Avaiabe onine at www.sciencedirect.com Procedia Engineering 9 (0) 45 49 0 Internationa Workshop on Information and Eectronics Engineering (IWIEE) Two Kinds of Paraboic Equation agorithms in the Computationa Eectromagnetics Kong Meng a,chen Mingsheng a,hu Yujuan b,wu Xianiang a a Department of physics and Eectronic Engineering,,Hefei Norma University,Hefei,3060,China. b Department of Pubic Computer teaching,hefei Norma University,Hefei,3060, China. Abstract Sip-step Fourier Transform (SSFT) and Finite Difference (FD) agorithms are introduced to sove Paraboic Equation (PE). SSFT is appied to anayze the probem of radio propagation in vacuum. Based on Spit-step Padé approximation, a high order FD agorithm is introduced to compute the Radar Cross Section (RCS). According to the resuts, the differences of the two kinds of PE method have been discussed. 0 Pubished by Esevier Ltd. Seection and/or peer-review under responsibiity of Harbin University of Science and Technoogy. Open access under CC BY-NC-ND icense. Keywords: Paraboic Equation, Sip-step Fourier Transform, Finite Difference, Radio Propagation, Radar Cross Section. Introduction As an approximated form of the wave equation, the Paraboic Equation (PE) method modes energy propagating in a cone centered on the paraxia direction. Recenty, many researchers have appied the paraboic equation to sove the probems of radio propagation and eectromagnetic scattering []. It was reaized that PE method may aso be adopted as an efficient numerica too of eectromagnetic fied cacuation, and it may bridging the gap between rigorous numerica methods ike the method of moments (MOM) or finite difference time domain (FDTD) [], and asymptotica methods based on ray-tracing or physica optics (PO). Using PE method can avoid the imitation of Corresponding author. Chen Mingsheng Te.: 3866394; E-mai address: cmsh@ahu.edu.cn 877-7058 0 Pubished by Esevier Ltd. doi:0.06/j.proeng.0..735 Open access under CC BY-NC-ND icense.
46 Kong Meng et a. / Procedia Engineering 9 (0) 45 49 CPU time and memory required by these rigorous methods. It aso can reduce the error caused by the high-frequency approximation. Recenty, the usua toos for soving the PE are spit-step Fourier transform (SSFT) and finite difference (FD) agorithms. SSFT using the FFT technique is a frequency-domain agorithm, when it is used to sove the PE, the step size of SSFT is amost free from the restrictions and can be seected as a arge discritized eement. Therefore, SSFT agorithm of PE is suitabe for arge-scae wave propagation probems. SSFT agorithm is compicated when deaing with irreguar boundary, so it is not easy to be used in the cacuation of eectromagnetic scattering probems with compex structures. FD agorithm is carried out through the mesh of the computationa domain, and accompishes the computation according to the eectromagnetic fied on the adjacent grid points. Since precise mesh is used, FD agorithm can directy cacuate the fied on arbitrary boundaries, therefore it is more convenient to dea with irreguar boundaries, but the discritized step in FD agorithm is restricted by the waveength. Therefore one must take very sma step, which wi resut arge-scae matrix operations. Soving the high-frequency, arge-scae wave propagation probems based on PE, FD agorithm is sow and the computer memory consuming is aso high. FD agorithm is mainy used to cacuate the target RCS and forecast the eectromagnetic distribution of urban and other sma-scae wave propagation. [3]. SSFT agorithm of Paraboic Equation As we know, froward paraboic equation can be soved by marching techniques and has the forma soution as [4] ikδx( Q ) u( x+δ xz, ) = e u( xz, ) () We assume the wave function of pane wave is of the form as beow (, ) exp( ( cosα ) sin α) u x z = ikx + ikz () Equation () can be approximated by using first-order Tayor expansions of the square root Q, which yieds the Standard Paraboic Equation (SPE) when set n=, one wi get u u + ik + k ( n ) u = 0 z x (3) u u + ik = 0 z x we referred equation (4) as SPE in vacuum. Using the properties of the Fourier transform, we can write the Fourier transform of Equation (4) as U( x, f) 4 π f U ( x, f ) + ik = 0 x This is now an ordinary differentia equation which can be soved in cosed form, giving f x U( x, f) e k U(0, f) We can write the inverse Fourier transform of Equation (4) as = (6) (4) (5)
Kong Meng et a. / Procedia Engineering 9 (0) 45 49 47 where u(0,z) is initia fied. The forma soution of SPE is given as f x uxz (, ) e k = I I( u(0, z)) ik x Q Δ u x+δ xz = e u xz (8) (, ) (, ) Then one wi get the compete spit-step Fourier transform soution of equation (8) Where Δx is step. 3.FD agorithm of Paraboic Equation 3. Spit-step Padé PE f Δx ux ( xz, ) e k +Δ =I I( uxz (, )) In this paper, we approximate the wide-ange/exponentia pseudo-differentia operator directy with a sum of Padé(,) functions as ( Q ) N ( ) ~ b ( Q ) ikδx Q ikδx + + a Q e = e + (0) = + in which ccompex coefficients (a, b) are determined by the constraints of Tayor series expansion and stabiity conditions, whie its accuracy is aso dependent on step Δ x size. The approximation accuracy equivaent Padé (N, N), which has N-order accuracy [5]. Substituting (0) into equation (), we wi get the form of SSPPE: where N u( x+δ xz, ) = u( xz, ) + v( x xz, ) +Δ () = ( +Δ, ) = + ( ) ( ) (, ) v x xz b Q a Q u xz Discretizing in z, we obtain finite-difference equations for the where j j v = v( x+δx, jδ z), s = kδ z, u = u( x, jδ z) v s in the form: { } j+ j s j ( ( ) ) a j+ j j v + v + + s n v = u + u + ( s ( n ) ) u b b Combined Equation () and (3), fied can be obtained recursivey. To simuate the infinite space of the eectromagnetic scattering probem, we added Non-oca boundary conditions [6] (NLBC) as the truncation boundary conditions. (7) (9) () (3)
48 Kong Meng et a. / Procedia Engineering 9 (0) 45 49 3. Near-Fied/Far-Fied Transformation Recaed that the -D bistatic RCS of an object in direction θ is given as ( cos, sin ) r s ψ r θ θ σ( θ) = im πr i r ψ ( rcos θ, rsin θ) (4) If the incident fied is a pane wave with unit ampitude, after compicated mathematics derives, equation (4) can be expressed as s ikzsinθ σ ( θ ) = k cos θ u ( x, z) e dz (5) In which x 0 can usuay be chosen as 0λ (λ is the waveength of incident wave). 4.Numerica Resuts 4. Large-scae radio wave propagation + We assume the incident fied is of the form in equation (), and the frequency of wave is 0.3GHz in a vacuum. The rea and imaginary parts of the fied u(x, z) were cacuated using the SSFT method,and set α = 0 o,x=000m, Δ x =00 m. The computation is accompished on a Pentium IV 3.0GHz computer using Matab7.0, the CPU time for this simuation is 0. seconds, and the resuts is shown in Figure. It can be seen that the resut of SSFT method agrees we with that of anaytica resuts. When the FD method is used to hande the probem, it is difficut to compete the operation because of the imited step and arge-scae matrix operations. Due to arge step desirabe, SSFT method is more appropriate for deaing with the probem. 0 Figure. (a) Figure. (b) Figure. Radio waves propagation in cacuating the vacuum vaue with SSFT :(a)the rea part of fied,(b) the imaginary part of fied 4. Eectromagnetic scattering of eectricay arge objects In this numerica exampe, the incident fied is a pane wave with its ampitude set to be.0, the incident source with its waveength of 0. m is horizonta poarized and propagating in a vacuum, which iuminated a PEC circuar cyinder of radius 5λ. The computation domain is set to be X Z=[0, 0λ] [0, 40λ], we use PML absorbing boundary conditions at the top and bottom of the domain. As shown in Fig. the ampitude of the scattered near fied is computed with the SPE and SSPPE (N=4) methods. In Fig.3 bistatic RCS resuts obtained by SSPPE method is compared with the resuts of method of moments (MOM), they agree we from 0 to 50. And the CPU consumed by this exampe is 5.8 seconds. When we use the SPE method [4] to hande the same probem, the CPU time is 3.0s, but
Kong Meng et a. / Procedia Engineering 9 (0) 45 49 49 the anges accuracy up to 5.To obtain the fu bistatic RCS, we need ony two rotated PE runs for SSPPE whie in SPE method the number of rotation is six [4]. The resuts which show in Figure 4 proved the efficiency of the present method. Figure.(a) Figure.(b) Figure.3 Figure.4 Figure. The ampitude of the scattered near fied computed with the SPE (a) and SSPPE(b). Figure.3 Bistatic RCS of PEC radius 5λ Cyinder. Figure.4 Bistatic RCS of PEC radius 5λ cyinder obtained with rotation paraboic axis. 5.Concusions Two agorithms for soving Paraboic Equation, Sip-step Fourier Transform and Finite Difference, are presented and compared. Meanwhie the appication of the two agorithms to radio propagation and eectromagnetic scattering probems is discussed. According to numerica resuts, we suggest to hande arge-scae radio wave propagation probems with SSFT method to obtain good resuts, whie FD method is suitabe for deaing with eectromagnetic scattering probems. Acknowedgement This work is supported by the Key Program of Nationa Natura Science Foundation of China (609300) and Science and Technoogica Fund of Anhui Province for Outstanding Youth (0040606Y08) and the Key Scientific Research Base program of Hefei Norma University(0jd06) and partiay by the Natura Science Foundation of the Anhui Higher Education Institution of China (KJ0A40). References [] Levy M F,Borsboom P P.Radar cross-section computations using the paraboic equation method.eectron Lett;996,3(3),p.34 36. [] A. Tafove and S.C. Hagness. Computationa eectrodynamics: the finite-difference time-domain method. Norwood MA, Artech House, 000. [3] Zaporozhets A A, Levy M F.Bistatic RCS cacuations with the vector paraboic equation method.ieee Trans on AP,999,47(),p.688-696. [4] M. F. Levy. Paraboic equation methods for eectromagnetic wave propagation. London: The Institution of Eectrica Engineers, 000. [5] Coins M D, Generaization of the spit-step padé soution.acoust.soc.amer,994,96,p.38-385. [6] Dan Givoi. Non-refecting Boundary Conditions. Journa of Computationa Physics, 99; 94, p.-9.