Neighboring-Patch Integrals in Transient Electromagnetic Scattering

The integrals over patches that are close to the self-patch are cslcolated by expanding the factors in the integrand in power series. The values are computed analytically up to first order in the linear sue of the patch. This procedure applies to patches for which the distance between the centers is of the same order of magnitnde as the size of the patch. The same formulas are useful in steady-state scattering problems. I


I. INTRODUCTION
N THIS PAPER we continue to consider the problems associated with the numerical solution of the integral equations that arise in the theory of the electromagnetic scattering of transient fields.These integral equations involve vector and scalar functions defined on the surface of the scatterer, here assumed to be sufficiently smooth so that the necessary derivatives are well defined.
The integrands in these integral equations are singular, and the contributions of the self-patch, where the singularities occur, have to be treated separately from the rest of the contributions.We previously determined [l] these contributions to first order in the linear dimension of the patches for orthogonal curvilinear coordinates on the surface.
The contributions to the integrals from neighboring patches also deserve special attention 121.The functions of the distance R between the field point and the source point can hardly be considered constant as the source point ranges over the patch; R varies by a factor greater than three over a nearest neighbor of the self-patch.The integrals we have to consider are no longer singular, as is the case for the self-patch, and here we have zerothorder contributions that vanish for the self-patched integrals due to symmetry.The linear size of the patch is of the same order of magnitude as the distance between the centers of the patches.We should calculate these neighboring-patch integrals accurately if we determine that the contributions from these integrals to the overall surface integral are significant.
In Section I1 we present the expansions of the integrals that occur in these equations to terms that are of fust order in the linear size of the patch.In Section I11 we present an example of these terms for the simple case of a sphere.Most of the calculations and formulas that permit the actual evaluation of the integrals are shown in the Appendix.The notation is similar to the one we used in [ 11 .
The integrals and the results are essentially the same as those needed for monochromatic waves.

II. NEIGHBORING-PATCH INTEGRALS
The magnetic field integral equation (MFIE) and electric field integral equation (EFIE) for perfect conductors show the typical terms found in the integral equations of electromagnetic scattering.The MFIE and EFIE are where js is the surface current density, ps the surface charge density, 7 the retarded time ( 3 ) eo the permittivity of free space, p o its permeability, c the s eed of light, S the surface of the conductor, iz the unit n,o.rmal,B 5, the magnetic induction of the incident pulse, and E'" its electric field.The field point i! is in a patch S, and the source point 2' is in a neighboring patch S 2 , as shown in Fig. 1.The center of the patch is defined by the midvalue of the curvilinear coordinates.
We fxst consider the integral We expand the functions in the integrand about the center ap,@,,, 7,) I"+ du, lU+ dv' p b ~$ + abii;tu at' where the limits of integration are variables defined in (28), and f , , K 2 , and K3 are polynomials in u' and 5 defined in ( 31), (33), and (38).The integrals can be carried out by changing the variables of integration to E and v' and then using the appropriate C(Z, m , n) from the Appendix.The first term, as expected, is of zeroth order, and the others that we have kept are of fust order.We write down the first few terms in the expansion to show some of the contributions; we have similar to (9), and = c(o, 0, 1)aJS(;;, T o y a t ' .(12) Other integral equations related to electromagnetic scattering have the same type of terms.
For monochromatic fields, the expansions are somewhat different because the fields are independent of t h e and differentiation with respect to time is replaced by multiplication by -io.The surface integrals remain unchanged and the contributions from neighboring patches involve the same C(1, m, n).

NEIGHBORING-PATCH INTEGRALS ON A SPHERE
The next term is Similar to but of higher order than (4), To obtain an estimate of the size of the error made by assumand it gives a fist-order contribution only.We have ing that the integrand is constant over the patch, we compute some of the expressions that occur in these integrals.We con- We compare this result with the zeroth-order terms in (7), --q I , O , 3)2;/4 -C(0, 1,3)2;/& = -0.036;-0.424; a vector that is close to the one in ( 16).A complete comparison would involve also+the fn2t-order terms, which include terms proportional to 0: J , and aJ,/at'.
with -C(l, 0, 2)ji0/ab -C(0, 1,2);;/& = -0.0082-0.099;The importance of the discrepancies in these individual terms depends on the total number of patches that contribute to the integral and the relative size of the field and its derivatives in the different patches.In a stepping-in-time procedure, the number of.patches that contribute to the integrals increases gradually starting from the time the wave first hits the scatterer, and we can have sigdkant errors at early times.

N. CONCLUSION
The method developed here allows us to compute more accurately the contributions to singular surface integrals from patches close to the self-patch in an arbitrary system of orthogonal curvilinear coordinates on the surface.
The error made by approximating the integrand by its value at the center of the patch is not neghgible, and the need to take these corrections into account depends on the overall accuracy that is required, the method of integration, and the nature of the fields.

APPENDIX
In this Appendix, we give the details of the expansions of the surface fields and the computations of the neighboring-patch integrals.
We consider two patches, SI and S2, close together on a sur- We also expand 3, about 2; and obtain where The quantities bo, Au,, U, and T are all small, and we assume that they are of the same order of magnitude for neighboring patches.We combine ( 24) and ( 25) to obtain We use the expansion (25) for Jo because otherwise t h i s constant vector remains in the integrals that have to be evaluated.These integrals become more complicated and they involve small constants that may cause problems in numerical computations.
An expansion based on (24) instead of ( 27) may be useful in an intermediate region where the expansion ( 25) is inaccurate but i? stu varies significantly as x" ranges over ~2 .
Fig. 1.Neighboring patches SI and S, showing the ditferent variables used.To simplify the notation, we have suppressed the scale factors ct and b.
ABo = AO' = n/20, The terms that come from the EFIE are If we assume that the integrand of in (4) is constant, we ANTENNAS AND PROPAGATION, VOL.AP-33, NO. 7 , JULY 1985 where M2 = s 5 ObAO'A@' is the area of the patch S 2 .The factor multiplying J, is A S 2 z o I R i = -0.099;-0.4063 + 0.296k (16) wxch are yery clo,se.The forms of ?3 and f4 are essentially those of I I and I2 .For Is we get excellent agreement between AS2 fRo z 0