025921-25-T TECHNICAL REPORT for NASA Grant NAG-2-541 NASA Technical Monitor: Alex Woo Grant Title: Development of 3D Electromagnetic Modeling Tools for Airborne Vehicles Report Title: Alternative Field Representations and Integral Equations for Modeling Inhomogeneous Dielectrics Institution: Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor MI 48109-2122 Period Covered: September 1991 - February 1992 Report Authors: J.L. Volakis Principal Investigator: John L. Volakis Telephone: (313) 764-0500

Alternative Field Representations and Integral Equations for Modeling Inhomogeneous Dielectrics John L. \olakis Radiation Laboratory Department of Electrical Engineering and Computer Science University of Michigan Ann Arbor, MI 48109-2122 Abstract New volume and volume-surface integral equations are presented for modeling inhomogeneous dielectric regions. The presented integral equations result in more efficient numerical implementations and should therefore be useful in a variety of electromagnetic applications.

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1 Introduction The modeling of inhomogeneous dielectrics via an integral equation approach is traditionally accomplished via the introduction of equivalent volume electric and magnetic currents [1] - [8]. For a dielectric with non-trivial permittivity and permeability this type of modeling implies six scalar unknowns at each volume location. As a result, the implementation of the resulting integral equation is computationally intensive and has excessive storage requirements. In this paper it is demonstrated that any inhomogeneous dielectric material, regardless of its permittivity and permeability profile, can be modeled by a single electric or magnetic current density. Alternatively, either the electric or magnetic fields within the dielectric can be used as the unknown quantities. It appears though that one must pay a price for resorting to these reduced-unknown and/or kernal-singularity representations. Specifically, because they involve derivatives of the unknown quantities, a higher (at least linear) basis function is required for discretizing the resulting integral equations. However, it is possible to relax this requirement by resorting to a new volume-surface field representation. In this case, either the undifferentia.ted electric or magnetic field within the dielectric is the unknown quantity along with the corresponding tangential electric or magnetic fields on the outer boundary. Provided the dielectric volume is not composed of a single thin layer, this volume-surface integral equation still represents a nearly fifty percent reduction in the number of unknowns relative to traditional implementations. 2 Volume Representations Let us consider the dielectric/ferrite volume Vd, shown in Fig. 1, having relative constitutive parameters 6e and I'r which are arbitrary functions of position. Assuming some exterior excitation, (Ei, HZ), the total field can be written as E = Et + Es H = Hz + Hs (1) where (Es, H8) are the scattered fields caused by the presence of the dielectric. Traditionally [1] the scattered fields are formulated in terms of the

equivalent currents Jeq -j koY (er - 1)E, Meq = j koZo(ir - 1)H (2) with ko and Zo = 1/Yo being the free space wavenumber and intrinsic impedance, respectively. In terms of these effective or equivalent current densities, the scattered field is given by Es=JJ- [Vxro(r, r'). Meq(r')+ j0Zoor(r,r') Jq(r)]dv' (3) in which r and r' denote the observation and integration points, respectivcly, o(r, r') = - [I + 2] G(r, r'), (4) is the free space dyadic Green's function, V x ro(r, r') = -VGo(r, r') x I, (5) e-jko r-r'| Go(r, r') 4rxlr - r'l (6) I = xx + +zz is the unit dyad and Hs is given by the dual of (3). By substituting (3) and its dual into (1) and then into (2), we obtain the coupled set of integral equations Jeq(r) ) - E = Ei r C Vd (7a) Meq(r) (H-) Hs r E Vd (7b) for a solution of the equivalent currents Jeq and Meq. The aforementioned formulation appears to be the only approach that has so far been utilized for three dimensional implementations. However, as noted in the literature [4, 5, 9], the singularity of the kernal in (3)

presents numerical difficulties. Also, for non-trivial permeability, six scalar unknowns are involved in the solution of (7). The first of these difficulties can be alleviated by resorting to higher order basis functions and expressing, for example, Es as Es = I {Meq x VGO(r, r') - jkoZoJeqGo(rr) jZo'. Jeq(r')Go(r, r)}dv (8) which is the volume equivalent to the Stratton-Chu surface integral equation. Likewise, the scattered magnetic field to be substituted in (7b) can be expressed by the dual of (8). Although this appears to be the most popular approach in modeling three-dimensional dielectrics, it can be shown that there are several other ways to formulate the problem. Most importantly, it can also be shown that (7) can be replaced with an equivalent system which involves only three (not six) scalar unknowns. Specifically, from Maxwell's equations [10] the radiation of Meq is indistinguishable from the radiation of the electric current J= VxMq (9) eq jkoZo This can be combined with (2) giving a single equivalent electric current J/q = jko o(6r - 1)E + V x [(V / - 1)H] _(rl)7 x H + V7 x [(r - 1)H] (10) Er for representing the scattered fields (ES, Hs). From the dual of (3) we then obtain that the scattered magnetic field due to the current density (10) is HS - ]]' [VGo(rr') x I] je6(r')V lx H(r') + V' X [(/r(r')- 1)H(r')] }dv' (11)

in which V' implies differentiation with respect to the primed/integration coordinates. When this is used in (1) we deduce the integral equation Ei(r) - 1 H[(r) = H(r)- j1 [VGo(r, r') x (r') V xH(r') er(r' ) + V x [(,(r') )- 1)H(r')] }dv' r E Vd (12) where the unknown quantity is now the magnetic field within Vd. Using a similar procedure it can be also shown that the scattered field may instead be represented by the radiation of a single magnetic current density M/q (t 1)V X E- V x [(r - 1)E] (13) From the first of (1) and (3), we then deduce the integral equation E(r) r) = E(r) - / / [VGo(r, r) x I] { (r)-) lVx E(r') + V' x [(e~r(r') - 1)E(r')] }dv' (14) which as expected is the dual of (12). We observe that the kernel singularity associated with (12) and (14) is the same as that associated with (8). In addition, as in the case of the integral equation (7) in conjunction with (8), linear expansion functions such as those in [3] or [4] are required for the discretizing (12) and (14). Thus, even though the new integral equations (12) and (14) h-ave half the unknowns, this was not achieved at the expense of increasing the kernel's singularity or the order of the expansion basis required in their implementation. It is remarked that special forms of these integral equations have already been successfully implemented for two dimensional applications [11, 12]. 3 Volume-Surface Representation The requirement to employ linear basis in connection with the implementation of (12) and (14) can be relaxed by resorting to a volume-surface

integral equation (VSIE) such as that derived in [13] and [14] for two dimensional simulations. To do so we begin with (3) which in conjunction with (2) can be rewritten as ES = E + Es = -k [Er(r') - 1] E(r') (r r) -j kZ0V x J J [Ir(r') - 1] H(r')G0(r, r')dv' (15) where Es is associated with the second integral and represents the field due to the magnetic equivalent current defined in (2). Setting H =V x E/jkoZour in this integral, and invoking the identities V x [V'x BE] Vx [V'O x E] + V x [qV'x E] we obtain Es =F + + F 3 (16) with =Vx J V X (r'x ) Go(r, r')E(r')} d' (17) = vx IXIV{{(l - r(r')V Go(r,r')xE(r')} dv' (18) = V x JJJ, {G~(r')sV (tin)) x E(')}dv' (19) F - VX G,) (r,,,)x(r' xE(r' (19) These integral expressions can be simplified through the use of various integral and differential identities. The volume integral in (17) can be transformed to a surface integral by invoking Stoke's identity IV' X A)d' = Sd(' X A)cls' (20)

where Sd is the surface enclosing Vd and i' = i(r') denotes the outward unit normal to the surface Sd. We have =s' Vx d1 ( r' ) GO(r, r') [fn' x E(r')] ds' = fJSd 1(-'1 n) [h' x E(r')] x VG,(r,r')ds' (21) which is an integral involving the undifferentiated tangential electric field over the surface enclosing Vd. Turning now to the integral in (18) we first rewrite it a~s = JfI [- /(r')] V x [V'Go(r,r') x E(r')] dv' (22) and we note that [15, p. 487] V x [V'Go x E(r')] = E(r')V2Go - E(r'). VVGo (23) Then, upon invoking the differential equation V2Go(r, r') + ko2G(r, r') = -6(r - r') (24) where 6(r') denotes the Dirac delta function, it follows that = -k i [ r(r')] E(r) F(r, r')dv' + [I ] E(r') (95) Again, this involves only the undifferentiated electric field within the dielectric's volume. Finally, the last integral in (16) can be readily simplified and written as = I J V X {Go(r,r')V' (r) x E(r') dv' =JJjv VGo(r r(') x E(r)} dv (26) 7 ~ rI.

When (21), (25) and (26) are substituted into (16) and then into (15), we find that the total scattered field can be expressed as E I = j [er(r') - E(r')- 1L(r'](r,r')dv + J J VGo(r, r') x {V [(r] x E(r') dv' -flSd [1- (r)] [n x E(r')] x VG,(r, r')ds' 1- (r')] (r) (27) For two dimensional simulations where the material parameters and the fields are invariant with respect to z, this expression can be readily shown to reduce to the VSIE given by Jin, etc. [13, equs. 28 and 31]. Expression (27) is also similar to the VSIE given by Tai [16]. However, Tai's expression was left in terms of differentiated field quantities and is only applicable for homogeneous dielectrics. To obtain an integral equation on the basis of (27) we substitute this into the first of (1) and upon taking the principal value of the appropriate integrals we have -kJ JJ eor' ) - (r I(r, r')dv' -k0fff;:.-Vo L r(r')] E~r F o lr - sfd-So [1- (r') [n' x E(r')] x VGo(r, r')dv' E(r) r not in Vd +Ei= [1 + r] E(r) r on Sd (28) I [Er(r) + 2/r(r)] E(r) r in ld ~~~~

In this, Vo is a vanishingly small spherical volume whereas SO is a vanishingly small hemispherical surface both having their centers at r. As given, (28) can be discretized via the moment method or some other technique for a solution of E(r) within the dielectric. Its kernal has, of course, the same singularity as (7a) but involves only a single unknown vector field in comparison with the two vector unknowns appearing in (7). If linear rather than pulse basis are employed for the solution of (28), then it may be desirable to rewrite the first integral of (28) in the form given by (8) with Meq = 0 and eq Zo = _-rir-r) E(r) (29) J jko [er(r)/lr((r)- ]Er) However, in this case one could also resort to the alternative integral equa.tions (12) or (14). Of course, the dual of (28) is another integral equation. Further, linear combinations of (28) and its dual or (12) and (14) can be utilized if so desired. In closing, we remark that if p,r and/or er are discontinuous within Vd, the surface integral in (27) and its dual must then be replaced by Z ffSd, [u+(r')- u_(r')] [nfi(r') x F(r')] x VGO(r, r')ds' where F = E or H. Here, Sdi denotes the ith discontinuous surface within Vd, nfi(r) is the unit normal to Si pointing from the - side to the + side (outermost side) and u_ denotes the inverse relative dielectric constant at the + or - side of the surface Sdi. In particular ui = 1//,' for the E-field integral equation (27) and u. - 1/e' for the H-field integral equation. 4 Conclusion Some alternative formulations were proposed for modeling three-dimensional inhomogeneous dielectrics. These are summarized in figure 2 and the aim of the investigation was to generate integral equations for the fields within the dielectric scatterer utilizing the minimum number of unknowns and the least singular kernels. A purely volume integral equation was derived involving half the unknowns required with traditional equations for ferrite

materials. The implementation of this reduced-unknown volume equation implies use of (at least) linear basis functions and to relax this requirement a volume-surface integral equation was derived. All of the integral equations presented here appear to be more efficient than the traditional ones without compromising the kernel's singularity. They should thus be found useful in a variety of radiation, scattering or SAR applications. References [1] R.F. Harrington, Time Harmonic Electromagnetic Fields, McGraw Hill: New York, 1961, (p. 126). [2] D.E. Livesay and K.M. Chen, "Electromagnetic fields induced inside arbitrarily shaped biological bodies," IEEE Trans. Microwave Theory Tech., vol. MTT-22, pp. 1273-1280, Dec. 1974. [3] D.H. Schaubert, D.R. Wilton, and A.W. Glisson, "A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies," IEEE Trans. Antennas Propagat., vol. AP-32, pp. 77-85, Jan. 1984. [4] C.T. Tsai, H. Massoudi, C.H. Durney, and M.F. Iskander, "A procedure for calculating fields inside arbitrarily shaped, inhomogeneous dielectric bodies using linear basis functions with the moment method," IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp. 1131-1139, Nov. 1986. [5] MI.J. Hagmann, H. Massoudi, C.H. Durney, and M.F. Iskander, "Comments on'Limitations of the cubical blocks model of man in calculating SAR distribution'," IEEE Trans. Microwave Theory Tech., vol. MITT33, pp. 347-350, Apr. 1985. [6] R.G. Rojas, "Scattering by an inhomogeneous dielectric/ferrite cylinder of arbitrary cross-section shape - Oblique incidence case," IEEE Trans. Antennas Propagat., vol. 36, pp. 238-246, Feb. 1988.

[7] M.F. Catedra, E. Gago, and L. Nufio, "A numerical scheme to obtain the RCS of three-dimensional bodies of resonant size using the conjugate gradient method and the fast fourier transform," IEEE Trans. Antennas Propagat., vol. 37, pp. 528-537, May 1989. [8] B.J. Rubin and S. Daijavad, "Radiation and scattering from structures involving finite-size dielectric regions," IEEE Trans. Antennas Propagat., vol. 38, no. 11, pp. 1863-1873, Nov. 1990. [9] R.F. Harrington, Field Computation by Moment Methods, R.E. Kreiger Publishing Co.: Matabar, FL, 1968. [10] P.E. Mayes, "The equivalence of electric and magnetic sources," IEEE Trans. Antennas Propagat., vol. AP-6, pp. 295-296, 1958. [11] A.F. Peterson and P.W. Klock, "An improved MIFIE formulation for TE-wave scattering from lossy, inhomogeneous dielectric cylinders," IEEE Trans. Antennas Propagat., vol. AP-36, pp. 45-49, Jan. 1988. [12] E. Mllichielssen, A.F. Peterson, and R. Mittra, "Oblique scattering from inhomogeneous cylinders using a coupled integral equation formulation with triangular cells," IEEE Trans. Antennas Propagat., vol. AP-39, pp. 485-490, April 1991. [13] J.M. Jin, V.V. Liepa, and C.T. Tai, "A volume-surface integral equation for electromagnetic scattering by inhomogeneous cylinders," J. Electromagnetic Waves and Appl., vol. 2, pp. 573-588, 1988. [14] M.A. Ricoy, S.M' Iilberg and J.L. Volakis, "Simple integral equations for two-dimensional scattering with further reduction in unknowns," IEE Proceedings, part H, vol. 136, pp. 298-304, August 1989. [15] J. rVan Bladel, Electromagnetic Fields, Hemisphere Publishing Corp.: New York, 1985. [16] C.T. Tai, "A note on the integral equations for the scattering of a plane wave by an electromagnetically permeable body," Electromagnetics, vol. 5, pp. 79-88, 1985.

(E', H) n Sd Fig. 1. Illustration of the inhomogeneous dielectric volume Vd enclosed by the surface Sd.

n (n YAt~rllF g)E AxVZ, ErlrH fi Fig. 2. Different volume equivalent currents for modeling the scattering by the inhomogeneous dielectric volume Vd in figure 1.

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