Full-Wave Analysis of Electrically Large Structures on Desktop PCs Branko Kolundzija 1 , Miodrag Tasic 1 , Dragan Olcan 1 , Dusan Zoric 2 , and Srdjan Stevanetic 2 1 Dept. of EE, Univ. of Belgrade, 11120, Serbia 2 WIPL-D d.o.o., Belgrade, 11070, Serbia kol@etf.rs Abstract—Method of moments applied to surface integral equations combined with higher-order basis functions enables full-wave analysis in frequency domain of complex and relatively large structures. The electrical size of solvable problems can be further extended using different techniques: symmetry of the problem, "smart reduction" of expansion orders, physical optics driven method of moments, iterative methods, multilevel fast multipole algorithm, out-of-core solver, and parallelization at CPU/GPU. Results are presented for: (1) monostatic RCS of cube of side 50A (100 A), (2) beam steering of array of 40 by 40 microstrip patch antennas at 9.2 GHz, and (3) beam steering of 4 by 4 patch antennas at 2 GHz (5 GHz), placed on a 19-m long helicopter. I. INTRODUCTION In the last few decades, there is continuous interest in the analysis of composite metallic and dielectric structures based on the method of moments (MoM) solution of surface integral equations (SIEs). According to basic MoM theory [1],[2] induced currents over metallic surfaces and equivalent currents over material boundary surfaces are approximated by series of known basis functions multiplied by unknown coefficients and the SIE is transformed from the linear operator form into the system of linear equations, which is solved for unknown co- efficients. By increasing the number of unknown coefficients, TV, the memory occupation and matrix-fill time increase as N 2 and the matrix-solution time (in case of direct methods as Gaussian elimination or LU decomposition) increases as TV 3 . Hence, the size of the solvable problem in terms of N is limited by memory and time resources of the computer used for the simulation. For example, 8 GB of operative memory on personal computers enables solution of systems up to about ^Vmax = 30000 unknowns. However, the electrical size of the solvable problem (in A 2 of surface area) is also dependent on flexibility of basis functions. In case of the Rao-Wilton- Glisson (RWG) basis functions, typical edge length of triangles is A/10, resulting in about 300 unknowns per A 2 for metallic surfaces [3]. In particular, if 7V max = 30000, the maximum electrical size is limited to about 100A 2 . Many techniques have been developed to increase the electrical size of the solvable structure within the limits of PC computer resources. In what follows, we shall focus on techniques applied in this paper, which are implemented in commercial software package WIPL-D Pro v9.0 [4]. Gen- erally, these techniques can be grouped into three classes. In the first class, there are techniques, which decrease the number of unknowns: (a) usage of higher-order basis functions (HOBFs) [5],[6], (b) exploiting symmetry of the problem [4], (c) "smart reduction" of expansion order [7], and (d) usage of macro-basis functions by physical optics (PO) driven MoM [8]. In the second class, there are techniques that decrease the memory resources and matrix-fill/solution time for given number of unknowns: (a) iterative techniques [9]-[12], and (b) fast multipole method (FMM) and multilevel fast multipole algorithm (MLFMA) [13]—[16]. Finally, in the third class, there are techniques that enable efficient usage of nowadays hardware resources: (a) out-of-core solution of matrix equation [17], (b) parallelization at CPU based on OpenMP [18], and (c) parallelization at GPU based on CUDA [19],[20]. The goals of the paper are: (1) to shortly present various techniques for increasing the electrical size of the solvable structure within the limits of PC computer resources, (2) to compare these techniques and discuss optimal usage of these techniques, and (3) to show numerical results for some typical electrically large structures. II. TECHNIQUES THAT REDUCE NUMBER OF UNKNOWNS A. Higher-Order Basis Functions (HOBFs) The basic way to decrease the number of unknowns is to use HOBFs. For interpolatory HOBFs defined over triangles [6], the maximum edge length of triangles can be extended to 0.5 ~ 1A, resulting in 40-70 unknowns per A 2 [14]; for polynomial HOBFs defined over quadrilaterals [2],[5] the maximum edge can be extended to 1 ~ 2A, resulting in 20-35 unknowns per A 2 [5]. In both cases, the expansion orders are chosen according to electrical size of patches. Thus electrical size of solvable structure within limit of computer resources is increased for an order of magnitude when compared with RWG basis functions. B. Usage of Geometrical Symmetry of the Problem In case when the geometry of a structure is symmetrical with respect to one plane, and the excitation is either sym- metrical or anti-symmetrical with respect to this plane, the unknown coefficients on one side of the plane are equal to the coefficients from another side of the plane multiplied by ±1, so that original number of unknowns is halved. The memory requirements are decreased four times, the matrix fill time is halved and the matrix solution time is decreased eight times. 978-1-4577-1686-4/11/$26.00 ©2011 IEEE 122