Advances in Radio Science

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1 dances in Radio Science, 3, , 2005 SRef-ID: /ars/ Copernicus GmbH 2005 dances in Radio Science Combining the multileel fast multipole method with the uniform geometrical theory of diffraction. Tzoulis T. F. ibert FGN-Research Institute for High-Frequency Physics Radar Techniques (FHR), Wachtberg-Werthhoen, Neuenahrer Str. 20, Germany bstract. The presence of arbitrarily shaped electrically large objects in the same enironment leads to hybridization of the Method of Moments (MoM) with the Uniform Geometrical Theory of Diffraction (UTD). The computation memory complexity of the MoM solution is improed with the Multileel Fast Multipole Method (MLFMM). By exping the ˆk-space integrals in spherical harmonics, further considerable amount of memory can be saed without compromising accuracy numerical speed. Howeer, until now MoM-UTD hybrid methods are restricted to conentional MoM formulations only with lectric Field Integral quation (FI). In this contribution, a MLFMM-UTD hybridization for Combined Field Integral quation (CFI) is proposed applied within a hybrid Finite lement - Boundary Integral (FBI) technique. The MLFMM-UTD hybridization is performed at the translation procedure on the arious leels of the MLFMM, using a far-field approximation of the corresponding translation operator. The formulation of this new hybrid technique is presented, as well as numerical results. 1 Introduction xact numerical solutions of electromagnetic radiation scattering problems with composite dielectric/metallic objects are actually obtained with the Finite lement Boundary Integral (FBI) method. The inoled integral equation (I) is discretized by the Method of Moments (MoM) using Rao-Wilton-Glisson (RWG) basis functions a Galerkin s type approach (Rao et al., 1982). The resulting hybrid linear equation system is efficiently soled with iteratie solers in combination with appropriate preconditioning (ibert, 2003). Computation complexity memory requirements of large scale MoM solutions for high frequencies increase rapidly with the dimensions of the objects. This problem is oercome by the Multileel Fast Multipole Method (MLFMM) resulting in low computation memory complexity of O(N log N) (Coifman et al., 1993; Chew et al., 2001). Further saing of a considerable amount of memory Correspondence to:. Tzoulis (tzoulis@fgan.de) without compromising accuracy numerical speed is achieed by exping the FMM ˆk-space integrals into spherical harmonics (ibert, 2004). lectrically large conducting objects are hled efficiently with ray-based high-frequency methods. Such methods gie accurate asymptotic solutions of the electromagnetic fields for objects with large dimensions as compared to the waelength no discretization is needed. The electromagnetic fields are described according to classical ray concepts of Geometrical Optics (GO) (Lo Lee, 1993) Uniform Geometrical Theory of Diffraction (UTD) (Kouyoumjian Pathak, 1974). Full electromagnetic coupling between dielectric arbitrarily shaped UTD objects within a common enironment is obtained with the hybrid FBI-UTD method (laydrus et al., 2001), which is howeer restricted to conentional MoM formulations only with FI. In this contribution, the hybrid FBI-UTD technique is extended by combining MLFMM with UTD for CFI. The MLFMM-UTD hybridization is performed in the translation procedure on the arious MLFMM leels. The amplitudes of the UTD rays are determined according to the far-field MLFMM in Chew et al. (2001). Due to different ray directions for outgoing incoming waes, appropriate interpolation anterpolation routines must be applied to consider the UTD contributions. In the following, first the formulation of the FBI-UTD for CFI after that the formulation of the MLFMM-UTD approach with expansion of the ˆk-space integrals in spherical harmonics is presented. Finally, numerical results for arious large scale examples are gien. 2 Formulation 2.1 FBI-UTD hybrid method Consider the configuration of Fig. 1, where inhomogeneous dielectric FBI large conducting UTD objects exist in the same enironment in free space. The electromagnetic fields in the interior region of the FBI objects are expressed according to the Finite lement Method (FM) (Volakis et al., 1998) are exped in terms of edge element basis functions α n using tetrahedral olume meshes.

2 184. Tzoulis T. F. ibert: Combining the MLFMM with the UTD O Source group r R UTD Object r D r r metal metal inc, H inc metal Volume V FBI Object closed surface dielectric O r r n r m r nn n Receiing group kˆi m d r Q rm Q r Qn kˆr, d Q UTD Object Fig. 1. Hybrid FBI-UTD concept. Fig. 2. Hybrid MLFMM-UTD concept. The fields in the exterior regions are expressed with I s oer the boundaries of the FBI objects, being soled by a Galerkin-type MoM with RWG basis functions (Rao et al., 1982). The presence of UTD objects is taken into account by modifying the Green s functions, as well as the incident fields of the I s, with additional high-frequency contributions receied at the testing or obseration points, respectiely (laydrus et al., 2001). In this contribution, 1st 2nd order reflections diffractions on flat structures are taken into account. The following formulations will be gien for clarity only for 1st order mechanisms. The formulation for 2nd order mechanisms can be easily gien in the same manner. The UTD ray contributions are taken into account for the CFI Z(1 α)mf I αf I = 0, (1) where α is the combination parameter with alues from 0 to 1 Z the wae impedance of the considered solution space. The FI is gien by ˆn [ ˆn ( [ G J,tot (r, r ) J (r ) ] ) + G M,tot (r, r ) M (r ) da + inc tot (r) + 1 ] 2 M (r) = 0 (2) the Magnetic Field Integral quation (MFI) by ( [ G H ˆn M,tot (r, r ) M (r ) ] ) + G H J,tot (r, r ) J (r ) da + H inc tot (r) 1 2 J (r) = 0, (3) where G /H J,tot (r, r ) G /H M,tot (r, r ) are the total Green s functions of the electric or magnetic field due to electric magnetic surface currents, respectiely. lso, inc tot (r) H inc tot (r) are the total incident electric magnetic fields at the obseration point r. The total Green s functions of the hybrid problem are the superposition of the Green s functions of the direct coupling of the currents G /H J/M (r, r ) of G /H J/M,UT D (r, r ) = s Rs R /H s G /H J/M (r R s, r ) + D D /H G /H J/M (r D, r ) +, (4) which is the Green s functions of the high-frequency raybased fields due to UTD objects. Rs D are the diergence phase factors for reflection diffraction, respectiely, R /H s D /H the dyadic reflection diffraction coefficients for electric or magnetic field defined by basic GO (Lo Lee, 1993) UTD concepts (Kouyoumjian Pathak, 1974). Direct coupling is not taken into account if UTD objects lie between source testing currents. Similarly, the high-frequency contributions of the incident electric magnetic field at the obseration point r hae the form inc UT D (r) = s Rs R s inc (r Rs ) + D D inc (r D ) + (5) H inc UT D (r) = 1 Z ˆk r s Rs R s inc (r Rs ) + 1 Z ˆk d D D inc (r D ) +, (6)

3 . Tzoulis T. F. ibert: Combining the MLFMM with the UTD 185 The matrix-ector product computations are accelerated by MLFMM. In order to retain low complexity in the hybrid approach, UTD contributions are taken into account in the matrix-ector product MLFMM computations. In the following, the MLFMM formulation for CFI with expansion of the ˆk-space integrals in spherical harmonics, as well as the hybrid MLFMM-UTD formulation is presented. 2.2 MLFMM-UTD hybrid method The hybrid MLFMM-UTD concept is shown in Fig. 2. The source group n the receier group m belong to the FMM model of the FBI object at a specific leel. The CFI matrix elements in FMM representation can be written as (Coifman et al., 1993; Chew et al., 2001) CF I Zmn,J = c 1 β m (ˆk) T L (ˆk ˆr m n ) ( I ˆk ˆk ) β n (ˆk) d ˆk 2 ) +c 2 (ˆk α m (ˆk) T L (ˆk ˆr m n ) β n (ˆk) d ˆk 2 (7) Fig. 3. Surface current density magnitude of trihedral 90 corner reflector for 15 GHz. bistatic rcs in dbsm f = 64 GHz MoM PO 22.1 Mio unknowns angle w.r.t. aperture Fig. 4. Bistatic RCS of trihedral 90 corner reflector for 64 GHz. where ˆk r ˆk d are the propagation directions of the reflected diffracted rays, respectiely. gain, no direct field contribution is taken into account if the obseration point is in the shadow of an UTD object. The MoM solution of the I s (1), (2) (3) is achieed by exping the surface currents in terms of RWG basis functions β n using triangular surface meshes. The exterior fields are coupled with the interior ones ia field continuity conditions at the boundaries of the FBI objects. The resulting hybrid linear equation system is soled by an iteratie solution technique, using a multileel iteratie preconditioning strategy (ibert, 2003). CF I Zmn,M = ) +c 3 (ˆk β m (ˆk) T L (ˆk ˆr m n ) β n (ˆk) d ˆk 2 +c 4 α ( I ) m (ˆk) T L (ˆk ˆr m n ) ˆk ˆk β n (ˆk) d ˆk 2 where c 1 = j ωµ 4π α, c k 2 = jz 0 4π (1 α), c 3 = j 4π k α ωε c 4 = jz 0 4π (1 α). Further, β m (ˆk) = β m (r)e jk r mm da (9) α m (ˆk) = (8) α m (r)e jk r mm da (10) are the ˆk-space representations of the basis functions T L (ˆk ˆr) = L (j) l (2l + 1)h (2) l (kr)p l (ˆk ˆr) (11) l=0 is the translation operator. lso, h (2) l is the second kind spherical Hankel function of degree l, P l is the Legendre polynomial of degree l * denotes complex conjugation. The MLFMM is implemented with memory efficiency by exping the representations β m (ˆk) of the basis functions in spherical harmonics according to β m (ˆk) = P p=0 q=p p f n pq Y pq(ϑ, ϕ), (12)

4 186. Tzoulis T. F. ibert: Combining the MLFMM with the UTD rcs in dbsw D = 1 m = Mio unknowns CFI with = 0.5 MoM Mie rcs in dbsw D = 1 m = Mio unknowns CFI with = 0.5 MoM Mie theta in degree theta in degree Fig. 5. RCS near forward direction of conducting sphere with diameter D = 1 m= 146.7λ by CFI solution. Fig. 6. RCS near backward direction of conducting sphere with diameter D = 1 m= 146.7λ by CFI solution. where Y pq (ϑ, ϕ) = (2p + 1)(p q)! P q p(cos ϑ)e jqϕ, (13) 4π(p + q)! are the orthonormalized spherical harmonics (Harrington, 1961). lso, P q p is the associated Legendre polynomial of degree p order q. ccording to this approach, the matrixector product computations within the iteration loop of the iteratie soler are performed by first aggregating all expansion coefficients at the center of each MLFMM group at the finest leel. fter that, the outgoing waes are computed at the quadrature points for all groups on the finest MLFMM leel. The translations of outgoing waes into incoming waes as well as the aggregations disaggregations between different MLFMM leels including interpolation anterpolation are performed as in stard MLFMM using the numerical quadrature samples. Howeer, when all incoming waes are collected in a certain group on the finest leel, the spherical harmonics expansion ( I T L (ˆk ˆr m n ) ˆk ˆk ) β n (ˆk) = P p p=0 q=p g n pq Y pq(ϑ, ϕ) (14) is carried out by utilizing the orthogonality of the spherical harmonics, the closed integral oer the wald sphere is simplified to the series Z mn = j ωµ 4π P p=0 q=p p (f m pq ) g n pq. (15) The number of expansion coefficients needed with this approach is in general less than the corresponding number of quadrature samples for numerical integration in the ˆk-space, resulting in saing of memory requirements. The ray contributions due to UTD objects are taken into account in the MLFMM matrix-ector product computations by assuming that the ray path from the source group to the UTD object r Qn is much greater than the dimensions of the group itself. This means, that only one ˆk direction is needed to express the radiated field from the source group. ssuming far-field conditions, the scalar Green s function from the source current to the local point on the UTD object is expressed with the far-field MLFMM approximation G(r Q, r ) = ejk r Qr r Q r = e jk i r nn T F F L, (16) where k i = k ˆk i, ˆk i = ˆr Qn is the direction of ray incidence T F F L = ejkr Qn r Qn (17) is the associated far-field translation operator (Chew et al., 2001). Thus, the translation of the high-frequency contributions is performed only for the direction of ray incidence ˆk i. The receied high-frequency fields are taken into account only for the reflection ˆk r or diffraction ˆk d directions from the local point on the UTD object to the receier groups. The following MLFMM-UTD formulations are gien for simplicity for the case of the FI contributions in (7). Howeer, they apply to all contributions in (7) (8) as well. The ray contributions of the matrix elements of the hybrid MLFMM- UTD approach can be written as F I Zmn,J,UT D = j ωµ β m 4π (ˆk r ) s j ωµ β m 4π (ˆk d ) ( Rs R I ) s TL F F ˆk r ˆk i β n (ˆk i ) ( D D I ) TL F F ˆk d ˆk i β n (ˆk i ) +, (18) where k r =k ˆk r k d =k ˆk d. gain, no direct field contribution is taken into account if the receier group is in the shadow of an UTD object.

5 . Tzoulis T. F. ibert: Combining the MLFMM with the UTD 187 f = 30 GHz 0 =100 V/m PC z inc 26 x y PC Fig. 7. Conducting cone cylinder in front of conducting plate at distance λ. Fig. 9. Magnitude of surface current density on conducting cylindrical parabolic reflector with X-b horn excitation for f =10 GHz FBI-MLFMM-UTD 3 h, 360 MB RM FBI-MLFMM 6.5 h, 1600 MB RM FBI-MLFMM-UTD 6 plates, 2 h, 1 MB FBI-MLFMM-UTD 10 plates, 3 h, 1 MB FBI-MLFMM-UTD 20 plates, 7.5 h, 1 MB FBI-MLFMM 11.5 h, 1900 MB RM tot (V/m) (db) zy-plane x(m) Fig. 8. lectric field in direction of propagation for conducting cone cylinder in front of conducting plate Fig. 10. Co-polar radiation pattern on principal -plane of conducting cylindrical parabolic reflector with X-b horn excitation for f =10 GHz. ( ) In the numerical implementation of MLFMM, a limited number of sampling points is used to ealuate the ˆk-space integrals. In general, the ray directions of incidence ˆk i, reflection ˆk r or diffraction ˆk d do not match with any of these sampling points. For this reason, the required direction of incidence must be interpolated from the neighboring sampling points. Similarly, after reflection or diffraction, the appropriate ray direction must be anterpolated to the neighboring sampling points. 3 Numerical examples In this section, numerical results of arious large scale examples are presented. First, a trihedral 90 conducting corner reflector with 1.4 m edge length has been computed, as shown in Fig. 3. In the same figure, the magnitude of the surface current density on the corner reflector can be seen, computed for 15 GHz, using a triangular surface mesh. For the same corner reflector, RCS computations hae been carried out for 64 GHz, using H-polarized plane wae excitation with normal incidence. In this case, the reflector was discretized with 22.1 Million unknowns the computation time of the solution was about 122 h on an Opteron 2.2 GHz CPU with about 28 GB memory requirements. In Fig. 4, the co-polar bistatic RCS can be seen, compared to a PO solution. It is noticed, that to our knowledge this computation is currently the world-wide largest MoM solution performed on a single CPU. Next example is a conducting sphere with diameter D=1 m=146.7λ. The triangular surface mesh on the sphere resulted in 15.3 Million unknowns the solution was

6 188. Tzoulis T. F. ibert: Combining the MLFMM with the UTD obtained with FBI for CFI with α=0.5. In Figs. 5 6, the computed RCS is shown, for small regions near forward backward direction, respectiely. The numerical results are compared to the corresponding Mie-series analytical solution. It can be seen, that numerical results hae excellent agreement with the exact analytical solution. specially at backward direction, ery small ripple of the numerical solution can be seen. Next example is the problem of a conducting cone cylinder in front of a conducting plate at distance λ, as shown in Fig. 7. Two computations hae been carried out. First, both objects where discretized with triangular surface mesh, resulting in unknowns. In the second computation, the same surface mesh was used for the cone cylinder the plate was treated by UTD. In this case, the number of unknowns was In Fig. 8, the calculated electric field in direction of propagation for both computations is shown. In the same figure, computation time memory requirements on an MD thlon PC are shown as well. It can be seen, that the hybrid FBI-UTD results agree ery well with the exact numerical FBI solution, needing much less computation time memory requirements. Last example is a conducting cylindrical parabolic reflector with X-b horn excitation for f =10 GHz, as shown in Fig. 9. In the same figure, the magnitude of the surface current density on both objects can be seen, using a triangular surface mesh for both objects with unknowns. For the hybrid FBI-UTD computation, the reflector was approximated by flat plates computations were done for 6, plates. In Fig. 10, the co-polar radiation pattern on the principal -plane for all computations is shown. In the same figure, the computation time memory requirements on an MD thlon PC are shown as well. It can be seen, that for 6 10 plates the hybrid FBI-UTD solution presents significant side lobes. For 20 plates, the solution shows ery good agreement with the exact numerical FBI solution, needing less computation time memory requirements. with expansion of the ˆk-space integrals in spherical harmonics was presented. Finally, numerical results for arious large scale antenna scattering problems were gien. References laydrus, M., Hansen, V., ibert, T. F.: Hybrid 2 : Combining the three-dimensional hybrid finite element boundary integral technique for planar multilayered media with the uniform geometrical theory of diffraction, I Trans. ntennas Propagat., ol., no. 1, 67 74, Jan Chew, W. C., Jin, J.-M., Michielssen,.: Fast efficient algorithms in computational electromagnetics, Boston: rtech House, Chew, W. C., Cui, T. J., Song, J. M.: FFF-MLFM algorithm for electromagnetic scattering, I Trans. ntennas Propagat., ol., no. 11, , No Coifman, R., Rokhlin, V., Wzura, S.: The fast multipole method: pedestrian prescription, I ntennas Propagat. Mag., ol. 35, no. 3, 7 12, Jun ibert, T. F.: dances in hybrid finite element boundary integral modelling of three-dimensional radiation scattering problems, Intern. ITG Conf. on ntennas, , Berlin, Sep ibert, T. F.: multileel fast multipole method with spherical harmonics expansion of the k-space integrals, URSI M-Theory Symposium, Pisa, Italy, 1 153, May Harrington, R. F.: Time harmonic electromagnetic fields, New York: McGraw-Hill, Kouyoumjian, R. G. Pathak, P. R.: uniform geometrical theory of diffraction for an edge in a perfectly conducting surface, Proc. I, ol. 62, no. 11, , No Lo, Y. T. Lee, S. W.: ntenna hbook, ol.1, Chapman & Hall, New York, Rao, S. M., Wilton, D. R., Glisson,. W.: lectromagnetic scattering by surfaces of arbitrary shape, I Trans. P, ol. 30, 9 418, May Volakis, J. L., Chatterjee,., Kempel, L.-C.: Finite lement Method for lectromagnetics, I, Inc., New York, Conclusions In this contribution, the hybrid FBI-MLFMM-UTD method for numerical radiation scattering computations was presented. Due to memory efficient implementation of the MLFMM, large scale numerical computations can be carried out. The FBI-UTD hybridization allows optimum treatment of problems including arbitrarily shaped composite dielectric/metallic objects electrically large conducting objects of simple shape in the same enironment. Low computation complexity memory requirements are retained in the hybrid FBI-UTD approach, due to MLFMM-UTD combination. The MLFMM-UTD hybridization is performed in the translation procedure on the arious MLFMM leels, using a far-field approximation of the appropriate translation operator. The hybrid FBI-UTD formulation for CFI, as well as the formulation of the MLFMM-UTD combination