ANALYSIS OF REFLECTOR AND HORN ANTENNAS USING MULTILEVEL FAST MULTIPOLE ALGORITHM

Transcription

1 European Congress on Coputational Methods in Applied Sciences and Engineering ECCOMAS 2 Barcelona, Septeber 2 ECCOMAS ANALYSIS OF REFLECTOR AND HORN ANTENNAS USING MULTILEVEL FAST MULTIPOLE ALGORITHM Alex Heldring *, Juan M. Rius, Angel Cardaa and Leo P. Ligthart * * International Research Centre for Telecounications and Radar, Delft University of Technology Mekelweg 4, 2628 CD Delft, The Netherlands Eail: irctr@its.tudelft.nl Universitat Politècnica de Catalunya, Dept. of Signal Theory and Counications, C/ Jordi Girona 1-3, 834 Barcelona, Spain Eail: heldring@voltor.upc.es Key words: Coputational Electroagnetics, Antennas, Method of Moents, Fast Multipole Algorith. Abstract. The MultiLevel Fast Multipole Algorith (MLFMA) has been ipleented in a Method of Moents code for arbitrarily shaped etal surfaces discretized in Rao, Wilton and Glisson (RWG) basisfunctions. The accuracy of the algorith has been assessed by coparing the results with the exact theoretical solution for a Perfectly Conducting Sphere. Subsequently it has been applied to a parabolic reflector and and X-band horn, coparing the results with easureents. Syetric properties of the antennas have been exploited and an efficient Incoplete LU preconditioner for the iterative solver has been applied. 1

2 1 INTRODUCTION Electrically large reflectors and horn antennas are usually analysed with high frequency approxiation techniques (Physical Optics, Physical Theory of Diffraction) which can adequately predict paraeters like the gain and the beawidth. Soeties, however, very accurate predictions of the entire radiation pattern are required. An exaple is the TARA atospheric radar syste currently under construction at the IRCTR in Delft, The Netherlands [1]. One of the requireents for the TARA syste, which involves two parabolic reflectors of approxiately 3 wavelengths in diaeter, is a far sidelobe level of 8 db. To achieve this, the reflectors are shielded by large conical etallic ris. A accurate nuerical siulation of this structure is a ost welcoe optiisation tool. The ost reliable and generally applicable ethod to obtain such accuracy is to solve the associated integral equation by way of the ethod of oents (MoM). Until recently, the excessive coputational requireents prohibited this approach to probles of this size, but the advent of very efficient acceleration ethods for the nuerical solution of electroagnetic integral equations and the fast growth of coputer capacity has brought the within reach. Aong the ost efficient acceleration ethods is the Multilevel Fast Multipole Algorith or MLFMA [2]. The present paper gives a short theoretical description of the MLFMA, describes an ipleentation in an existent MoM code for perfectly conducting objects of arbitrary shape, and assesses its perforance by coparisons to an exact theoretical solution for a etal sphere and to easured antenna results. The necessary iproveents to tackle the TARA proble are shortly adressed in the conclusion. 2 THEORY 2.1 Method of Moents The appropriate integral equation to describe large open objects like antenna reflectors is the Electric Field Integral Equation (EFIE). The object is considered built up of electrically thin sheets (d << λ) and the calculated quantity is the su of the induced surface currents on both sides, due to a known incident electric field. A finite set of basis functions is chosen to represent the current over the surface, and a set of testing functions is chosen to weigh the incident field over the surface. In the present case both basis and testing functions are Rao, Wilton and Glisson triangular rooftop basis functions [3]. This leads to a atrix representation [ E ] = [ Z][ I] inc, (1) where [ E inc ]and [I] are vectors of length N representing the incident field and the current respectively, and [Z] is an N by N atrix with eleents given by (three diensional vectors are represented by lower case boldface characters) 2

3 In (2), k, ω and ε are the wave nuber, angular frequency and free space perittivity, respectively. The vector functions f n (x) are the basis and testing functions. The double integration is over the doain of the basis and testing functions, and g is the scalar free space Green s function. Equation (1) ust be solved for [J], after which the radiation pattern can be obtained straightforwardly. Solving (1) directly has a coputational coplexity of N 3. Alternatively, an iterative algorith can be eployed to obtain an approxiate solution. A coon iterative algorith is the Biconjugate Gradient (BiCG) ethod, which takes two atrix-vector products (2N 2 operations) per iteration step, but generally converges to sufficient accuracy in far less than N steps. Nevertheless, both coputing tie and storage becoe prohibitive once the object surface exceeds certain liits. 2.1 Multilevel Fast Multipole Algorith Z = 1 n k g d d j 2 2 f n (x) ( ' ) ( x, x')) f (x') x x'. ωε The MLFMA yields an approxiate representation of the atrix [Z] in (1). The accuracy of the approxiation can be chosen at will, to be traded against the coputational costs. The basis of the algorith is an expansion of the Green s function around the source and field points [2] (2) jk rc+ dn d e jk j j g( r d d k d ) e T( k,r $ k d n ) e d k$ c + n = r r + 2 c, r + d d 4π c n (3) L l T( $ ( ) k,r ) j ( l ) hl ( k rc ) Pl ( $ c = k rc ). l= (4) In (3), the integration is over the unit sphere. The expansion is valid for r c >> d n and r c >> d. In (4), h l (2) is the spherical Hankel function of the second kind of order l and P l is Legendres polynoe of order l. The first step is to subdivide the basis functions into geoetrical sub groups. To this ai, the object is enclosed inside a box. The box is then recursively split into eight saller boxes, down to a fixed level of sallest boxes (the resulting structure is called an octal tree). At every level, the interactions between basis functions inside nearby boxes (typically touching boxes) is represented by the interactions between their child boxes. At the lowest level, these are siply the interactions between basis functions, as in (2). For the interactions between far enough boxes, the subatrix [Z s,f ], where the subscripts s and f denote the source and field boxes respectively, is replaced by the product [Z ] V T V. (5) s,f f s, f s i [ ][ ][ ] 3

4 The substitution (5) is obtained using (3) in (2), with an appropriate discretisation of the integral in (3). [V] is a plane wave expansion of the basis functions around the box centre; the suation index i is over the three spatial coponents plus the derivative of the basis functions. It represents the inner product in (2). [T] is a diagonal atrix with the factors defined in (4). The nuber of directions on the unit sphere necessary to represent the interaction depends linearly on the geoetrical size of the concerned boxes. Therefore, twice as any directions are needed with every step up the octal tree. They need not be recoputed directly fro the basis functions however. Instead, they can be found using shifting (to the new group centre) and interpolation of the child box plane wave expansions, L+ 1 L [ ] [ L ][ L ][ ] V Q W V, c (6) where L denotes the level in the octal tree. The suation index c is over all the child boxes. [Q L ] is the (diagonal) shifting operator, and [W L ] interpolates to a new set of unit vectors on the sphere. If a sparse (local) approxiation schee is used, the overall storage requireent and the coputational coplexity of a atrix-vector product is reduced to order Nlog(N). The error in the ultipole approxiation depends on three paraeters, the order of the expansion L in (4), the nuber of ultipole directions and the iniu distance between boxes. Furtherore, the overall error also depends on the interpolation operator [W] that is used. The first three are linked. Reference [2] presents the as functions of a single paraeter p. A choice of p=5 yields 32 bits accuracy, but if an error in the surface current of 1 25% is acceptable (this typically leads to an error of less than 1% in the far field pattern) p=1.5 is sufficient. In the latter case, a low order interpolation schee (based on second degree polynoials) is adequate. For 32 bits precision, a high order (global) interpolation is necessary, leading to full interpolation atrices [W]. For this case, fast interpolation operators based on FFT and a Fast Legendre Transfor have been proposed [4]. Since these have an Nlog(N) coplexity, this leads to an overall coplexity of Nlog 2 (N) per atrixvector product. Another issue of iportance is the convergence of the BiCG algorith. The convergence speed depends on N and generally, it is unacceptably slow for large probles (N > 1e4). Therefore, a preconditioned syste (7) M 1 1 E = M Z J [ ][ inc ] [ ][ ][ ] with an advantageous choice for [M], is solved instead. Good results are obtained when taking for [M] a sparse approxiation of [Z], retaining only the 5 largest eleents per row. This is easily obtained fro [N]. Then [M -1 ] is coputed as an incoplete LU decoposition (ILU) of [M], where the threshold is chosen to obtain as large as possible [L] and [U], given the available coputer storage. 4

5 3 RESULTS 3.1 Perfectly Conducting Sphere The accuracy of the MLFMA was assessed by coparing the coputed RCS of a large conducting sphere (diaeter D = 8.84 λ) with the exact (Mie series) result. The sphere was discretised into triangular facets, yielding RWG basis functions. The MLFMA precision factor was set to p=1.5 with second degree polynoial interpolation. The total size of the MLFMA decoposition was 64 MB, plus a 77 MB ILU preconditioner. The MLFMA atrices are swapped fro the hard disk in chunks during the iteration. Total coputing tie was 6h 49in on a Pentiu II 4 MHz, 256 MB RAM (the convergence to 1% relative error took 13 iterations, 12.5in per iteration). The relative RMS error in the current was 11%, in the RCS:.8%. Fig. 1 shows the RCS. 35 Bistatic RCS, E plane 3 exact f 25 2 dbs θ 35 Bistatic RCS, H plane 3 exact f 25 2 dbs φ Figure 1: RCS of PEC sphere, diaeter 8.84λ, unknowns 5

6 3.2 Parabolic Reflector Subsequently, a sall parabolic reflector was odelled and copared with easureents. The specifications of the reflector and the siulation are given in Table 1. Diaeter 5 λ F/D.376 Feeding Dipole + ½ λ circular disk Nuber of RWG basis functions Using 4-fold syetry 2954 Table 1 : Specifications of Parabolic Reflector The reflector was fed with an infinitesial dipole in the focal point, backed by a ½ λ circular disk placed at ½ λ above the focal point. Using the 4-fold syetry of the proble, the MoM linear syste could be solved by direct inversion. This allowed for a coparison of of direct MoM and MLFMA. The MLFMA decoposition was done with precision factor P=1.5 and 2 nd degree polynoial interpolation. It contained 5 levels. The difference in the coputed surface current between MLFMA and direct MoM was saller than 3%. Fig. 2 shows the radiation patterns obtained fro the coputed currents copared with the easured pattern. The direct inversion took 59 inutes (of which 37 inutes for building the ipedance atrix), the MLFMA coputation took 24 inutes (of which 15 for building the MLFMA decoposition and a 7 MB ILU preconditioner). -1 parabola, E plane easured MoM lf -1 parabola, H plane easured MoM lf -15 ra di ati -2 on pa -25 tte rn -3 (d B) ra di ati -2 on pa -25 tte rn -3 (d B) observation angle (deg) observation angle (deg) Figure 2: Radiation pattern of parabolic reflector, diaeter 5λ, unknowns 6

7 3.3 X-band Horn The next siulation concerned an X-band horn with the specifications described in Table 2. Aperture 3 x 4 λ Horn height 8.2 λ Waveguide section.8 x.3 x 2 λ Feeding Dipole in centre of waveguide RWG basisfunctions 69 (17781 using syetry) Table 2 : Specifications of X-band Horn 4-fold syetry was eployed. The MLFMA decoposition had precision paraeter P=2 and 4 th degree polynoial interpolation. The decoposition had 7 levels. The entire MLFMA decoposition had a size of 1.3 GB. A 15 MB ILU preconditioner was used. The atrix setup tie was 11h6in, the iteration (to 1% relative error) took 22 steps of about 13in each adding up to 4h4in. Fig. 2 shows the MLFMA coputed, Aperture Theory coputed and easured radiation patterns. Beyond 4 degrees off boreside, where Aperture Theory fails, the MLFMA pattern still atches well with the easureent as the figure shows. Fig. 3, a shows the horn with the coputed current distribution (aplitude). Horn Antenna, H plane easured theory lf Horn Antenna, E plane easured theory lf radiation pattern (db) radiation pattern (db) observation angle (deg) observation angle (deg) Figure 3: Radiation pattern X-band horn, aperture 4 x 3 λ, 69 unknowns 7

8 Figure 4: X-band horn, MLFMA coputed surface current aplitude 8

9 4 CONCLUSIONS The presented results show that the MLFMA cobined with the EFIE and Method of Moents yields good results for large etal objects, two probles of well over 1. unknowns have been presented. In its present stage of developent probles of over 1. unknowns should not pose any probles on a coon PC like the one that has been used here. The size of the MLFMA decoposition is not a fundaental liitation, due to the hard disk swapping. The ain bottleneck for larger probles is the preconditioner. Since the Incoplete LU-decoposition needs to be perfored in eory, this poses a liit to the size of the preconditioner. Alternative ethods of preconditioning, that do not have a size liiting proble, are under investigation. REFERENCES [1] S.H. Heijnen and L.P. Ligthart, TARA: Transportable Atospheric Radar, European Microwave Conference, Asterda, 1998 [2] R. Coifan, V. Rokhlin and S. Wandzura, `The Fast Multipole Method for the Wave Equation: A Pedestrian Prescription,' IEEE A\&P Magazine, Vol 35, No 3 June 1993 [3] R. M. Rao, D. R. Wilton and A. W. Glisson, `Electroagnetic Scattering by Surfaces of Arbitrary Shapes,' IEEE A&P Magazine, Vol AP-3, No 3, May 1982 [4] M.F. Gyure and M.A. Stalzer,`A Prescription for the Multilevel Helholtz FMM,' IEEE Cop. Sci.& Eng. July-Sept