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J. Electromagn. Eng. Sci > Volume 23(1); 2023 > Article
Yoo and Koh: Comparison of Linear Iteration Schemes to Improve the Convergence of Iterative Physical Optics for an Impedance Scatterer

Abstract

The conventional iterative physical optics (IPO) method updates the surface current based on the Jacobi iteration scheme, which typically diverges for large objects. To control the convergence property of the IPO method, other iteration schemes, such as Gauss–Seidel and successive over-relaxation, can be used. In this study, we compare the convergence properties of three iteration schemes for scatterings by five scatterers comprising electrically perfect or imperfect conductors modeled with an impedance material. The accuracy of the IPO method is compared with that of the multi-level fast multipole method.

I. Introduction

Iterative physical optics (IPO) has been widely used to analyze scattering by large and complex objects, and it iteratively calculates the surface currents on an object based on the magnetic field integral equation and physical optics (PO) approximation [1]. Near-field corrected IPO (NC-IPO) may be an efficient method for this type of computation of an impedance scatterer [2]. The conventional NC-IPO updates the current based on the Jacobi iteration scheme, but other iteration schemes, such as Gauss–Seidel (GS) and successive over-relaxation (SOR), may be more efficient. In [3], an IPO method, along with the GS and SOR schemes, was applied to solve a perfectly electrical conductor (PEC) problem, but the convergence properties of each scheme were not addressed. Therefore, we compare the convergence properties of the Jacobi, GS, and SOR schemes for scattering by several impedance objects.

II. Linear Iteration Scheme

The conventional IPO update equation, known as the Jacobi iteration, for an impedance object is shown in (1). The IPO update procedure is described in detail in [2].
(1)
Jn=Jn-1-12πn^×S1+jk0RRe-jk0R(R×Jn-1)dS-Zsj2πk0Z0S1+jk0R-k02R2R3e-jk0R(n^×Jn-1)dS+Zsj2πk0Z0S3+3jk0R-k02R2R5e-jk0R[(n^×Jn-1)·R](n^×R)dS,
where R=rr', R=|R|, and Jn denote the nth-order electric current. k0, Z0, Zs, and n^ denote the free-space wavenumber, impedance, surface impedance, and unit normal vector of the scatterer surface, respectively. The update starts with the zeroth-order PO current, J0=2n^×Hi, in the lit region, where Hi denotes the incident magnetic field. The Jacobi iteration can be simply represented in terms of an infinite series as follows:
(2)
J=J0+ΔJ1+ΔJ2,
where Jn=Jn-Jn-1, and the integral terms in (1) can be accurately calculated using the near-field correction (NC) scheme [2].
Because (2) may diverge for a large scatterer, the computation of (2) should be terminated by two stop criteria: the nth residual error (ɛn) is less than a given tolerance (δ) and ɛnɛn–1. The first and second cases pertain to the IPO procedure converging and diverging, respectively. The nth residual error is defined as follows:
(3)
ɛn=S|ΔJn|dSS|Jn|dS.
During the iteration, the GS scheme directly uses the updated current to calculate Jn. The SOR scheme is a linear combination of Jacobi and GS with a weighting factor w, typically chosen as a value between 0 and 2. For w=1, SOR is reduced to GS.

III. Numerical Examination

To investigate the convergence properties of the three linear iteration schemes, five scatterers were considered: boat, vessel, tank, and homogeneous and inhomogeneous aircraft. The five scatterers consisted of the PEC or impedance material. The dimensions, the number of meshes, and the normalized surface impedance (η) of each scatterer are summarized in Table 1, where λ0 denotes the free-space wavelength. The frequency was fixed at 2 GHz. For the inhomogeneous aircraft, η of the aircraft body, the canopy window, and the radome were 0.39–j0.06, 0.71 – j0.01, and 0.54, respectively. Fig. 1 shows the shape of each scatterer and incident wave direction. The incidence direction and observation line to compute the bistatic radar cross-section were defined by the elevation and azimuth angles, θinc and φinc, and θ and φ, respectively, which are summarized in Table 2. φ and δ were fixed as 0° and 10−3, respectively.
The convergence of the three iteration methods for the PEC boat versus the iteration number is shown in Fig. 1(a). The results of the IPO without the NC scheme were added to 30 for a clear comparison. Owing to the strong interaction among the surfaces, GS without NC and SOR with w = 1.4/1.8, with/without NC, diverge at the first iteration. However, the error of the NC-IPO was slightly less than that of an IPO without NC. For the remainder of the simulation, NC-IPO was used. The SOR weight factor was assumed to be 0.6 if not specified by a number.
Fig. 1(b) and 1(c) show identical comparisons for the impedance boat, tank, and (in)homogeneous aircraft, in which the convergence of SOR with four different weight factors is also compared. The results for the tank and inhomogeneous aircraft were added to 30. Because the reflection by the impedance surface can be less than that by the PEC, more iterations among the surfaces can be calculated for the impedance object, as shown in Fig. 1(b). In addition, the SOR convergence property can be controlled by varying the weight factor; a factor between 0 and 1 can increase the number of iterations. For inhomogeneous aircraft, SOR with a small w of 0.2 can converge to the given tolerance, 10−3.
Table 3 summarizes the number of iterations and the normalized root mean squared error (NRMSE) of each iteration scheme for the five PEC or impedance scatterers. In the “# of iterations” row, “D” and “C” indicate the schemes finally diverging and converging, respectively. The NRMSE is calculated as follows:
(4)
NRMSE=NΣi=1N(σiMLFMM-σiIPO)2Σi=1NσiMLFMM,
where σMLFMM and σIPO are the RCS computed using the multi-level fast multipole method (MLFMM) and IPO methods, respectively. σMLFMM is calculated using FEKO. N is the total number of observation points. Notably, GS and SOR can provide an almost identical NRMSE to that of Jacobi at fewer iterations.
Table 4 lists the ratio of the computational time of GS and SOR to that of Jacobi. For this comparison, the ratios for the vessel and tank were omitted because the iteration number was small; therefore, the ratio was almost unity. For this comparison, the tolerance was reduced to 10−2, which could provide an almost identical NRMSE to that of 10−3. GS and SOR can accelerate the IPO convergence speed, particularly for PEC objects.

IV. Conclusion

The convergence properties of three linear iteration schemes were compared for five PEC or impedance scatterers. Among the three schemes, SOR provided an improved convergence property—a smaller number of iterations for the identical error level. In addition, SOR convergence can be controlled by varying the weight factor. Therefore, NC-IPO, in conjunction with the SOR method, could be a robust and efficient scattering analysis method for large-scale scattering problems. Based on the simulations, 0.5 and 10−2 were initially recommended for the SOR weight factor and tolerance, respectively.

Acknowledgments

This work was supported by the Laboratory of Computational Electromagnetics for Large-scale Stealth platform (No. UD200047JD).

Fig. 1
Residual error of the IPO for a PEC and impedance objects. (a) Without NC and with the NC scheme for the PEC boat. (b) Impedance boat and tank. (c) (in)Homogeneous aircraft.
jees-2023-1-l-12f1.jpg
Table 1
Geometry information for scatterers
Model Dimension # of mesh η
Boat 40λ0×8λ0×17λ0 9,724 0.39 – j0.06
Vessel 5.3λ0×0.6λ0×1.2λ0 28,292
Tank 16λ0×7λ0×4.8λ0 78,504
Aircraft 106λ0×88λ0×28λ0 60,244
Table 2
Incidence and observation angles
Model Incidence angle Observation angle
Boat/vessel θinc =45°,φinc =180° θ=90° to −90°
Tank θinc =45°,φinc =0° θ=90° to −90°
Aircraft θinc =45°,φinc =0° θ=0° to 360°
Table 3
Accuracy comparison of Jacobi, GS, and SOR for a PEC and impedance objects
Model Method PEC Impedance


# of iterations NRMSE # of iterations NRMSE
Boat Jacobi 16 (C) 3.22 8 (D) 4.53
GS 2 (D) 2.96 6 (D) 4.53
SOR 2 (D) 2.98 8 (C) 4.52
Vessel Jacobi 12 (D) 0.98 1 (D) 1.70
GS 2 (D) 0.99 1 (D) 1.29
SOR 2 (D) 0.99 2 (D) 1.29
Tank Jacobi 15 (D) 0.61 3 (D) 1.89
GS 1 (D) 0.59 2 (D) 1.86
SOR 1 (D) 0.59 3 (D) 1.90
Homogeneous aircraft Jacobi 10 (C) 0.92 10 (C) 0.81
GS 2 (D) 0.96 2 (C) 0.71
SOR 2 (D) 0.95 2 (C) 0.71
Inhomogeneous aircraft Jacobi NA NA 6 (D) 1.99
GS NA NA 6 (D) 1.90
SOR NA NA 8 (D) 1.89

D = diverging, C = covering.

Table 4
Time comparison of GS and SOR versus Jacobi
Model Method PEC Impedance
Boat GS, SOR 3.7, 3.7 1.5, 1.5
Homogeneous aircraft GS, SOR 1.9, 1.7 2.0, 2.2
Inhomogeneous aircraft GS, SOR NA 1.0, 1.2

References

1. F. Obelleiro-Basteiro, J. L. Rodriguez and R. J. Burkholder, "An iterative physical optics approach for analyzing the electromagnetic scattering by large open-ended cavities," IEEE Transactions on Antennas and Propagation, vol. 43, no. 4, pp. 356–361, 1995.
crossref
2. J. W. Rim and I. S. Koh, "Convergence and accuracy of near-field-corrected iterative physical optics for scattering by imperfectly conducting and dielectric objects," IET Microwaves, Antennas & Propagation, vol. 14, no. 10, pp. 999–1005, 2020.
crossref pdf
3. R. J. Burkholder, C. Tokgoz, C. J. Reddy and W. O. Coburn, "Iterative physical optics for radar scattering predictions," The Applied Computational Electromagnetics Society Journal (ACES), vol. 24, no. 2, pp. 241–258, 2009.

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