistive surface. To discretize the integral equation (2), the surface is subdivided into straight segments as shown in Fig. 5 and as in the case of the flat strip we again choose the expansion and testing functions to be the triangle functions shown in Fig. 3(a). To define these more explicitly, let us assume that one of them is centered at the node (Xm, yn,), as shown in Fig. 6, with the left and right segments being of length Sm = (n m - m-i)2 + (ynm - Yn-1 )2 (8) Sm2 = \/(m+l - Xm)2 + (Ym+l - ym)2 3

respectively. We can then express Wm(s) and Lm(s) as - < s < Sml Wm(s) = Lm(s) = S (9) j-m --- 0 < Sm2 Sm2 which are suitable for integration along the s coordinate. In the subsequent calculations we shall also make use of the parameters Xm - Xm-1 aml = i(,,- ( Vml ~ m (10) Om2 = tanI ( + --- Y) (11) Xm-+l - X-1 nml = X COS Oml + Y sin 0,nl, (12) Sm2 = X COS Om2 + ~ sin 0m2, (13) where the last expressions represent unit vectors which are tangent to the segment on either side of the point (Xm, Ym). Using these parameters, the parametric equations for the linear segment to the left of (Xm, ym) are X = Xm-1 + S COs Oml, Y = ym-1 + S sin O91, (14) and those for the segment to the right of (Xm, ym) are X = Xm + S COS Om2, Y = -m + s sin 0m2. (15) Thus, given N samples of the strip's or cylinder's surface we can proceed with the computation of all parameters describing the linear segment which make the discrete version of that surface. Substituting (9) into (2) we obtain the discrete system [Amn] [Jn] -= [bm] (16) where the elements of the matrix [Anm] and those of the excitation column can be defined in terms of the parameters introduced in (10)- (15). For the 4

excitation matrix elements we have bn - (sin qo cos 0, - cos qo sin nl) Snl S ejko{(Xn-l+s cos nl)cos o+(Yn-l +sinl)sin ol)si} ds O Snl + (sin Co cos On2 - cos 4o sin,n2). Sn2 Sn2 — S eJko{(xn+scosOn2)cos o+(yn+s sin 2) sin o} ds (17) dO Sn2 which after integration can be written as b = (sin Oo cos nl - cos O0 sin nl) ejk[xn-1 COSO+Yn-1 sin lo] SnI -O ) (jkgl -) o + - (sin 05o cos n2 - cos So sin 9n2) ejko(xn cOs o0+Yn sin 0o) Sn2 [^^j3kosgn2 6Jkosgn2 Sn2 jkog,2 + k2(g2 (jkog2 - ) (18) in which gni = cos Onl COS 0 + sin nl sin;o (19) and 92 is similarly defined upon replacing the subscripts in (19) from nl to n2. The impedance matrix elements can be expressed as Anm = anm + anm + am (20) where anm = R(s)Lm(s)Wn(s)ds, (21) nn anm = - (S) Lm(s) {(ss')HO2)(kor)} ds'ds, (22) 4 n?FC 3 — 1 J d/TV (s) dLm(s') 2)( ) dsds. (23) ~5ads. (23),ds ds'

The first of these integrals can be readily integrated once the resistivity is specified over the mth segment. Given that the resistivity will be discretely defined as a constant within each segment, with the mth segment (lefined as that between the (m - l)th and mth node, it follows that Rn (3 - )+Rn+i( 3) n m RnSnl n=mm n= m ~ 1 Unm6 (24) Rmn -- n = -1 6 0 otherwise, where Rm denotes the average resistivity of the mth segment. To evaluate amn we proceed with the substitution of the expansion and weighting functions as given in (9). Doing so we obtain 2 Jo I i SO 1 Smi J ( anm J(Snl sm nl) {-} k {S- } H2)(kor-l) dds ko Snl) { S}, (m22) + (S m2) jfl S }o 2 - H2 (k0r2) ds' ds - ko (S2 Sn2 -lS sm2 s Hm(2)(o ) d 1 d + (^n2 'm1 jSn2 f 5n2 - 'Sm2 S' } H~2)(kor21) ds'd (1 4 5n2 Sm1 + 2 Sm2) S { Sn2 { 2 } H) (kor22) ds ds 4 Sn2 Sm2 21 22 23 24 anm — + anm + anm anm, (25) where rll [(Xn-1 + S COS 0nl - Xm-1 - S COS Oml )2 1 + (Yn-1 + s sin nl -Ym-i-1 - st sin {ml)2] 2 (26) r12 [(Xn-1+ s cos Onl - Xm - s cOs 2m2)2 + (Yn-1 + s sin nl - Ym - s' sinm2)2] 2 (27) 6

r21 - [(Xn + S COS 0n2 - Xm1 - S COS ml)2 + (Yn + s sin n2 - Ym-1 -s l sin ml)2] (28) r22 [(Xn + 2 COS n2 - Omm -S COS )2 + (yn + s sin On2 - Ym - s sin 9m2)2]. (29) All of the integrals in (25) can be evaluated numerically except when n = 77 and n = m ~ 1. In this case some of the integrands become singular and although their singularity is integrable, the associated integrals must, nevertheless, be evaluated with care. When n = m, the integrand of the first (a,21) and fourth (a2 ) integrals is singular at s s'. To evaluate them we can rewrite a m as in (27) of [1]. However, a more accurate procedure is to regularize the approximate integrands by adding and subtracting a term which can be integrated analytically. Applying this procedure to the first term of a2 gives nn4 Onl Sn1/ -21 = 0 j0$lS js () [() H)0(k s!) - - In kols -s I - 1)] ds'ds + jk (S) {-35 + 241n ( ) (30) 27r 72 2 and for the fourth term we obtain 24 k[n fSn1 Sn 2- S\ [ 8 Sn2 - s 2 / a = T Ho2)(koIs - s\) 4Jo Jo Sn2 Sn2 sn2 'J2 (s,) - { - In (-ko s - s ds' ds + {44 (-61 + 48 In() + 12 n(sn2)) Sn2 -/Sn2 {sn2-s }2 (s-sf2)ln Is-Sn2 ds}(31) The integrands of all integrals appearing in (31) and (32) are now nonsingular and can be evaluated numerically. We note that in obtaining the 7

analytical portions of (30) and (31) we employed the integral identities and a ln (clx - x'l) dx' =a ln(c) - a + xn xl -(x- a)ln Ix - al Jxlnx -dldx= (x2 -d2) ln(x-d) -d2 + 2 () - 2 -d+ d (32) (33) The second (a22) and third (a, ) term of anm have singular integrands when n = m + 1 and n = m- 1, respectively. Regularizing their integrands as above yields 22 a(m+l)m 23 a(m-l)m - T/2 /m2 -- \ ---o ko0 jSm2jSm2 { [{Sm2 - StHo(kols-s '1) 4 Sm2 Sm2 Sm2 - S j2( ds' + r {( 2 (-19+ 121n (kfOSm2))} Smi2 7 2 [ Sn2 fS2 Sn2 - [r { S H(2) (sS - S S)I 4 Sn2 SLn2 S2 j l ln}- (ols )) ds'ds + 2 -19 + 121n 2 -x 72 2 (34) (35) It remains to evaluate the integrals belonging to the term appears in (23). We have that a3 which nm a3 nm -1 1 S'nl fSml H(2) (kor11) ds ds 4ko Snl1 J~l ~ +1 1 sl j2 Ho) (korl2) ds' ds 4ko snlSm2 ~ 1O +1 1 JS22 jm H(2) (kor2) ds' ds 4ko Sn2Sml J O 1 1 JS72 J-m2 H2H (kor22) ds' ds 4ko n2Sm2 0 0 31 32+ 33 34 = nm + anm + anm ~ anm (36) 8

where rmn are the same as for a2 and were defined in (26)-(29). As in the case of a2m all of these integrals can be evaluated numerically except when n = m and n = m ~ 1. When n = m, the integrands of a 31 and an34 are singular and by regularizing them we obtain the alternate expressions 31 -1 1 fSnl/ nl (2) (o S) a I Ho kols - sD1) nn 4k0 (Snl)2 J -- In( kols - s' ds'ds 1 1 j2 (nl )2 3+21 4ko (snl )2 7r 2 1) + 21n(snl,) } (37) and 34 -S1 1 Sn2 Sn2 F(2)(kl-S) a = I I H(kols - s'I) 4ko (Sn2)2 Jo Jo -2- n 2ols-s'l)] ds'ds 4k0 2 (-2 ) 2 (-3 + 21n 2 )+ 21n(Sn2) (38) 4k0 (8,2)2 7 2 2 These are now suitable for numerical evaluation. When n = m + 1, the second term of a3 has a singular integrand and must be evaluated as anm 32 a(m+l)m - 1 1 jfS2 j7n2 H2) 4ko( m2) 2 o o (2) ol - I 1) ln ( kols -s'l) ds' ds 4 0 (S2)2 7 -3 + 2In + 21n(s,,2) )(39) 4koo(SM2)2 7r _ 2 2 / Finally, when n = m - 1, the third term in (37) should be evaluated as 33 a(m-l)m - 1 1 Sn2 jSn2 [HO2)(kols - sl 4ko(sn2)2) Jo o -jn n( I kols - s'l ds'ds 1 1 j2 (S 2)2 (-3 +2 )+2 nk()1(40) ~ 4ko_3s+)21n 2 - +21n(sn2) (40) 4ko (sn2)2 7 22 9

By comparing (37)-(40) it is evident that a(+3 =) -a3m and that a(31)m = -a34m. We further note that the simpler result in (34) of [1] could be used for evaluating a31 a4, a32)m and a(33 ann a(+l) an a (ml)m. 4 Solution of the Charge Integral Equation Using Pulse Bases Let us again consider the curved strip or cylinder shown in Fig. 4. We are now interested in solving the charge integral equation for this geometry. To do so we shall use pulse basis for expanding the quantity +(s) and not the doublet function used in [1]. Thus, the charge conservation requirement must now be imposed explicitly. At first this would appear to yield an overdetermined system of equations. However, the usual Galerkin's or point matching technique leads to ill-conditioned systems and cannot be employed in the standard manner. In particular, on using point matching (at the center of the segment), the self-cell term of the last integral (4) vanishes making this testing/weighting procedure completely inappropriate since it zeros the most important term of the integral equation. Also, the usual Galerkin's testing leads to a similar situation. To avoid this, we can shift the weighting/testing pulse one-half of a subdivision as shown in Fig. 7. This type of weighting retains the dominance of the self-cell term but leads to N - 1 equations for open surfaces, if N denotes the number of expansions employed in the discretization of (4). Consequently, the natural condition for a unique solution of the charge distribution is the conservation of charge equation N E n = 0 (41) n=l where (n/(-j w) denotes the charge amplitudes at the nth segment. That is, they appear in the expansion N q - E ~>nP(s - Sn) (42) n=1 where P(s) 1 m -1 < s < (43) P(s) { 0 elsewhere (43) 10

with so = 0. Alternatively, in the case of point matching the test point can be placed halfway between the segment midpoint and its beginning or end point. To discretize (4) as stated above, we introduce (42) in place of +(s) and this yields the integral equation YoE = RJ(s) -, GI( x, y; xy) ds' 4fo n=1 8n-1 os To generate a system of equations from this, we shall employ a dual set of test points located one-fourth of the mth and (n + l)th segment lengths from either side of (Xm, y,,) as illustrated in Fig. 8. This testing procedure is simpler than the shifted pulse approach and should not compromise the accuracy of the solution in view of the reduced singularity of the kernels in (4) versus those in (2). That is, each equation to be generated will be the sum of the equations obtained by testing at (xm-1/4, Ym-1/4) and (Xm+1/4, Ym+l/4), and consequently the resulting system will retain certain symmetry with respect to the sample points (Xm, Ym). To generate the matrix elements of the system resulting from (44) we must evaluate the integrals JSn-1 TIn (Xt, = Gi(xt, yt; X,y) ds' (45) and T2n(t, yt)= XJ (Ho2) (kor) ds' (46) -n-1 OS where Xt = Xm~1//4, Yt = Ym~1/4, GI(xt x', y') = (' s(t )HO2)(kods", (47) rt -= (t - x')2 + (yt - y)2, (48) t = (x - ")2 + (yt - y 2, (49) and s(x, y) denotes the cumulative distance up to (x, y). 11

Assuming equal segment lengths As, T1n can be approximated as Tin(xt, yt) = As Gj(xt, Yt; Xn-1/2, yn-1/2) A= E S* )H)(ko)ds' p=2~ n A= s SIip(xt,Yt) (50) p=2 where Spo = As p < n Sp S= 1 AS (51) 2 2 pn and rtp = Xt - Xp_- - 'cos Op)2 + (yt- Yp-1 - S sin pi)2}2 (52) It is important to note that in generating the coefficients of )n, it is not necessary to perform the entire summation in (50) for every n. Instead the nth coefficient should be generated by adding one term to the (n - l)th coefficient, thus retaining our O(N2) operation count to fill the matrix. Also, there is substantial overlap in computing the coefficients for each of the two testing points and this could be exploited to further reduce the CPU matrix fill time. All of the integrals IYp(xt,yt) appearing in the sum (50) have wellbehaved integrands except when t = tip = p - 1/4 or t = t2p = p + 1/4. In this case the integrands are singular and Ilp(t, yt) must be evaluated analytically. We have (p < n) Ilp(Xtip, =p) j H (k Is0plJ - sl) ds' J 10 = Ls 'oH(2)(kols'I)ds' 4 { sln )-s]} (53) - Spl where y = 1.781 is Euler's constant. Also 1p(Xt2py) | Ho )(kol - s') ds' Jo 4 12

4 {^(ko s)]}.SpSp2/4 7r [ 2 ) ] sp2/4 where p2 = As and sp = As if p < n or p s= if p = n. The evaluation of 12 is rather straightforward. We have T2.(x, yt) = snl 0Ho2(kortH) ds' = -ko j 5 * rtnH)(kokrtn) ds' (55) where rtn = {(Xt -Xn-1 - COSnl )2 + (t - Yn-1 - s sin9nl)2}2 (56) and (Xt -Xn-1 - coS nl); + (Yt - Yn-1 - s'sinnl)Y rtn For t 7 n - or t (n - 1) + - the integrand of (55) is non-singular, and T2n can then be evaluated numerically. When t tln n- -, T2n becomes T2n(xtln, Ytln) = - Ho — (kortn) ds' - -[H0(2) (kortn)];nl [ (2)k s) - H2) (3k )] (58) Similarly when t = t2n = (n - 1) + 1 we have T2n (t2n, t2n) = -] 0 1HO (kortn) ds' = -[H(2)(kortn)]On2 -[ ( 3koAs 2) (k oAs) = -Tln(xtln,ytln). (59) 13

Finally, before completing the discretization of (44) it is necessary to also consider the integral given in (7). From the Appendix we find that 1 N [ N Js(Sm~l /4) n + (n + 1 — m r /)As As n=l + (N + 1-n + mn - )As] N -.nBmn n=l with As (n + -m T) >:~ 1 ")rm < 2(n + -mT ')As In + -m~ T - < 2 1 1 -As n+ — m:T- < -. The matrix system resulting from (44) can now be written as [Zmn] [n] [Vm] where Vm = [EnC(m -l/4, Ym-1/4) + ESn (xm+1/4, Ym+1/4)] Zmn = R[Bn+ + B ]- - [Tln(Xm+/4, Ym+1/4) Tln(Xm-1/4, Ym —1/4)] 1 + 4[T2n(Xm+1/4, Ym+1/4) + T2n(Xm-1/4, Ym-1/4)] 4k0o Provided Tln is computed via the recursive procedure noted above, the fill time of [Z,,n] will be of O(N2). 14

5 Numrnerical Implementation In this section we present some numerical results based on the solution of the systems given in sections 3 and 4. We have numerically verified that the two systems give identical numerical results and thus only patterns based on the solution of the current integral equation will be presented. Figure 9 compares the bistatic echowidth at 2 GHz of a 4 cm square metallic cylinder as computed using a pulse basis-point matching moment method program and a linear basis moment method program which employed Galerkin's technique. Figure 10 illustrates a similar comparison for a circular cylinder of radius 5 cm also at 2 GHz. Clearly, the results based on the two formulations are identical validating the given formulations for metallic surfaces. Let us now look at scattering results for resistive surfaces. As an example, let us consider the flat resistive strip. Figure 11 compares the backscatter patterns for a 5 cm strip at 2GHz having a normalized resistivity of R -= 1.1 - j'0.2. A similar result is given in figure 12 for a flat strip whose left side is metallic whereas the right side has a normalized resistivity of R = 2. Once again the agreement between the two different formulations serves as validation for resistive surfaces. We have been very interested in simulating infinite structures with finite models. One method of hiding the undesirable edge of a finite structure is to gradually taper the resistivity of a strip. An example of this is shown in figure 13 where a metallic halfplane is modelled by a 100A long tapered card whose normalized resistivity varies as E x 16 R(x = 20 oo] -100<x<0 (60) In this figure, the scattering by the tapered card is compared to the known metallic halfplane diffraction coefficient. These results are also compared to those obtained by a range gating procedure which is used for removing the contribution from the trailing edge. The range gating procedure involved the computation of the scattered field from a 0.5m wide strip for 128 frequencies between 100MHz and 10GHz. By inverse Fourier transforming this data, one is then able to isolate and gate the first order contribution of the strip's unwanted edge. The shown pattern is simply obtained by 15

applying a Fourier transform to the gate profile for each observation angle. This technique is very effective for E-polarization but for H-polarization, we are still hampered at grazing incidence by the difficulties discussed by Hermann [2]. A more pertinent application of the code's range gating feature to this project is the analysis of S-shaped surfaces. Previous work [3,4] presented a uniform physical optics (PO) diffraction coefficient for this geometry and employed the pulse basis-point matching implementation of the current integral equation to validate the PO analysis. Because the scattering return from the S-shaped geometries is rather small in certain regions, it is of interest to re-examine this geometry using the more accurate program presented in this report. The particular S-shaped surface to be considered is shown in figure 14 and its frequency response was computed as described above and processed via inverse Fourier transformation at each angle. Figure 15 illustrates the resulting range profile at grazing incidence. The trailing edge is pronounced and we observe a non-local behavior about the inflection point. The inflection point behavior effects low frequency returns and is not as pronounced for E-polarization. Figure 16 illustrates the range profile after removing the contribution from the termination edges of the finite model which simulates the inflection surface. The scattering pattern resulting from a Fourier transformation of this gated profile(repeated for each angle) is shown il figure 17. Clearly, the scattering near grazing incidence is negligible. Nevertheless, it is important to note that the current distribution near the inflection region is not necessarily negligible. In fact, as shown in figure 18, the current distribution at grazing incidence is much larger than the PO current near the inflection point and it drops off rather rapidly to small values past the lower and upper knees. The presence of the strong current near the inflection point and at the knees implies that a small perturbation of the surface in that region could cause a substantial return by unbalancing the canceling contributions from the surface currents on the otherwise smooth surface. This conjecture was examined by placing a small depression centered at the location of the lower knee corresponding to the peak of the range profile shown in figure 16, i.e. at x = -6.8cm. The actual depression was a groove 0.01 cm deep and 4 cm wide and because of its extremely small depth such a groove can be thought of as a small scratch(almost invisible 16

to the naked eye). At the computation frequency of 5 GHz, this groove is 1 A deep! Surprisingly, the numerical results for this blemished surface are substantially different in the non-specular region for H-polarization(the E-polarization was not affected by the blemish). In particular, as shown in figure 19 the H-polarization pattern from the blemished surface has a distinct lobe near grazing incidence. A similar pattern is given in figure 20 when the same blemish is placed on the upper knee at x = 6.8cm. A lobe again appears near 165 degrees which is not as broad as that caused by the blemish on the lower knee. Tests were performed to verify the validity of the patterns in figures 19 and 20 and interestingly the same patterns were generated by our older code which was based on a different formulation. This provided some confidence that the results are not caused by numerical inaccuracies. Results were examined when the blemish was placed at other locations and it was observed that these grazing lobes persisted but were of reduced strength. At this point, an interesting task would be to examine the effect of the groove shape and position on the scattering pattern. Certainly the position will play a major role on the scattering strength. Nevertheless, the smoothness and shape of the blemish will likely play a role as well. 17

Appendix: The current in terms of charges Consider the integral expression (see [1]) Js = -2 [A(s + L ) [- ((s - 6)] de 2Jo \ L / (61) where O(s) represents a quantity proportional to the charge on a strip of length L/2 and Js is the corresponding current density. In evaluating the integral, +(s) must be assumed to satisfy the conditions +(s) = q(-s) = q(L - s) (62) and consequently it can be considered as a periodic function of period L (see figure). Because O(s) = /(-s), only the portion of +(s) in the region -L -L 2 L L+ L 2 2 S 0 < s < L/2 is unique. It is desirable to write (61) so that the integrand is a function of the unshifted O(s). To this end, we write (61) as s) = - ( - ) (+ 2L/ ( + ) s () - -- (s+6)d6 + -- ( + s +6)d8 2 Jo L 2 J-L/2 L 18

1 L/2+s 2( - s )d 2s -2 --- s) + L/2+S 2(s ] ( )ds (63) Let us now assume that q(s) can be expanded as a summation of pulses in the form N q(s) -= O P(s - s.) (64) n=l ( ) 1 Sn- A/2 < s< n + A/2 -P(s-s) = j0 elsewhere (65) where N denotes the linear elements/segments comprising the L/2 long strip. Substituting (64) into (63) and making use of the relation 6(s) - q(-s) = q(L - s) yields 1 L/2 2(' - s)] N Js (sm ) = - Es - )- [ ] m) >n P(s' -s n)ds' 2 J0 L n=l - s - S) qN+[1-nP(S - L/2 Sn)-ds 1 rO I 2(s' O - S m) - - 1- 2 - )]E N+-nP(S + L/2 - sn)d('66) 2 L/2+. L n=1 In this f 1 S > Sm E(s' -Sm) - = U(S'- Sm) (67) [ -1 S, <Sm ) and the upper summation limit m~ implies that the integration of the pulse centered at Sn T L/2 with 1sn - Sml < A/2, will only be over a portion of it which corresponds to the crosshatched region in the figure. From (66) and (65) we now have 1 N 2mJS ) -- E nI (1)(Sm)- E N+l-n (Sn) n=l n=l 1 N - E N++l-nln3( ) (68) 2n=m+ 19

The integral I~)~)is given by I'(1)(Sn~~t = U(S' - Sm~) ds' — 8M s Jsn-2 2 n-A/2 ds' where U(s) denotes the unit step function, and can be evaluated to give Ins~ =y(Sn - Sm) - -j(Sn - Sm)A (69) where 'Y(Sn-srnV{= 2(s,, Similarly, the integrals I,2)(Sm ) an' 1,(2) (,Sm)= Jf2 (Smds' - '(s m ) = In /2 -1 s 3 (sn,sm,)~ - in which -n Sm > A/2 - Sm) ISn~ S < A/2 Sfl - m < A/2 I I,3)(sm) are given by ~I~hI7 (L s'- sm) ds' 2 snA2 L L e(sn,sm) 2 5 S A/2 Sn Sm > A/2 -~ Sm IK A/2 A/2 Sn Sm < A/2 ISn -Sm I < A/2. (70) (71) (72) (73) (74) 2 (Sn, Sm ) and f3(S,, Sm) = Evaluating the integrals we I7(2)(s5, = Sn Sm {n::!obtain j(Sn -Sm),A Sn -m > A/2 n~mA2)2Sn -SmI < A/2 L (75) 20

and 2 -(S - (Sn-m)A,- S- < A/2 I(3)(m)= (76) (Sn - Sm+ A/2)2 - Isn- Sm- < A/2. L Using (75) and (76), the last two sums in (68) can be combined to yield i N i N Js(sm) = -1 In nl)(sm) - m z ON+l-In(4)(Sm) (77) n=l n=l where I(4)(sm) = In2)(Sm) + I3)(Sm) = (- - -S )A (78) More explicitly, J,(s) can be written as Js(-) = - n Y(Sn -- ) + (n s- ) + (SN+1 -n -) S) (79) L n=l 2A Note that s can have any value and in fact it will be later chosen to be some distance away from s.n 21

References [1] John L. Volakis and Jian-Ming Jin, "On the Formulation and Solution of the E-field Integral Equation," University of Michigan Technical Report, 390573-2-T, 1991. [2] Gabriel F. Herrmann, "Numerical Computation of Diffraction Coefficients," IEEE Trans. Antennas Propagat.,vol. AP-35, No. 1, pp. 53-61, 1987. [3] Leo C. Kempel and John L. Volakis, "Numerical Simulation of the Scattering by S-shape Surfaces," University of Michigan Technical Report, 390285-1-T, 1990. [4] J.L. Volakis, L.C. Kempel, and T.B.A. Senior, "A Uniform Physical Optics Approximation for Scattering by S-shaped Surfaces," University of Michigan Technical Report, 390285-2-T, 1990. 22

A n A S (x, y) Js = Ks+ - Ks 0 Ks Figure 1. Geometry of the 2D curved surface (strip). y) A S An n t* p'(x', y') 0 Figure 2. Illustration of the observation and integration point parameters.

A (S) S j- A s Sj+ A S (a) P(S-Sj) Sj-A/2 Sj+A/2 s Sj I (b) Figure 3. Expansion functions. (a) Linear expansion functions for the current (b) corresponding expansion functions for the charge.

Figure 4. Illustration of an arbitrary cross section cylinder and a curved strip. Figure 5. Discretized versions of the cylinder and the strip shown in Figure 4.

AL 10, I' J# I % I % le I % -I % owl# - I 01o -- Al,I I -, 10, I AIF Sm Alf I (xM, Yin) mn-*1 'Ym i) Figure 6. Geometrical parameters for adjacent segments. expansion pulse 0, a-.0 % 10 ".0 0. I.,.0.0 I It s It It m2 F,. (X It s 4 +1 " Ym+I I % ml 111- ) Overlapping itesting pulse (xm, Yin) (XmiP Ym-1) (x-1, I n (xn, (x~+,y~ Figure 7. Illustration of discretization for pulse expansion and testing with shifted pulses.

(Xn-1/2 Yn-1/2) A* Snl (xn, Sn2 Xn+l Yn+1 (a) (Xm- 1 / \ (Xm-1/4' Ym-1/4) '' " A 5m2 (Xm+1' Ym+l) (Xm+l/4, Ym+1/4) (b) Figure 8. Illustration of parameters for pulse expansion and point matching in connection with the charge integral equation.

0.0 0 -PQ ta:i L) V mS En.1 - -5.0 -10.0 -15.0 -20.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Observation Angle (4) [deg] Figure 9. Bistatic echowidth of a 4cm square metallic cylinder computed with 4i = 0 and f = 2 GHz.

10.0 - 4 -To c) * - (^3 OQ *f 5.0 0.0 -5.0 -10.0 0.0 30.0 60.0 90.0 120.0 Observation Angle (6) [deg] 150.0 180.0 Figure 10. Bistatic echowidth at a 5cm metallic circular cylinder computed with Oi = 0 and f = 2 GHz.

-10.0 C~ 1=1 0 -20.0 -30.0 -40.0 -50.0 -60.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Observation Angle (4) [deg] Figure 11. Backscatter echowidth of a 5cm resistive strip with normalized resistivity R = 1.1 + iO.2 computed at f = 2 GHz.

0.0 e.g *-= eo cO 4~ 0t c) urr -10.0 -20.0 -30.0 -40.0 -50.0 -60.0.e, -- - 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Observation Angle (~) [deg] Figure 12. Backscatter echowidth of coplanar joined metallic and resistive strips each 2.5cm wide computed at f = 2 GHz.

lc, C) U0 ct C)l C) 20.0 10.0 0.0 -10.0 -20.0 -30.0 -40.0 -50.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Observation Angle [deg] Figure 13. Backscatter echowidth of a simulated metallic halfplane whose trailing edge is suppressed by either a long tapered resistive card or by range gating.

0.4 W..I......................,.[...........I....... l. 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.. -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 x [meter] Figure 14. S-shaped surface which is described by f(x) = 0.1 erf(6x).

0.5 0.4 0.3 0.2 0o.1 0.0 -0.1 -oI.2 -0.3 -0.4 -100.0 -80.0 -60.0 -40.0 -20.0 0.0 20.0 40.0 60.0 80.0 100.0 Range [cm] Figure 15. Range profile of the surface (see fig. 14) at grazing incidence.

0.5 0.4,- 0.3 L- | i, ao — \,/ ---0.3 -0.2 0.1 oo,......... -0.1 - -0.2 -0.3 -0.4 -0.5 -. -...... 6.. 10 -100.0 -80.0 -60.0 -40.0 -20.0 0.0 20.0 40.0 60.0 80.0 100.0 Range [cm] Figure 16. Gated range profile of the surface (see fig. 14) at grazing incidence.

o Analytical - 10.0 0.0 0.0 -10.0.. -20.0 o -30.0 PQ -40.0 -50.0........ 90.0 100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 Observation Angle (4) [deg] Figure 17. H-pol backscatter echowidth of the surface (see fig. 14).

3.0 2.5 2.0 1.5 a, es 1.0 - 0.5 - 0.0 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 Position (x) [meter] Figure 18. Current magnitude for S-shaped(see fig. 14) at grazing incidence and Hpolarization( n',, v c / e-,- f4 An x -I tA^\ L ^ J

30.0 20.0 10. 1 \- |Defect: H-pol 10.0 - I - ------— Defect: E-pol 3 0.0 - \ o Control: H-pol Control: E-pol 0 -10.0 o -30.0 pq -40.0 - - 90.0 100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 Observation Angle (() [deg] Figure 19. Backscatter echowidth of the surface (see fig. 14) with a defect placed at x = -6.8 cm for both E- and H-polarization.

Defect: -6.8 cm 10.0 -------- Defect: +6.8 cm 0 Control 0.0 o -10.0 -20.0,o -30.0- - & -40.0-,,-50.0 90.0 100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 Observation Angle (4) [deg] Figure 20. Backscatter echowidth of the surface (see fig. 14) with a defect placed at x = ~ 6.8 cm for H-polarization.