Efficient RCS Computation Over a Broad Frequency Band Using Subdomain MoM and Chebyshev Approximation Technique

The analysis of the target’s broadband electromagnetic scattering characteristics plays an important role in radar stealth and recognition areas. The traditional method of moments (MoM) needs to calculate the currents at each frequency point when analyzing the target. With the increase of the target electrical size and complexity, the radar cross section (RCS) of the target changes drastically with frequency, it is necessary to calculate the accurate frequency response with a small frequency interval, which is very time-consuming in calculation and not feasible. This paper proposes to combine the subdomain method of moments (SMoM) with the Chebyshev approximation technique (CAT) to efficiently analyze the electromagnetic scattering of arbitrary shaped objects over a broad frequency band. In the SMoM technique, the whole object can be divided into several regions, each of which can be independently solved by the method of moments (MoM). The CAT technique is employed to expand the unknown current coefficients into a Chebyshev series for achieving fast frequency sweeping. Instead of direct point-by-point simulations, it is only necessary to calculate the currents by SMoM at several Chebyshev-Gauss frequency sample points. Furthermore, the Chebyshev series are matched via the Maehly approximation to a rational function to improve the accuracy. Numerical examples show that the hybrid technique can greatly reduce the computation time without loss of accuracy.


I. INTRODUCTION
Accurate radar cross section (RCS) prediction of complex objects over a broad frequency band plays an important role in radar imaging, target detection and recognition areas. The method of moments (MoM) has been a very useful tool in the past decades to solve electromagnetic scattering problems. However, the memory requirement is O(N 2 ) and the computational complexity is O(N 3 ) for a direct solver and O(N 2 ) for an iterative one in the traditional MoM, where N is the number of unknowns. For complex objects, the convergence of MoM based on an iterative solver may be very slow since The associate editor coordinating the review of this manuscript and approving it for publication was Mehmet Alper Uslu. the electrically large objects with small features will lead to the non-uniform meshes.
In order to efficiently analyze the multi-scale problems and the non-uniform meshes problems, many algorithms have been developed in recent years. One is a variety of precondition techniques, such as Block-diagonal [1], Incomplete LU (ILU) [2], Sparse approximate inverse (SAI) [3], Incomplete-Cholesky (IC) [4], and so on. The other is the Characteristic Basis Function methods (CBFM) [5], [6], the integral equation domain decomposition method (IE-DDM) [7] and subdomain MoM (SMoM) [8]. Among these methods, the SMoM introduces the idea of domain decomposition method (DDM), the electrically large size objects with small features is divided into several regions, each of which can be solved independently by MoM. Since only one subdomain needs to be calculated at a time, the amount of storage can be drastically reduced. However, if the wide-band RCS computation is of interest, the SMoM requires to repeatedly solving the electric field integral equations (EFIEs) at each frequency point, which leads to a lot of computation time. The asymptotic waveform evaluation (AWE) [9] and model-based parameter estimation (MBPE) [10] are proposed to achieve fast frequency sweeping. In the AWE and MBPE technique, the current coefficients in wide-band frequency response are interpolated by a low-order rational function, namely Padé approximation. Some other model order reduction (MOR) techniques, such as the impedance matrix interpolation technique [11] and the Cauchy method [12], are applied for fast frequency or angular sweeping as well. Recently, the Chebyshev approximation technique (CAT) is present to interpolate the frequency response by the Chebyshev polynomials [13]- [17]. Among the above techniques, the AWE is quite popular and applicable to the CBF and MoM its fast solvers [18]- [21]. However, the high-order derivative impedance matrices require to be calculated and stored first, which make the procedure difficult to implement and leads a lot of memory costs. While the CAT is much easier to integrate into the SMoM code and it does not require computing the high-order derivative matrices.
In this paper, a novel hybrid SMoM-CAT is proposed to achieve fast wide-band RCS prediction of arbitrary 3D objects. In the proposed SMoM-CAT technique, the whole object is divided into several subdomains and the initial surface currents on these subdomains are obtained at several Chebyshev-Gauss frequency points. The modified excitation vectors of each subdomain can be updated by coupling the other subdomains' contributions and the new currents on these subdomains are recomputed at these Chebyshev-Gauss points. An iterative procedure is implemented until the desired accuracy is satisfied. Finally, the currents at any frequency can be obtained by the Maehly approximation. Distinguished from the prior publications [7], [8], [13]- [17], the implementation of SMoM-CAT method is presented as follows: 1) The SMoM technique introduces the artificial surfaces. Therefore, the currents on the real surfaces and the artificial surfaces over a broad frequency band are explored.
2) Both the currents on the real surfaces and artificial surfaces are employed to compute the coefficients of Chebyshev series and Mahely approximation.
3) The wide-band RCS of SMoM-CAT with different orders L are explored and the memory and computation time of SMoM-CAT with different orders L are discussed.
The remainder of this paper is organized as follows. In Section II, the method of establishing the electric field integral equation (EFIE) for connected objects and the principle of SMoM algorithm are introduced; the formula and solution procedure of the proposed SMoM-CAT algorithm are given. Numerical results are presented in Section III to demonstrate the accuracy and efficiency of the hybrid technique and the conclusions are provided in Section IV.

A. ELECTRIC FIELD INTEGRAL EQUATION (EFIE)
For simplicity, we consider a combined object consisting of two cones which is illuminated by an incident plane wave E i , as shown in Fig. 1(a). In the proposed SMoM-CAT method, first of all, we divide the whole domain into two separated subdomains 1 + 12 and 2 +¯ 21 , as shown in Fig. 1(b). Two artificial surfaces which are denoted by the blue part are introduced to close these two subdomains. J 1 , J 12 , J 21 , and J 2 are the electric currents on the surfaces ∂ 1 , ∂ 12 , ∂ 21 , and ∂ 2 , respectively.
For subdomain 1, the electric field integral equation (EFIE) is given by where where the subscript α stands for 1, 12, 21, and 2. Similarly, for subdomain 2, the EFIE is given by For a numerical solution of the EFIEs, these two subdomains can be discretized into a number of small triangular patches, and the equivalent currents on each subdomain can be expanded by the RWG basis functions f k (r): I 21,k f 21,k (r), r ∈¯ 12 (5) where N 1 , N 12 , N 2 , N 21 are the number of basis functions, I 1,k , I 12,k denote the current coefficients on the real surface and artificial surface of subdomain 1, respectively, I 21,k , I 2,k stand for the current coefficients on the real surface and artificial surface of subdomain 2. Substituting (4), (5) into (1), (3) and applying the Galerkin's procedure results in the following linear systems for each subdomain: whereĨ 1 ,Ĩ 2 are the column vectors with the size of (N 1 + N 12 ) × 1, (N 2 + N 21 ) × 1 for the unknown amplitudes of the basis functions on subdomain 1 and subdomain 2, respectively. For subdomain 1, the impedance matrixZ 1 , the excitation vectorṼ 1 ,and the modified excitation vector Ṽ 1 can be expressed as: The elements of Z 1,1 , Z 1,12 , Z 12,1 , Z 12,12 , V 1 , V 12 , V 1 , V 12 are given by: where α denotes the subscript 1, 12, and β also stands for the subscript 1, 12.
Next, the Q + 1 Chebyshev-Gauss frequency pointsk q which are the roots of the Chebyshev polynomial T Q+1 (x) are computed and then the Chebyshev-Gauss sampling points k q are obtained by transforming thek q from the interval [−1, 1] to the desired band [k a , k b ] as: where q = 0, 1, . . . , Q. And Q denotes the truncated order of the Chebyshev series. Secondly, the initial currentsĨ at the certain Q + 1 Chebyshev-Gauss frequency points can be calculated by MoM: where the subscript α stands for 1, and 2. Next, the boundary conditions are enforced on the artificial surfaces¯ 12 and¯ 21 by: Then the new Q + 1 excitation vectorsṼ 1 1 k q andṼ 1 2 k q of subdomain 1 and subdomain 2 can be updated by: Thirdly, the new currentsĨ 1 1 k q = I 1 1 k q , I 1 12 k q T on 1 +¯ 12 andĨ 1 2 k q = I 1 2 k q , I 1 21 k q T on 2 +¯ 21 at the certain Q+ 1 frequency points can be obtained by: Next, the iterative procedure is carried out from (21) to (26) by using ''i'' instead of ''1''. The convergence criterion is that the maximum errors of the currents at Q + 1 frequency points on two subdomains are checked whether they are satisfied the desired accuracy: Fourthly, the currentsĨ 1 (k) andĨ 2 (k) on 1 +¯ 12 and 2 +¯ 21 at any frequency point can be approximated as: where T q (x) denotes the Chebyshev polynomial which satisfies the recursion relation in [8], m α,q are the expanding coefficients. The elements of m 1,q on 1 +¯ 12 can be expressed as: where N 1 and N 2 are the number of unknowns on 1 and¯ 12 , respectively.
The elements of m 2,q on 2 +¯ 21 can be obtained by: Finally, the Maehly approximation is employed to achieve a better accuracy by a rational function: where L and M are the order of unknown coefficients a α,β,i and b α,β,j , respectively.
The coefficients in the numerator and the denominator of the Maehly approximation on 1 +¯ 12 are obtained by: A flow chart is shown to help readers better understand the procedure of the hybrid SMoM-CAT method, as illustrated in Fig. 2.

III. NUMERICAL RESULTS
In this section, three examples are presented to show the accuracy and efficiency of the proposed SMoM-CAT method for fast wide-band scattering analysis of arbitrarily shaped 3D object. All the computations were carried out serially on a PC with 4.2 GHz Intel CPU and 8GB RAM.

A. A FOUR PATCH ARRAY
To show the accuracy of the SMoM-CAT algorithm for sparse objects, the first example we consider is a four PEC patch array, and each patch element has a size of 2.5 cm × 2.5 cm, as shown in Fig. 3. The gap between the two adjacent elements is 0.5 cm. The incident direction of the plane wave is θ inc = 60 • and ϕ inc = 0 • . In the SMoM and SMoM-CAT analysis, the whole array is divided into four subdomains. Fig. 4 shows the θ θ polarized monostatic RCS from 2 GHz to 12 GHz by using three different orders of SMoM-CAT (L = 1, 3, 5). The numerical results of SMoM-CAT are compared to that of point-by-point SMoM simulations, we can see that the 1 st order approximation cannot get the correct VOLUME 8, 2020 results, and the 3 rd order approximation becomes better but is still not accurate around 8.5 GHz. The 5 th order approximation can produce satisfactory results over the entire band. To further verify the accuracy of the proposed SMoM-CAT algorithm for sparse objects, we extract the currents at the non-Gaussian sampling point calculated by the SMoM-CAT method to calculate the bistatic RCS and compare it with the    the incident direction of the plane wave is θ inc = 60 • and ϕ inc = 0 • , the bistatic acceptance angle is θ sca = 0 ∼ 360 • and ϕ sca = 0 • , the polarization is θθ polarization. Compare its bistatic results with the results of MoM calculations we can see that the two are well fitted, and the accuracy of the proposed algorithm for sparse objects is further verified.

B. A MISSILE MODEL
To show the accuracy of the SMoM-CAT algorithm for connected objects, the second example we consider a missile model. The length, width and height of the missile model are 3.63 cm, 1.49 cm, and 0.95 cm, respectively. In the SMoM and SMoM-CAT technique, the whole missile model is divided into six subdomains and these subdomains are displayed in different colors, as shown in Fig. 7. The incident direction of the planewave is θ inc = 60 • and ϕ inc = 0 • . Fig. 8 shows the θ θ polarized monostatic RCS from 2 GHz to 18 GHz obtained from the SMoM-CAT with three different order (L = 4, 7, 10). Compared to the SMoM solutions, we can observe that the 3 rd order approximation gives incorrect results across the entire frequency band. The 5 th order approximation can give better results over the entire band but the results are not accurate at 4 to 10GHz. When the 7 th order approximation is applied, we can obtain the results with good accuracy over the entire band. Fig. 9 illustrated that the SMoM and SMoM-CAT results and they agree well with the MoM solutions. The frequency sweeping increment is 100 MHz for the SMoM-CAT, the SMoM and the MoM methods.
To further verify the accuracy of the proposed SMoM-CAT algorithm for connected objects, we extract the currents at the non-Gaussian sampling point calculated by the SMoM-CAT method at L = 7, the bistatic RCS was obtained by using the current at a frequency of 10 GHz and compared with the calculation result of MoM. As shown in Fig. 10, the incident direction of the plane wave is θ inc = 60 • and ϕ inc = 0 • ,  the bistatic acceptance angle is θ sca = 0 ∼ 360 • and ϕ sca = 0 • , the polarization is θ θ polarization, we can see from the figure that the results are in good agreement, VOLUME 8, 2020  and the accuracy of the proposed algorithm connected objects is further verified.

C. AN AIRCRAFT MODEL
To show the accuracy of the SMoM-CAT algorithm for connected objects, the third example we consider an aircraft model. The length, width and height of the aircraft model are 0.95 m, 0.69m and 0.195 m, respectively. In the SMoM and SMoM-CAT technique, the whole aircraft model is divided into four subdomains and these subdomains are displayed in different colors, as shown in Fig. 11. The incident direction of the planewave is θ inc = 90 • and ϕ inc = 0 • . Fig. 12 shows the θθ polarized monostatic RCS from 0.2GHz to 1.2GHz obtained from the SMoM-CAT with three different order (L = 4, 7, 10). Compared to the SMoM solutions, we can observe that the 4 th order approximation gives accurate results from 0.2 GHz to 1.2 GHz. It provides incorrect results out of these bands. The 7 th order approximation can give better results over the entire band except the peak around  0.4 GHz is not correct. When the 10 th order approximation is applied, we can obtain the results with good accuracy over the entire band. Fig. 13 illustrated that the SMoM   and SMoM-CAT results and they agree well with the MoM solutions. The frequency sweeping increment is 10 MHz for the SMoM-CAT, the SMoM and the MoM methods.
For example C, we also extract the currents at the non-Gaussian sampling point calculated by the SMoM-CAT method to calculate the bistatic RCS and compare it with the calculation result of MoM. Fig. 14 shows the bistatic results obtained at the non-Gaussian sampling point calculated at 0.7 GHz according to the SMoM-CAT method at L = 11, the incident direction of the plane wave is θ inc = 90 • and ϕ inc = 0 • , the bistatic acceptance angle is θ sca = 0 ∼ 360 • and ϕ sca = 0 • , the polarization is θθ polarization. Compare its bistatic results with the results of MoM calculations we can see that the two curves are well fitted.
The memory requirement and total CPU time of the SMoM-CAT technique with three different orders for the patch array, the missile model, and the aircraft model are illustrated in Table 1, Table 2, and Table 3, respectively. We can observe that the memory requirement and CPU time are gradually rising as we increase the order of the SMoM-CAT. The reason is that the higher order is chosen, the more currents at Chebyshev-Gauss sampling frequency points are needed to be computed and stored. VOLUME 8, 2020 Table 4 summarizes the number of unknowns, the average number of iterations, the memory requirement, and the total CPU time of the SMoM-CAT, the SMoM, and the MoM, respectively. From Table 4, we can see that, compared with the SMoM, the SMoM-CAT can achieve 84%, 87.8% and 69% reduction on CPU time for the patch array, the missile, and the aircraft, respectively.

IV. CONCLUSIONS AND DISCUSSIONS
The hybrid SMoM-CAT technique is presented for fast analyzing the scattering problem of 3D PEC objects over a broad frequency band. The SMoM is employed to flexibly model the object with small subdomains and each subdomain can be handled independently. For efficiently predicting the wide-band RCS, the Chebyshev approximation technique is combined with the SMoM to significantly reduce the computation time and maintain good accuracy. Since the surface currents of all subdomains over the desired frequency band have been obtained, the proposed SMoM-CAT can be also used to calculate the bistatic RCS at any frequency point. Till now, he teaches with the Electronic Engineering Institute, Xidian University, where he is currently an Associate Professor. His current research interests include theory and design of antennas and arrays, antenna array radiation calculation and pattern optimization, scattering calculation and electromagnetic stealth design of antennas and arrays, and research on radiation and scattering calculation and control methods for large-scale array antennas. He is currently a Professor with the Electronic Engineering Institute, Xidian University. His research interests are antenna design and target electromagnetic scattering. VOLUME 8, 2020