Overview of Hybrid Methods in FEKO: Theory and Applications U. Jakobus1 Abstract In order to solve complex electromagnetic radiation or scattering problems, there is often not a single method of choice. Different components or parts of such a complex problem are best solved with different numerical techniques, which then also need to be coupled within a hybrid framework. This paper provides an overview of a number of such hybrid techniques as available in the field solver package FEKO and presents real-world applications. 1 INTRODUCTION Real-world electromagnetic problems are often complex in nature combining different problem scales or characteristics (e.g. antenna mounted on a complex structure as shown in Fig. 1, or a complex human body phantom at some distance in front of a GSM base station antenna). While it is possible to use a single numerical technique to solve such problems, it is often advantageous to solve different parts with different techniques best suited for these parts, and then combining these together into one solution within a hybrid framework. wave solutions (MoM where the size still allows, or the Multilevel Fast Multipole Method MLFMM [6]). 2 GENERAL HYBRID FRAMEWORK The basis for all the hybrid methods in FEKO is the MoM, which this is then coupled with high frequency asymptotic techniques (PO, UTD, GO/SBR) for the solution of electrically large problems but also including small features (e.g. excitation of an aircraft fuselage with a patch antenna or similar), or also a hybridisation with FEM is possible in particular for complex dielectric media still taking advantage of MoM regarding not meshing free-space regions for instance (see Section 6 for details). Figure 2: Bi-directional coupling between two methods (here MoM with dielectric PO). Figure 1: Example for hybrid methods - Antenna placement on an electrically large structure. In FEKO [1] several hybrid methods are used combining the Method of Moments (MoM) with high frequency asymptotic techniques Physical Optics (PO) [2], Uniform Theory of Diffraction (UTD) [3], RayLaunching Geometrical Optics (GO), also known as Shooting-and-Bouncing Rays (SBR) [4] and also with the Finite Element Method (FEM) [5]. The aim of this paper is to provide an overview of these hybrid methods. Results are presented for real world applications with validations compared to full Characteristic to all these hybrid methods is the requirement to couple the various regions in a bi-directional manner, i.e. from MoM to the other method but in general also couple back. This is illustrated in Fig. 2 for the example of combining MoM (simple wire antenna) with PO for a dielectric body (then resulting in asymptotic surface currents JPO and MPO in the PO region): The coupling from MoM to PO (via radiated H-fields here) is always required; otherwise the solution would be as if the PO region was not there, i.e. PO region not excited. The coupling back (here from PO to MoM via radiated E-fields) is in general optional. If used it improves the solution accuracy (e.g. input impedance of the dipole will change depending presence of other region), but it requires extra processing time and 1 EM Software & Systems S.A. (Pty) Ltd, 32 Techno Avenue, Technopark, Stellenbosch 7600, South Africa, e-mail: jakobus@emss.co.za, tel.: +27 21 880 1880, fax: +27 21 880 1936. 978-1-4244-7368-7/10/$26.00 2010 IEEE 434
sometimes also extra memory (depending on the methods combined). larger the benefit of PO versus MLFMM (not done here so that ordinary MoM still possible to run). 3 HYBRID MOM/PO The first hybrid method to be discussed and presented in this paper (historically also the first available in FEKO) is the combination of MoM with PO. In the MoM formulation, current densities on surfaces are approximated by basis functions f n with unknown complex coefficients α n which are obtained by solving a system of linear equations. For electrically large problems (many basis functions required), the size of this system of linear equations might become prohibitive (not fitting any more the computer memory, solution time too long). This is why we proposed in [2] a hybridisation of MoM with PO, where PO currents are also expressed as superposition of RWG basis functions f n [7] Figure 4: Rectangular horn antenna in front of a parabolic reflector. (1) The coefficients α n are derived directly from the PO approximation avoiding the solution of a system of linear equations. The main advantage of the MoM/PO hybrid method is that both methods (rigorous MoM and asymptotic PO) are based on currents and model the same physical quantity, and thus domain decomposition (which region to model with MoM and which with PO) is easily possible even when solving one large scattering object as shown in Fig. 3. Figure 5: Far-field pattern of the horn antenna with reflector as shown in Fig. 4. Method Memory Run-time [h:min:sec] MoM 13.280 GByte 9:22:04 MoM/PO 309 MByte 29:33 MLFMM 461 MByte 22:25 Table 1: Memory and run-time for the horn antenna example with parabolic reflector. 4 HYBRID MOM/UTD Figure 3: Decomposition of a surface into MoM and PO regions with connection between the two. A simple example for the application of the MoM/PO hybrid method is shown in Fig. 4, where a horn antenna illuminates a parabolic reflector of 20λ diameter. In Fig. 5 the far-field pattern is shown comparing the MoM/PO hybrid method with full wave reference solutions based on MoM and MLFMM. Table 1 compares memory and run-time for these three solutions. It must be noted that the larger the reflector the As opposed to PO, UTD is based on rays and not currents, but has also been hybridised with MoM in FEKO. The biggest advantage for UTD is that memory and run-time are frequency independent. Fig. 6 shows the schematic hybrid approach with just one MoM basis function f n in front of a UTD region, where then different ray types are involved (and couple of course also back onto the MoM region). For instance, the electric near-field at any point in space is given by (2) 435
where δ indicates the visibility (direct ray), E is the MoM integral operator and EUTD considers all the UTD contributions (reflections, diffractions etc.). In FEKO also the MoM region can touch the UTD region (similar to Fig. 3 for PO); special MoM half basis functions are then used. Figure 6: MoM basis function fn with a UTD region. ing rays) based on meshed geometries (electrically large triangular patches). As compared to PO, the treatment of multiple interactions is much easier and also the transmission through dielectrics (e.g. lens antennas, radomes), and as compared to UTD (which in FEKO is based on ray-tracing for objects composed of canonical shapes like polygonal plates) arbitrarily shaped objects can be modelled (triangular patch mesh) and also far-field RCS can be computed (avoiding the UTD problem of caustics). For an RCS problem typically this method is used alone in FEKO (not hybridised with MoM), thus to show the hybrid method again a reflector antenna example has been selected, see Fig. 9 (reflector here 36λ diameter). Table 2 shows the savings this GO technique allows, here also faster than PO. Figure 9: Near-field of a reflector antenna solved with MLFMM (left) and MoM/GO (right). Figure 7: Antenna mounted on a satellite body. Method Memory MLFMM MoM/PO MoM/GO 5 098 MByte 97 MByte 2 MByte Run-time [h:min:sec] 1:59:45 25:15 0:39 Table 2: Memory and run-time for the reflector antenna example of Fig. 9. 6 Figure 8: Far-field pattern of antenna on satellite. As an example for the MoM/UTD hybrid method, the antenna mounted on a satellite body shown in Fig. 7 is analysed at an operating frequency of 6 GHz. Fig. 8 compares the results of the MoM/UTD hybrid with a full wave MoM solution as reference. 5 HYBRID MOM/GO (MOM/SBR) FEKO also includes the ray-launching GO (sometimes also referred to as SBR shooting and bounc- HYBRID MOM/FEM The last hybrid method to be discussed in this paper and available in FEKO is a combination of MoM with FEM. Unlike the previous hybrid methods with PO / UTD / GO-SBR, this combination of methods does not aim at solving electrically large problems with some finer details (e.g. antenna on large body). Rather, the MoM/FEM hybrid method is best applied to dielectric problems in open space optionally with some other object nearby (or also far away). See Fig. 10 for an example of a GSM base station (MoM region) with a human phantom (FEM region). Also the MoM is used on the surface of the FEM region as FEM absorbing boundary condition and to couple the two regions (optional, if coupling back is switched off, see Section 2, then the FEM region is terminated 436
with an ordinary absorbing boundary condition). The big advantage of using the MoM/FEM hybrid method for such applications is that the air region between the objects does not have to be meshed (as would be the case for a pure FEM modelling). Method Memory Run-time [min:sec] MoM/FEM 1.51 GByte 6:52 MoM(SEP) 2.01 GByte 14:00 FEM alone 0.04 GByte 0:14 Table 3: Memory and run-time for the waveguide example of Fig. 11. References Figure 10: Human phantom (FEM region) with GSM base station antenna (MoM region). Figure 11: Waveguide filter example from [8]. Figure 12: S 21 for the waveguide filter example in Fig. 11 comparing different FEKO solutions. Another application example for the MoM/FEM hybrid method is shown in Fig. 11: A waveguide filter with dielectric blocks inside (see [8] for dimensions etc.). In Fig. 12 three solutions are compared for S 21 : The MoM/FEM hybrid method (FEM for the dielectric blocks, MoM for the other metallic parts), a standalone MoM (using the SEP = surface equivalence principle for the dielectrics) and also FEM alone (i.e. full inner waveguide meshed with tetrahedra and PEC boundary condition on side walls). Memory and run-times are summarised in Table 1 showing the advantage of FEM versus MoM here. [1] EM Software & Systems S.A. (Pty) Ltd, Stellenbosch, South Africa, FEKO Field Computations Involving Bodies of Arbitrary Shape, Suite 5.5 July 2009, http://www.feko.info. [2] U. Jakobus and F. M. Landstorfer, "Improved PO- MM hybrid formulation for scattering from threedimensional perfectly conducting bodies of arbitrary shape," IEEE Transactions on Antennas and Propagation, vol. 43, pp. 162-169, Feb. 1995. [3] I. P. Theron, U. Jakobus, D. B. Davidson, and F. J. C. Meyer, "Recent progress on moment method / UTD hybridization," in Proceedings of the Progress in Electromagnetics Research Symposium (PIERS), Nantes, p. 459, July 1998. [4] U. Jakobus, M. Bingle, J. Marais, and F. Illenseer, Overview of the latest additions to FEKO Suite 5.4 enabling faster solutions to a larger variety of electromagnetic problems, in 25th International Review of Progress in Applied Computational Electromagnetics (ACES 2009), Monterey, USA, Mar. 2009. [5] U. Jakobus and M. Bingle, "Overview of the hybrid finite element / method of moments technology in FEKO - capabilities and applications," in 9.th International Workshop on Finite Elements for Microwave Engineering, (Bonn, Germany), May 2008. [6] U. Jakobus and J. van Tonder, "Fast multipole acceleration of a MoM code for the solution of composed metallic / dielectric scattering problems," in Advances in Radio Science, vol. 3, pp. 189-194, 2005. [7] S. Rao, D. Wilton, and A. Glisson, Electromagnetic scattering by surfaces of arbitrary shape, IEEE Transactions on Antennas and Propagation, vol. 30, pp. 409-418, May 1982. [8] H. Shigesawa, M. Tsuji, et al., Two-path cut-off waveguide dielectric resonator filters, IEEE Transactions on Microwave Theory and Techniques, vol. 37, pp. 1105-1112, July 1989. 437
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