Scattering of an Object with Impedance Surfaces Using IPO and MLFMM Ehsan Rashidi Ranjbar and Mojtaba Dehmollaian School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran. e.rashidi.r@ut.ac.ir m.dehmollaian@ece.ut.ac.ir Abstract— Scattering from an electrically large object having impedance boundaries is evaluated using the approximate high frequency technique, iterative physical optics (IPO) together with the multilevel fast multipole method (MLFMM), a fast hybrid technique. For example, radar cross section (RCS) of an airplane with size of about 78 85 28 λ λ λ × × λ being the wavelength at the operation frequency of 1GHz is computed within 20 minutes. The result is verified with that of method of moment (MoM)- MLFMM, carried out in about 5.5 hours. Keywords: RCS computation, Approximate high-frequency techniques, MLFMM I. INTRODUCTION The radar cross section (RCS) of a complex target (e.g., airplane) plays an important role in its detection and identification. Fast and accurate computational methods make RCS reduction and radar imaging possible. Due to the large size of such targets and multiple interactions involved, this problem is a time consuming task. Full-wave techniques such as method of moments (MoM) require N 2 operations for computation of a vector-matrix product where N resembles number of unknowns. However, using multilevel fast multipole method (MLFMM) this is reduced to O(NlogN)[1]. In this paper, combination of the MLFMM with an iterative physical optics (IPO) technique is presented to enhance the speed of calculations. Unlike [2, 3], in this paper, targets with non-zero surface impedances are considered. This method is capable of computing the multiple interactions in an object (e.g., jet engine) efficiently and is used for a wide range of complex targets [3-6]. II. THEORIES In [7] hybrid IPO and FMM is proposed for efficient calculations. Here, the method is modified by hybridization of IPO and MLFMM for even faster computations. In the following, a brief review of the IPO and MLFMM is given. A. Itertive Physical Optics The IPO is based on the physical optics (PO). In each iteration, it attempts to improve the surface current densities by using the magnetic field integral equation (MFIE). The number of iterations depends on the number of important reflections involved [8]. IPO was originally used for PEC open-ended cavities [8]. Later, this method was used for non- PEC cavities for which both electric and magnetic current densities J G and M G exist [9]. The MFIE is given by [9] ( ) 0 0 0 0 2 2 [ ] 2 [ ] [ ˆ [ ˆ ] 1 ] ˆ i s S s S J r n H n LJ n KM LJ J g dS KM M g dS jk η = × + × + × = ×∇ = ∇× ×∇ ∫ ∫ G G G G G G G G G (1) where ˆ n is the outward normal unit vector, i H G is the incident magnetic field and 0 g is the scalar free-space Green’s function. Equation (1) is solved iteratively by using the impedance boundary condition (IBC) given by [7] ˆ s M ZJ n = × G G (2) where s Z is the surface impedance. To do this, first the zeroth-order PO currents are estimated as those of an infinitely large resistive flat plate. For a ˆ ˆ ˆ ( , ) i i h v p ∈ polarized incident plane-wave ˆ exp( ) . i E p k r = G G G propagating along i k G and for a point r s G on a facet of the object, the zeroth-order equivalent surface electric and magnetic current densities are (0) 0 1 ˆ ˆ ˆ ˆ [( )(1 )( ) ˆ ˆ ˆ ˆ ( )(1 )( )] i s i v i ik r i h i J n h R pv Z n v R ph e ⋅ = × + ⋅ - × - ⋅ G G G (3) (0) ˆ ˆ ˆ ˆ [( )(1 )( ) ˆ ˆ ˆ ˆ ( )(1 )( )] i s i h i ik r i v i n h R ph n v R pv e M ⋅ =- × + ⋅ + × - ⋅ G G G (4) where ˆ ˆ ˆ ˆ ˆ | | i i i h k n k n = × × , ˆ ˆ ˆ i i i v h k = × and v R and h R are respectively the vertical and horizontal reflection coefficients