早稲田大学大学院
先進理工学研究科
博 士 論 文 概 要
論 文 題 目
Study on Acceleration of Electromagnetic
Wave Analysis Based on Integral Equation
Methods
（積分方程式法に基づく電磁波解析の高速化
に関する研究）
申 請 者千葉
英利
Hidetoshi
CHIBA
2011 年 11 月 （ 受 理 申 請 す る 部 科 主 任 会 開 催 年 月 を 記 入 ）
With the exponential growth of computer technology in recent decades, the demand for more accurate simulation on a larger scale has increased in scientific computing. In particular, in the field of electromagnetic wave analysis, numerical methods based on integral equation methods and a design scheme using an optimization algorithm have been intensively studied for large-scale and practical applications.
The objective of this dissertation is to develop a fast electromagnetic solver based on integral equation methods and an efficient design procedure using an optimization algorithm for solving large-scale and practical electromagnetic radiation and scattering problems. This dissertation includes four subjects of research.
In all of the subjects, the following two issues are fundamental:
(1) reduction of the computational cost of matrix-vector products in iterative linear system solvers
(2) improvement of the convergence properties of iterative linear system solvers and optimization algorithms
In the first subject, we address issue (1) for the development of a fast electromagnetic solver based on the method of moments (MoM). We use a combination of the wavelet transform method (WTM) and the MoM for alleviation of the matrix-vector product (MATVEC) computation. An important point here is that the effectiveness of sparsification of the dense coefficient matrix by the WTM depends on the geometry of the scatterer as well as on the nature of Green’s function. That is, one drawback of the WTM is that when the scatterer is discontinuous in the space domain, as in the case of a scattering problem with multiple scatterers, sparsification is not satisfactory. To overcome this drawback, we define an alternative wavelet transform by using an orthogonal transform in conjunction with the original wavelet transform. To be more precise, the first step in the proposed method involves the construction of a partial wavelet transform matrix and the generation of the full-size matrix by arranging the precomputed submatrices. Then, to convert the full-size matrix into an orthogonal matrix, the partial matrices are multiplied by the elements of a Hadamard matrix. The proposed method is efficient for solving problems with a finite periodic structure, which is common in antenna applications. Numerical experiments reveal that, compared to the conventional approach, the proposed method yields a sparser coefficient matrix than does the conventional approach, and achieves more significant saving in CPU time, with sufficiently accurate solutions.
Using the method explained in the previous subject, we can reduce the computational complexity of the MATVEC and accelerate the numerical analysis based on the MoM. However, since the method requires direct computation of the dense coefficient matrix, it still suffers from an unacceptable computational cost on the order O(N2). Consequently, this method cannot deal with significantly
large-scale problems. Besides, since the method works effectively only for problems with a finite periodic structure, its applicability is limited. Against the aforementioned context, the second subject aims at providing a solution to issue (2) and thus facilitating a more practical and generalized application. The solver in this part of the study is based on the MoM in conjunction with the fast multipole method (FMM) to expedite the MATVEC computation. For the linear system solver, flexible Krylov subspace methods (KSMs), a class of methods in which preconditioning can be different for each step, are employed. As the FMM does not explicitly generate a coefficient matrix, it is generally difficult to construct an efficient preconditioner by using only conventional static preconditioning techniques. On the other hand, KSMs can accept a variable preconditioner constructed from the FMM operator; hence, these methods can help maximize the efficiency of the FMM. In general, it is desirable to carry out the MATVEC computation within the variable preconditioner with the least possible effort. However, there is no definite rule for determining the optimal accuracy of the FMM in the variable preconditioner. The main contribution of this subject of research is to clarify the relationship between the overall efficiency of the inner-outer flexible generalized minimal residual algorithm (FGMRES), in which FGMRES is used for the outer solver and generalized minimal residual algorithm (GMRES) is used for the inner solver, and to increase the accuracy of the MATVEC within the variable preconditioner. To this end, we use the following feature of multipole techniques: control of the accuracy and computational cost of the FMM by choosing an appropriate truncation number that indicates the number of multipoles used to express far-field interactions. Accordingly, we construct two FMM operators with different levels of accuracy: a highly accurate version for the outer solver and a less accurate and cheaper version for the inner solver. To verify the proposed method, we perform numerical experiments on scattering and radiation problems with mixed dielectric and conducting objects. We employ the volume-surface integral equation (VSIE) formulation and the combined tangential formulation (CTF). The results of the numerical experiments reveal that there is an optimal accuracy for the FMM within the variable preconditioner, and a moderately accurate FMM operator serves as an optimal preconditioner.
The method described in the second subject can be used for generalized and practical electromagnetic wave analysis. However, since the proposed method has large memory requirements, a large-scale computing system is needed. As a result, to promote the utilization of the developed solver, we explore another approach to issue (2) for the development of a fast solver that can be applied in a small-scale computing environment. The numerical solver discussed in this subject is again based on the KSM and the FMM-accelerated MoM. We attempt to use a new type of KSM solver, i.e., the induced dimension reduction (IDR)(s) method. A unique feature of the IDR(s) method is that the residual vector converges to a zero vector because the dimensions of the spaces to which the residual vector belongs decrease monotonically in accordance with the IDR theorem. However, there is an inherent problem in the IDR(s) method, the so-called “spurious convergence.” That is, the real relative
residual norm (RRRN) at convergence becomes much larger than the predefined tolerance, and the phenomenon tends to frequently occur for problems with high geometrical complexity. Due to this problem, IDR(s) has not attained practical-use level in the field of electromagnetic wave analysis. In this study, using electromagnetic scattering problems with different geometrical complexities, we investigate the convergence performance of the IDR(s) and conduct in-depth experiments on the relationship between the parameter s and the tolerance to spurious convergence. In the numerical experiments, we observe that when s is set to a high value, spurious convergence frequently occurs even with the acceleration of the convergence. Hence, for high geometrical complexity, s should be set to a low value to avoid spurious convergence. In the real-life example of airplanes, the IDR(s) shows better convergence than do generalized product-type based on bi-conjugate gradient methods (GPBiCG) and GMRES(m), although s has to be set to a value smaller than 8. In addition, it is confirmed that the IDR(s) is more advantageous than GMRES in terms of memory requirements and well suited for small-scale computing environments.
According to all the methods and results presented in the preceding three subjects, accurate and fast electromagnetic wave analysis can be carried out in several scale computing systems. However, in a practical design scenario, we require not only an analysis technique but also a systematic design scheme that is independent of the designer’s preference and subjectivity. Hence, as a part of the fourth subject, we establish a design method having reproducibility, reproductivity, and systematicity. To be more specific, we deal with issue (2) for the development of an efficient and practical radome design scheme. Layered dielectric radomes are frequently required in many applications to protect antenna systems from damage by meteorological and environmental conditions. Designing a radome is generally a formidable problem since several operating conditions such as operating frequency, beam scan angle, and arbitrary shape of the radome have to be taken into account. In practice, rigorous analysis and a simple design procedure are possible only for a few radomes with simple geometries. Therefore, we develop a design method based on an optimization algorithm. For the optimization method, we adopt the particle swarm optimization (PSO) algorithm, which has recently attracted considerable attention because of its preferable convergence property. However, since the diversity of the population is inevitably degraded owing to the ability of the PSO algorithm for information sharing, it has lower stability and reliability than does genetic algorithm (GA). To remedy this problem, we introduce a mutation in the original PSO algorithm. This simple modification provides diversity to the particles and causes perturbation of the PSO algorithm, and the reliability of the algorithm is enhanced without sacrificing its inherent simplicity. In the numerical experiments, we have successfully applied the proposed algorithm to the practical radome design problem; the transmission coefficient determined using the algorithm is consistent with the target value under all operating conditions of frequency bandwidth and beam scan angle considered.
Ｎｏ.1
早稲田大学 博士（工学） 学位申請 研究業績書
氏 名 千葉 英利 印 （２０１１年１０月 現在） 種 類 別 題名、 発表・発行掲載誌名、 発表・発行年月、 連名者（申請者含む） Paper Itn. Conf. Domestic Conf.[1] H. Chiba, K. Nishizawa, H. Miyashita, and Y. Konishi, “Acceleration of wavelet-based method of moments for periodic structures,” IEEJ Trans., vol. 129-A, no. 10, pp. 729–736, Oct. 2009.
[2] H. Chiba, T. Fukasawa, H. Miyashita, and Y. Konishi, “Efficient implementation of inner-outer flexible GMRES for the method of moments based on a volume-surface integral equation,” IEICE Trans. Electron., vol. E94-C, no. 1, pp. 24–31, Jan. 2011.
[3] H. Chiba, T. Fukasawa, H. Miyashita, and Y. Konishi, “Convergence property of IDR(s) method implemented along with method of moments for solving large-scale electromagnetic scattering problems involving conducting objects,” IEICE Trans. Electron., vol. E94-C, no. 2, pp. 198–205, Feb. 2011.
[4] H. Chiba, T. Fukasawa, H. Miyashita, and Y. Konishi, “Acceleration of flexible GMRES using fast multipole method for implementation based on combined tangential formulation,” IEICE Trans. Electron., vol. E94-C, no. 10, pp. 1661–1668, Oct. 2011.
[5] H. Chiba, K. Nishizawa, H. Miyashita, and Y. Konishi, “Optimal design of broadband radome using particle swarm optimization,” TEEE Trans., vol. 7, no. 4, July 2012 (to be published).
[6] H. Chiba, Y. Inasawa, Y. Konishi, and S. Makino, “Evaluation of propagation characteristics for antenna array in MIMO system,” 2005 IEEE/ACES conference, Apr. 2005.
[7] H. Chiba, Y. Inasawa, Y. Konishi, and S. Makino, “Evaluation of propagation characteristics for antenna array in MIMO system,” ISAP 2005, FC3-4, pp. 1113–1116, Aug. 2005.
[8] H. Chiba, Y. Inasawa, N. Yoneda, and S. Makino, “A Modification of wavelet-based method of moments,” 2006 ACES conference, pp. 378–381, March 2006.
[9] H. Chiba, Y. Inasawa, N. Yoneda, and S. Makino, “Acceleration of wavelet based method of moments in scattering problems of periodic structures,” ISAP 2006, Nov. 2006.
[10] H. Chiba, Y. Inasawa, H. Miyashita, and Y. Konishi, “Optimal radome design with particle swarm optimization,” 2008 AP-S, 411.4, July 2008.
[11] H. Chiba, K. Nishizawa, H. Miyashita, and Y. Konishi, “Efficient implementation of inner-outer flexible GMRES using the fast multipole method,” ISAP 2008, Oct. 2008.
[12] H. Chiba, T. Fukasawa, H. Miyashita, and Y. Konishi, “Convergence property of IDR(s) method in method of moments for large-scale electromagnetic scattering problems,” 2nd international Krylov forum, pp. 79–84, March 2010.
[13] H. Chiba, T. Fukasawa, H. Miyashita, and Y. Konishi, “Convergence property of inner-outer flexible GMRES for solving electromagnetic scattering problems with method of mements,” 2010 AP-S, 523.8, July 2010.
[14] H. Chiba, Y. Inasawa, Y. Konishi, and S. Makino, “Evaluation of propagation characteristics in indoor environment for MIMO System,” IEICE Technical Report, AP2005-27, pp. 103–108, May 2005.
Ｎｏ.2
早稲田大学 博士（工学） 学位申請 研究業績書
種 類 別 題名、 発表・発行掲載誌名、 発表・発行年月、 連名者（申請者含む） Domestic Conf.[15] H. Chiba, Y. Inasawa, Y. Naofumi, and S. Makino, “An application of wavelet-based method of moments to scattering problems with discontinuous impedance matrix,” IEICE Technical Report, AP2006-20, pp. 65–70, May 2006.
[16] H. Chiba, Y. Inasawa, Y. Naofumi, and S. Makino, “An application of wavelet-based method of moments to scattering problems of periodic structure,” IEICE Technical Report, EMCJ2006-23, pp. 29–34, July 2006.
[17] H. Chiba, Y. Inasawa, H. Miyashita, and Y. Konishi, “Optimal design of radome with particle swarm optimization,” IEICE Technical Report, AP2007-126, pp. 21–25, Jan. 2008. [18] H. Chiba, K. Nishizawa, H. Miyashita, and Y. Konishi, “Efficient implementation of
inner-outer flexible GMRES using the fast multipole method,” The Papers of Technical Meeting on Electromagnetic Theory, IEE Japan, EMT-08-126, pp. 73–77, Nov. 2008.
[19] H. Chiba, T. Fukasawa, H. Miyashita, and Y. Konishi, “Convergence property of IDR(s) method in FMM-MoM method,” The Papers of Technical Meeting on Electromagnetic Theory, IEE Japan, EMT-09-151, pp. 29–24, Nov. 2009.
[20] H. Chiba, T. Fukasawa, H. Miyashita, and Y. Konishi, “On the convergence property of IDR(s) variant methods in implemented with the method of moments,” The Papers of Technical Meeting on Electromagnetic Theory, IEE Japan, EMT-10-115, pp. 77–82, Nov. 2010.
[21] H. Chiba, T. Fukasawa, H. Miyashita, and Y. Konishi, “Convergence property of IDR(s) variant methods in large-scale electromagnetic scattering problems,” 14th Symposium of Industrial and Applied Mathematics at Setouchi-Rim, pp. 1–6, Jan. 2011.
[22] H. Chiba, T. Fukasawa, H. Miyashita, and Y. Konishi, “Efficient implementation of inner-outer flexible GMRES using the fast multipole method for solving electromagnetic scattering problems involving composite conducting and dielectric objects,” IEICE Technical Report, AP2010-177, pp. 19–24, March 2011.
[23] H. Chiba, T. Fukasawa, H. Miyashita, and Y. Konishi, “Acceleration of FGMRES for method of moments based on combined tangential formulation,” The Papers of Technical Meeting on Electromagnetic Theory, IEE Japan, EMT-11-156, pp. 35–40, Nov. 2011.
[24] H. Chiba, Y. Inasawa, Y. Konishi, and S. Makino, “Evaluation of propagation characteristics in indoor environment for MIMO System,” Proc. IEICE Gen. Conf. '05, B-1-242, March 2005.
[25] H. Chiba, Y. Inasawa, Y. Konishi, and S. Makino, “Comparison between ray-trace and FDTD in analysis of indoor radio propagation,” Proc. Commun. Conf. IEICE '05, B-1-21, Sep. 2005.
[26] H. Chiba, Y. Inasawa, N. Yoneda, and S. Makino, “An application of wavelet-based method of moments to scattering problems with discontinuous impedance matrix,” Proc. IEICE Gen. Conf. '06, B-1-40, March 2006.
[27] H. Chiba, Y. Inasawa, Y. Konishi, and S. Makino, “Acceleration of wavelet based method of moments in scattering problems of periodic structures,” Proc. Electron. Conf. IEICE '06, CS-1-6, Sep. 2006.
[28] H. Chiba, Y. Inasawa, N. Yoneda, and S. Makino, “Basic study on broadband radome,” Proc. IEICE Gen. Conf. '07, B-1-163, March 2007.
Ｎｏ.3
早稲田大学 博士（工学） 学位申請 研究業績書
種 類 別 題名、 発表・発行掲載誌名、 発表・発行年月、 連名者（申請者含む） Domestic Conf.[29] H. Chiba, Y. Inasawa, H. Miyashita, and Y. Konishi, “An evaluation method of boresite error caused by the influence of radome,” Proc. Commun. Conf. IEICE '07, B-1-113, Sep. 2007. [30] H. Chiba, Y. Inasawa, H. Miyashita, and Y. Konishi, “Optimal radome design with particle
swarm optimization,” Proc. IEICE Gen. Conf. '08, B-1-181, March 2008.
[31] H. Chiba, K. Nishizawa, H. Miyashita, and Y. Konishi, “On the inner-outer flexible GMRES using the fast multipole method,” Proc. Commun. Conf. IEICE '08, B-1-66, Sep. 2008. [32] H. Chiba, K. Nishizawa, H. Miyashita, and Y. Konishi, “Convergence property of IDR(s)
method in FMM-MoM method,” Proc. IEICE Gen. Conf. '09, B-1-173, March 2009.
[33] H. Chiba, T. Yanagi, I. Koji, T. Fukasawa, H. Miyashita, Y. Konishi, Y. Urata, T. Endo, T. Kishida, and T. Kurokawa, “Convergence property of IDR(s) method in FMM-MoM method based on volume and surface integral equations,” Proc. Commun. Conf. IEICE '09, B-1-106, Sep. 2009.
[34] H. Chiba, T. Fukasawa, H. Miyashita, and Y. Konishi, “Convergence property of inner-outer flexible GMRES for the method of moments based on a volume-surface integral equation,” Proc. IEICE Gen. Conf. '10, CS-1-4, March 2010.
[35] H. Chiba, T. Fukasawa, H. Miyashita, Y. Konishi, H. Okazaki, and T. Kaneko, “Convergence property of IDR-based iterative methods for solving large-scale electromagnetic scattering problems,” Proc. Commun. Conf. IEICE 10, B-1-151, Sep. 2010. [36] H. Chiba, T. Fukasawa, H. Miyashita, and Y. Konishi, “Convergence property of IDR-based
iterative methods implemented along with the method of moments based on PMCHWT formulation,” Proc. IEICE Gen. Conf. '11, CS-1-1, March 2011.
[37] H. Chiba, T. Fukasawa, H. Miyashita, and Y. Konishio, “Acceleration of FGMRES for method of moments based on combined tangential formulation,” Proc. Commun. Conf. IEICE 11, B-1-54, Sep. 2011.
