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A Hybrid Finite Element/Moment Method for Solving Electromagnetic Radiation Problem of Arbitrarily-Shaped Apertures in a Thick Conducting Screen ABUNGU N ODERO1, DOMINIC B O KONDITI1 ALFRED V OTIENO2 1Jomo Kenyatta University of Agriculture and Technology 2University of Nairobi Abstract – In this paper a hybrid numerical technique is presented for characterization of the transmission properties of a three-dimensional slot in a thick conducting plane. The slot is of arbitrary shape and is excited by an electromagnetic source far away from its plane. The analysis of the problem is based on the "generalized network formulation" for aperture problems. The problem is solved using the method of moments(MOM) and the finite element method(FEM) in a hybrid format. The finite element method is applicable to inhomogeneously filled slots of arbitrary shape while the method of moments is used for solving the electromagnetic fields in unbounded regions of the slot. The cavity region has been subdivided into tetrahedral elements resulting in triangular elements on the surfaces of the apertures. Validation results for rectangular slots are presented. Close agreement between our data and published results is observed. Thereafter, new data has been generated for cross-shaped, H-shaped and circular apertures. Index Terms – Finite–Element Method, Method of Moments, Perfect Electric Conductor(PEC) I. INTRODUCTION T HE problem of electromagnetic transmission through apertures has been extensively investigated[1-6]. Many specific applications, such as apertures in a conducting screen, waveguide-fed apertures, cavity-fed apertures, waveguide-towaveguide coupling, waveguide-to-cavity coupling, and cavity-to-cavity coupling have been investigated in the literature. Both intentional and inadvertent apertures of various shapes are encountered in the many applications afforded by the advances in technology and increased scale of utilization of microwave and millimeter-wave bands for communications, radar, industrial and domestic applications. Examples of undesirable coupling are leakage from microwave ovens and printed circuit boards, electromagnetic penetration into vehicles, and electromagnetic pulse interaction with shielded electronic equipment. These lead to problems of Electromagnetic Compatibility (EMC) and Electromagnetic Interference (EMI) which should be minimized or eliminated. Desirable coupling exist in slotted antenna arrays, microstrip-patch antennas, directional couplers, cavity resonators, etc. For the purposes of either minimizing or enhancing electromagnetic coupling through apertures, it is desirable to quantify the electromagnetic penetration through apertures. The past seventeen years have witnessed an increasing reliance on computational methods for the characterization of electromagnetic problems. Although traditional integral-equation methods continue to be used for many applications, one can safely state that in recent times the greatest progress in computational electromagnetics has been in the development and application of hybrid techniques, such as FEM/MOM and (finite-difference timedomain)FDTM/MOM . The FDTM requires a fine subdivision of the computational domain for good resolution and so is computationally intensive just like the MOM. The method of moments is an integral equation method which handles unbounded problems very effectively but becomes computationally intensive when complex inhomogeneities are present. In contrast, inhomogeneities are easily handled by the finite element method, which requires less computer time and storage because of its sparse and banded matrix. The matrix filling time is also negligible when simple basis functions are used [7]. However, it is most suitable for boundary value problems. Since the investigation to be carried out encompasses unbounded problems with complex inhomogeneities, a procedure combining the finite element method and the method of moments would be effective. It is the objective of this paper to develop and demonstrate the validity and accuracy of a new hybrid FEM/MOM method. Apertures having various shapes, such as rectangular, circular, cross and H have been studied. II. GENERAL FORMULATION OF THE PROBLEM Consider the 3–dimensional ( aperture–cavity– aperture ) structure illustrated in Fig.1. Using an approach similar to that presented in [8], one can replace the tangential electric fields in the aperture planes with equivalent magnetic currents M 1 in the z 0 aperture plane and M 2 in the z d aperture plane. In the interior region, the equivalent magnetic currents M 1 and M 2 in the z 0 and the z d aperture planes, respectively. Perfect conductors can be placed in the z 0 S1 and the z d S 2 planes, as illustrated in Fig. 2, without altering the fields. The problem geometry is then decomposed into 3 regions as illustrated in Fig. 3. y cavity wall Region A ( z > 0 ) upper aperture o S1 M1(x) x x -M1(x) Region C lower aperture d M2(x) -M2(x) (a) Ji, Mi Region A Region B ( z < -d ) z y o x S2 Fig. 2: Equivalent currents introduced in the aperture planes. z A, A Ji, Mi Region C S1 M1(x) x conductor Region B conductor z =0 S1 (b) -M1(x) Fig. 1. Problem geometry: (a) Top view (b) Cross-sectional view The specific configuration consists of a cavity in a thick conducting plane having an aperture at the top surface and another one at the bottom surface. The free space region above the plane ( z > 0 ) is denoted as region A and that below the plane ( z < -d ) as region B. Also, the volume occupied by the cavity ( -d < z < 0 ) will be referred to as region C. It is assumed that the cavity is filled with an inhomogeneous material having a relative permittivity c r and relative permeability M2(x) C , C S2 z = -d x -M2(x) S2 c r . B , B -z Fig. 3: Problem geometry decomposed into three regions; A, B, and C. The total magnetic field in regions A and B can be expressed as a superposition of the short-circuited magnetic field H sc ( i.e., the incident magnetic field plus the specular magnetic field H scat , produced by the equivalent surface currents. The tangential components of the total magnetic field in the aperture planes are then expressed as (1) nˆ H tot nˆ H sc nˆ H scat In the interior region C, the closed cavity is perturbed by the equivalent currents M 1 and M 2 , as illustrated by Fig. 3. The total tangential magnetic field that is produced on S 1 is expressed as the superposition of the fields produced by each current (2) nˆ H1tot M1 H1tot M 2 with a similar expression on S 2 . To ensure the uniqueness of the solution, the tangential magnetic field must be continuous across the aperture planes. To this end, continuity of the total tangential magnetic fields across the aperture planes is enforced. Therefore, on S1 (3) nˆ H1sc nˆ H1tot M 1 H1tot M 2 H1scat M 1 At this point, the equivalent currents are still unknown. Their approximate solution is derived using the method of moments. To this end, the equivalent magnetic currents are expanded into a series of known basis functions weighed by unknown coefficients. M 1 x, y, z 0 N V 1n M 1n P V 2n M 2n Y V Y V H 1sc Y11C sc C H 2 Y21 (4) x, y (5) n 1 Y A 0 2n C 12 C 22 1n 0 YB V1n V2 n (8) The advantage of using this function is that the admittance matrices of the interior and the exterior regions can be constructed independently. The aperture admittance matrices Y A and Y B are computed by solving the problem of the equivalent magnetic currents radiating into a half-space. The aperture admittance matrix Y C is computed by solving the interior cavity problem for each equivalent magnetic current basis function to compute in the aperture planes, and then taking the inner products with the appropriate testing functions. The following section describes the solution of this interior problem using the FEM. III. x, y n 1 M 2 x, y, z d V2 n can be the unknown coefficients V1n and obtained. The first two inner products of equations (6) and (7) are referred to as the aperture admittance matrices for region C [10]. These matrices relate the tangential magnetic fields in the aperture to the equivalent currents, which are equivalent to the tangential electric fields. The last inner products appearing in equations (6) and (7) are referred to as the interior aperture admittance matrix for regions A and B respectively. In terms of the aperture admittance matrix, the coupled equations (6) and (7) are expressed as COMPUTING THE INTERIOR ADMITTANCE MATRIX BY THE FEM A. SOLUTION OF THE INTERIOR PROBLEM The approximate equivalent currents are substituted into equation (3). Two sets of vector testing functions are also introduced : W1m x, z 0 and W2m x, z d Within the cavity region, the magnetic fields must satisfy the vector Helmholtz equation in the S1 and S 2 planes, respectively. Taking the inner advantage of the linearity of the operators results in where H is the total magnetic field. The cavity region is defined as a closed space confined by PEC walls, product of equation (3) with W1m x, z 0 and taking W1m , H1sct N V 1n V W1m , H1tot t M 1n n1 N V 1n P 2n W1m , H1tot t M 2n n1 (6) W1m , H1scat M 1n t n1 where, a similar expression is derived on S 2 W2 m , H 2sct P V 2n n 1 P V 2n W2 m , H 2tott M 2 n N n 1 t W2 m , H 2sca M 2n t 1 r H k 2 r H 0 which will be referred to as the domain V . A functional described by the inner product of the vector wave equation and the vectors testing function, H , is expressed as H FH V1n W2 m , H 2tott M 1n V (7) n 1 where the subscript t denotes the tangential fields. By introducing N P vector testing functions, equations (6) and (7) provide N P linear equations from which (9) * . H k 2 r H * .H dV r 1 (10) is defined. The solution of the boundary-value problem is then found variationally, by solving for the magnetic field at a stationary point of the functional via the first variation F H 0 Wej being the Whitney basis with the spacial vector (11) function defined by The functional has a stationary point ( which occurs at an extremum ) and equations (9) and (10) can be used to derive a unique solution for the magnetic field. For an arbitrary V the solution of the above variational expression cannot be found analytically and is derived via the FEM. To this end, the domain V is discretized into a finite number of subdomains, or element domains, Ve ( Fig. 4). It is assumed that each element domain Ve has a finite volume and that the material profile within this volume is constant. Within each element domain the vector magnetic field is expressed as H e . If there are N e element domains within V , the functional can be expressed in terms of the approximate magnetic fields as FH Ne e 1 * 1 H e k 02 r H e* .H e dVe H e . Ve r FH Ne Ve e 1 * * 1 2 H e . H e k 0 r H e .H e dVe r Wij i j j i and as 1 r rb . Ai i r 4 3Ve (13) position vector of the barycenter of the tetrahedron defined as r0 r1 r2 r3 rb 4 in which ri is the position vector of node i . Using equations (17) in (16) leads to: (14) Also k 1 * H e .nˆ H e j 0 H e* .M r (15) 0 implying contribution from only regions of implied magnetic current sources, i.e, at the apertures only. M = surface magnetic current source. So that F H Ne e 1 Nb b 1 S * * 1 2 H e . H e k0 r H e .H e dVe Ve r k0 * H b .M n dS j (16) 0 The approximate vector field in each element domain is represented as m H ejWej j 1 Ne e 1 * b Nb m m * ej j 1 ei i 1 Ve 1 Wej . Wei k 2Wej .Wei dVe r j W .M dS b 1 * * 1 2 V H e . H e k 0 r H e .H e dVe N ji e r F H 1 * e 1 ˆ H . n H dS e e e S e and r (20) k0 0 S b (21) n Letting and the divergence theorem, (12 ) can be written as (19) where Ai is the inwardly directed vectorial area of the tetrahedron face opposite to node i , Ve the element volume, and r the position vector. Also, rb is the F He 1 1 1 H e* . H e . H e* H e H e . H e* r r r is the barycentric function of node i expressed (12) Applying the vector identity Ne i (18) (17) 1 Wej . Wei k 2Wej .Wei dVe Ve r Pnb j (22) k0 0 W b .M n dS S (23) equation (21 ) becomes Ne m m Nb F H e ej* ei N ji b* Pnb e 1 j 1 i 1 b 1 (24) By enforcing the continuity of the magnetic field, a global number scheme can be introduced. The following symmetric and highly sparse matrix results N eb e 0 (25) N bb b Pnb where : b are coefficients weighting the edge elements lying in the aperture plane S1 and S 2 . e are N ee N be coefficients weighting the remaining edge elements in V. B. EVALUATION OF N ji Pib j k0 0 W S pq .M i1dS ,for the i th common non- Consider the tetrahedron in Fig ( ) : boundary edge whose end nodes are p and q . 0 l l 01 l 02 D0 03 3 l li 2 A i1 , r in Ti i l M i1 (r ) i i1 , r in Ti 2 Ai 0 , otherwise l 13 23 A0 1 2 l 12 W pq p q q p 1 r rb . Ap p r 4 3Ve Fig( 4): Tetrahedron simplex 3-dimensional finite element K12 M12 N12 K 2 K1 M 2 M1 N 2 N1 D KK3KK1MM 3MM1 NN3NN1 K4 K1 M 4M1 N 4N1 51 5 1 5 1 K 6 K1 M 6M1 N6 N1 R12 Q12 P12 R2 R1 Q2Q1 P2 P1 K RR3RR1QQ3QQ1 PP3PP1 R4 R1 Q4Q1 P4P1 5 1 5 1 51 R6 R1 Q6Q1 P6 P1 K1K 2 M1M 2 N1N 2 K 22 M 22 N 22 K 3 K 2 M 3M 2 N 3 N 2 K4K2 M 4M 2 N4 N2 K1K 3 M1M 3 N1N 3 K 2 K3 M 2 M 3 N 2 N3 K 32 M 32 N 32 K 4 K3 M 4 M 3 N 4 N3 K5 K 2 M 5M 2 N5 N 2 K6 K 2 M 6M 2 N6 N 2 K5 K3 M 5M 3 N5 N3 K 6 K3 M 6 M 3 N 6 N3 R1R2 Q1Q2 P1P2 R2 2 Q2 2 P2 2 R3 R2 Q3Q2 P3 P2 R4 R2 Q4Q2 P4 P2 R5 R2 Q5Q2 P5 P2 R6 R2 Q6Q2 P6 P2 R1R3 Q1Q3 P1P3 R2 R3 Q2Q3 P2 P3 R32 Q32 P32 R4 R3 Q4Q3 P4 P3 R5 R3 Q5Q3 P5 P3 R6 R3 Q6Q3 P6 P3 N ji are the elements of matrix N D K C. Pnb j EVALUATION OF k0 0 S Wb .M n dS Pnb K1K 4 M1M 4 N1N 4 K2K4 M 2M 4 N2 N4 K1K 5 M1M 5 N1N 5 K 2 K5 M 2 M 5 N 2 N5 K 3 K 4 M 3M 4 N 3 N 4 K 42 M 42 N 42 K5 K 4 M 5M 4 N5 N 4 K 3 K 5 M 3M 5 N 3 N 5 K 4 K5 M 4 M 5 N 4 N5 K52 M 52 N52 K 6 K5 M 6 M 5 N 6 N5 K6 K 4 M 6M 4 N6 N 4 R1R4 Q1Q4 P1P4 R2 R4 Q2Q4 P2 P4 R1R5 Q1Q5 P1P5 R2 R5 Q2Q5 P2 P5 R3 R4 Q3Q4 P3 P4 R4 2 Q4 2 P4 2 R5 R4 Q5Q4 P5 P4 R3 R5 Q3Q5 P3 P5 R4 R5 Q4Q5 P4 P5 R52 Q52 P52 R6 R4 Q6Q4 P6 P4 R6 R5 Q6Q5 P6 P5 K 3 K 6 M 3M 6 N 3 N 6 K 4 K6 M 4M 6 N 4 N6 K5 K 6 M 5M 6 N5 N 6 K62 M 62 N62 K1K 6 M1M 6 N1N 6 K 2 K6 M 2M 6 N 2 N6 R3 R6 Q3Q6 P3 P6 R4 R6 Q4Q6 P4 P6 R5 R6 Q5Q6 P5 P6 R6 2 Q6 2 P6 2 R1R6 Q1Q6 P1P6 R2 R6 Q2Q6 P2 P6 [Y B ] [ Tq , H 2scat M p ] PxP t The discretization of the MOM computational domain is based on triangular patch modeling scheme as proposed by Rao et al [12] and as implemented by Konditi and Sinha [ paper u carried ]. where rb is the position vector of the tetrahedron barycenter. k l Pib j 0 i 0 2 Ai A. W pq . i1 dS S Eliminating surface integration over each triangle by approximating W pq by its value at the centroid of each EVALUATION OF MATRIX ELEMENTS As explained in [paper u carr ], following Galerkin procedure ( Tm M m ), a typical matrix element for the rth region is given by r Ymn 2 M m , H t r (M n ) = 2 Pib j k0 c Ai W pq ri 0 li ic1 Ai W pq ri c 2 A i . li ic1 2 A i . lk j i 0 W pq ri c . ic1 W pq ri c . ic1 2 0 D. Tm triangle : COMPUTING Y C The computation of the nth column of Y C in ( ) and ( ) is performed by perturbing the cavity with an equivalent current basis function M n . Then, with the use of ( ) or ( ), the interior tangential magnetic fields in the aperture planes S1 and S2 are computed. The first N row C elements of the nth column of Y are then computed by taking the inner product of the aperture field on S1 successively with the N testing functions Tm . The next P row elements of the nth column are computed by taking the inner product of the aperture field on S 2 with with the P testing functions Tq . Similarly, the = 2 M IV. COMPUTING THE EXTERIOR ADMITTANCE MATRICES BY THE MOM [Y A ] [Tm , H1scat M n ]NxN t Tm m M m H t r ( M n ) ds H t r ( M n ) ds Tm where the notation ( ) ds has been introduced Tm for compactness. In terms of the electric vector potential F (r ) and the magnetic scalar potential (r ) , the magnetic field H r ( M n ) can be written as H r (M n ) j Fn (r ) n (r ) where Fn (r ) = r G (r / r ' ) M n ( r ' ) ds ' T n (r ) Fn (r ) j r r In Eqn. (14), G ( r / r ' ) denotes the dyadic Green's function of the half space. Substitution of Eqn. (13) in Eqn. (12) and use of twodimensional divergence theorem leads to r Ymn 2 j M Tm remaining P columns of Y C can be computed by perturbing the cavity with equivalent current basis functions M p . M m H t r ( M n ) ds m Fn ds 2 m m n ds Tm (2.19) where mm Mm j (2.20) Eqn. (16) contains quadruple integrals; a double integral over the field triangles Tm and a double integral over the source triangles Tn involved in the computation of Fn (r ) and n (r ) . In order to reduce the numerical computations, the integrals over Tm can be c c I mi lm H t i (rmc ) m H t i (rmc ) m 2 2 approximated by the values of integrals at the centroids of the triangles. This procedure yields r Ymn mc mc c c j Fn (rm ) 2 Fn (rm ) 2 2l m c c n (rm ) n (rm ) where Fn (rmc ) r G (r Tn c | r ' ) M n (r ' ) ds' 1 j r (21) Following the procedure as outlined in [ paper u carried], transmission coefficient and transmission cross-section are evaluated for the aperture problem. V. (19) n (rmc ) (18) G (r c | r ' ) M n (r ' ) ds' Tn (20) In (18), mc are the local position vectors to the centroids of Tm and rmc = ( rm1 rm2 rm3 ) / 3 are the position vectors of centroids of Tm with respect to the global coordinate system. Similarly, as explained in[ paper u carried], using the centroid approximation in Eqn. (2.10), an element of excitation vector can be written as RESULTS AND DISCUSSION REFERENCES 1. 2. 3. 4. 5. 6. 7. Collin, R.E.: Field Theory of Guided Waves, New York: McGraw-Hill, 1960 Harrington, R.F.: Time-harmonic Electromagnetic Fields, New York : McGraw-Hill, 1961 N. Amitay. V. Galindo. and C. P. Wu, Theory and Analysis of Phased Array Antennas. New York: WileyInterscience, 1972 W. F. Croswell. "Antennas and wave propagation," Electronics Engineers' Handbook, D. G. Fink and A. A. McKenzie, Eds. New York: McGraw-Hill, 1975, pp. 1826. R.F. Harrington, J.R. Mautz, and D.T. Auckland: "Electromagnetic coupling through apertures," Dept. Elec. Comput. Eng., Syracuse Univ., Syracuse NY, Rep. TR-81-4, Aug. 1981. C. M. Butler. Y. Rahmat-Samii. and R. Mittra. "Electromagnetic penetration through apertures in conducting surfaces," IEEE Trans. Electromagn. . vol. EMC-20. pp. 82-93, Feb. 1978. Xingchao Yuan, “Three-dimensional electromagnetic scattering from inhomogeneous objects by the hybrid moment and the finite element method”, IEEE Trans Microwave Theory Tech., Vol. MTT-38, pp. 1053- 1058, 1990. Harrington, R.F. and Mautz, J.R. : "A generalized network formulation for aperture problems," IEEE Trans. Antennas Propagat., Vol. AP-24, pp. 870-873, 1976 9. Harrington, R.F. and Mautz, J.R. : "A generalized network formulation for aperture problems," IEEE Trans. Antennas Propagat., Vol. AP-24, pp. 870-873, 1976 10. S. K. Jeng, "Scattering from a cavity-backed slit in a ground plane TE case," IEEE Tram. Antennas Propagat., vol. 38, pp. 1523-1529, 9, Oct. 1990. 11. 8.