Hybrid Methods

Many practical problems are too complicated to be solved accurately by a single numerical or asymptotic method. It is often advantageous to combine two numerical modeling techniques in a single field solver in order to take advantage of the strengths of each technique to solve problems that neither technique alone could model efficiently. For example, a finite element method can be combined with a boundary element method to form a more powerful hybrid numerical technique for analyzing both open-region problems and complex inhomogeneous objects. A full-wave numerical technique can also be combined with an asymptotic method to model very large objects with small features that require detailed analysis, such as an antenna mounted on an airplane.

Hybrid methods are not simply two separate modeling codes with a common user interface. A hybrid method generally divides a problem into two parts and applies a different technique to each part while matching the currents or fields at the boundary to ensure a unique solution. Some hybrid codes solve one part first, and then use the boundary fields as the sources when solving the second part. Other hybrid codes solve both parts simultaneously allowing the solution of each part to influence the solution of the other.

Formulations for many hybrid techniques have been developed and reported in the literature. Some of the more common (and useful) hybrid techniques are described in the following sections.

Hybrid FEM/BEM

Finite element methods excel at modeling complex volumetric structures, but are weak when it comes to modeling thin wires and unbounded radiation problems. Boundary element methods excel at modeling wires and unbounded geometries, but do not model complex structures that include a variety of materials well. The complementary strengths of these two methods make them ideal candidates for hybridization. Full-wave FEM/BEM (also called FEM/MOM, FE-BE and FE-BI) techniques have been successfully used to model many problems that could not be modeled effectively using either of the two techniques alone [1-8].

Hybrid FEM/BEM techniques introduce a fictitious surface (which may or may not coincide with an actual material surface) that separates an interior volume from an exterior volume. The interior region is analyzed using a finite element method with unknown electric or magnetic surface currents establishing the boundary condition on the outer surface. The exterior region is analyzed using a boundary element method, with unknown electric or magnetic currents on the fictitious surface. Two sets of matrix equations are developed that share unknowns on the boundary between the interior and exterior volumes. By forcing the fields on both sides of the fictitious surface to be consistent with each other, the two matrix equations can be combined into one larger equation with a unique solution.

In practice, it is relatively inefficient to generate one large matrix that is partly dense and partly sparse. Inward looking techniques repeatedly solve the finite element portion of the problem, while populating the boundary element method matrix. Outward looking techniques repeatedly solve the boundary element method portion while populating the finite element matrix. Choosing the right hybrid technique for a particular application greatly increases the efficiency of a hybrid FEM/BEM approach [9].

Hybrid MOM/GTD, MOM/PO

Asymptotic methods can deal with objects whose overall dimensions are large in terms of the wavelength. However, if large objects contain features that are too fine to be analyzed by an asymptotic method, it becomes necessary to employ hybrid methods that combine asymptotic techniques with numerically rigorous methods. Techniques that combine a moment method with an asymptotic method can be broadly categorized as either ray-based or current-based. Ray-based methods, such as MOM/GTD, provide a considerable speed advantage, but can be difficult to implement for arbitrary and complex objects [18].

Hybrid MOM/GTD techniques were first described in the 1970s [10-11]. Since GTD fails in regions where the field cannot be approximated by a local plane wave, uniform solutions have been developed that overcome some of the limitations of GTD. Hybrid approaches combining MOM and UTD are discussed in [12-14].

Physical optics is a current-based asymptotic method. The hybridization of PO with MOM has two advantages over the combination of ray-based methods and MOM. First, since both MOM and PO are current-based, they can be easily blended on the same surface. Second, MOM/PO is relatively general in that there are no specific restrictions on the geometries that can be modeled [15]. Consequently, this hybrid technique has received considerable attention in the literature [15-22]. It has also been implemented in a commercial numerical code [23].

Hybrid FEM/PO

Hybrid methods that combine high-frequency asymptotic techniques with the method of moments are not suitable for solving problems with inhomogeneous or anisotropic materials. These types of problems are better suited for analysis by a hybrid FEM/PO technique. The hybridization of FEM and asymptotic techniques is described in [24-25].

References

[1] T.-K. Lee, S.-Y. Lee and J.-W. Ra, "A hybrid finite element-boundary element method and its application to inverse scattering," Proc. IEEE Antennas and Prop. Symp., vol. 3, pp. 1628-1631, June 1989.

[2] W. E. Boyse and A. A. Seidl, "A hybrid finite element and moment method for electromagnetic scattering from inhomogeneous objects," Proc. 7th Ann. Rev. Progress Appl. Computat. Electromag., Mar. 1991.

[3] J. M. Jin and John L. Volakis, "Electromagnetic scattering by and transmission through a three dimensional slot in a thick conducting plane," IEEE Trans. Antennas Propagat., vol. 39, pp. 543-550, April 1991.

[4] K. D. Paulsen, D. R. Lynch, and J. W. Strohbehn, "Three-dimensional finite, boundary, and hybrid element solutions of the Maxwell's equations for lossy dielectric media," IEEE Trans. on Microwave Theory and Tech., vol. 36, pp. 682-693, April 1988.

[5] J. Shen and A. Kost, "Hybrid FE-BE method for EMC problems in power cable systems," IEEE Trans. on Magnetics, vol. 32, no. 3, pp. 1493-1496, May 1996.

[6] Y. Ji, J. Chen, T. Hubing, J. Drewniak, "Application of a hybrid FEM/MOM method to a canonical PCB problem," Proc. 1999 IEEE Int. Symp. Electromagn. Compat., Seattle, WA, August 1999, pp. 91-96.

[7] Y. Ji and T. H. Hubing, "EMAP5: A 3D hybrid FEM/MOM code," Journal of the Applied Computational Electromagnetics Society, vol. 15, no. 1, March 2000, pp. 1-12.

[8] K. Zhao, M. Vouvakis and J.-F. Lee, "Solving electromagnetic problems using a novel symmetric FEM-BEM approach," IEEE Trans. on Magnetics, vol. 42, no. 4, pp. 583-586, April 2006.

[9] Y. Ji, H. Wang and T. Hubing, "A novel preconditioning technique and comparison of three formulations for hybrid FEM/MOM methods," Journal of the Applied Computational Electromagnetics Society, vol. 15, no. 2, Aug. 2000, pp. 103-114.

[10] W. D. Burnside, C. L. Yu, and R. J. Marhefka, "A technique to combine the geometrical theory of diffraction and the moment method", IEEE Trans. Antennas Propagat., vol. 23, no. 4, pp. 551-558, July 1975.

[11] G. A. Thiele and T. H. Newhouse, "A hybrid technique for combining moment methods with the geometrical theory of diffraction," IEEE Trans. Antennas Propagat., vol. 23, no. 1, pp. 62-69, Jan. 1975.

[12] U. Jakobus and F. M. Landstorfer, "A combination of current-and ray-based techniques for the efficient analysis of electrically large scattering problems," Proc. 13th Annu. Rev. Progress Appl. Computat. Electromagn., Monterey, CA, Mar. 1997, pp. 748-755.

[13] I. P. Theron, D. B. Davidson, and U. Jakobus, "Extensions to the hybrid method of moments/uniform GTD Formulation for sources located close to a smooth convex surface," IEEE Trans. Antennas Propagat., vol. 48, no. 6, pp. 940-945, June 2000.

[14] I. P. Theron, U. Jakobus, I. B. Davidson, and F. J. Meyer, "Recent progress on moment method / UTD hybridization," Proc. of PIERS98 - Progress in Electromagnetics Research Symposium, Nantes, France, vol. 1, p. 459, July 1998.

[15] H. J. Bilow, "Scattering by an infinite wedge with tensor impedance boundary conditions - a moment method/physical optics solution for currents," IEEE Trans. Antennas Propagat., vol. 39, pp. 761-773, June 1991.

[16] L. N. Medgyesi-Mitschang and D.-S. Wang, "Hybrid solutions for scattering from perfectly conducting bodies of revolution," IEEE Trans. Antennas Propagat., vol. 31, p. 570, 1983.

[17] M. Kaye, P. K. Murthy, and G. A. Thiele, "An iterative method for solving scattering problems," IEEE Trans. Antennas Propagat., vol. 33, pp. 1272-1279, 1985.

[18] D. S. Wang, "Current-based hybrid analysis for surface wave effects on large scatterers," IEEE Trans. Antennas Propagat., vol. 39, pp. 839-850, June 1991.

[19] C. S. Kim and Y. Rahmat-Samii, "Low profile antenna study using the physical optics hybrid method (POHM)," Proc. IEEE Int. Symp. Antennas Propagat. Soc. Meet., London, ON, Canada, June 1991.

[20] U. Jakobus and F. M. Landstorfer, "Improvement of the PO-MM hybrid method by accounting for effects of perfectly conducting wedges," IEEE Trans. Antennas Propagat., vol. 43, pp. 1123-1129, Oct. 1995.

[21] R. E. Hodges and Y. Rahmat, "An iterative current-based hybrid method for complex structures," IEEE Trans. Antennas Propagat., vol. 45, pp. 265-276, Feb. 1997.

[22] Johan Edlund, "A parallel, iterative method of moments and physical optics hybrid solver for arbitrary surfaces," UPPSALA University, Department of Information Technology, August, 2001.

[23] D. B. Davidson, I. P. Theron, U. Jakobus, F. M. Landstorfer, F. J. C. Meyer, J. Mostert, J. J. Van Tonder, "Recent progress on the antenna simulation program FEKO", Proc. of the 1998 South African Symposium on Communications and Signal Processing, 1998. COMSIG '98, pp. 427-430, 7-8 Sept. 1998.

[24] J. M. Jin, S. Ni, and S. W. Lee, "Hybridization of SBR and FEM for scattering by large bodies with cracks and cavities," IEEE Trans. Antennas Propagat., vol. 43, pp. 1130-1139, Oct. 1995.

[25] Dong-Ho Han, A. C. Polycarpou, C. A. Balanis, "FEM-based hybrid methods for the analysis of antennas on electrically large structures," Radio and Wireless Conference, 2000. RAWCON 2000, pp. 59-61.