1、电 子 科 技 大 学 UNIVERSITY OF ELECTRONIC SCIENCE AND TECHNOLOGY OF CHINA 硕士学位论文 MASTER THESIS （电子科技大学图标） 论文题目 学 科 专 业 学 号 作 者 姓 名 指 导 教 师 分类号 密级 UDC 注 1 学 位 论 文 （题名和副题名） （作者姓名） 指导教师 （姓名、职称、单位名称） 申请学位级别 学科专业 提交论文日期 论文答辩日期 学位授予单位和日期 电子科技大学 年 月 日 答辩委员会主席 评阅人 注 1：注明国际十进分类法 UDC的类号。 Caldern Technique Based In
2、tegral Equation Methods in Computational Electromagnetics A Master Thesis Submitted to University of Electronic Science and Technology of China Discipline: Electromagnetic Field and Microwave Author: XXX Supervisor: Prof. XXX School: School of XXXXXXXX 独创性声明 本人声明所呈交的学位论文是本人在导师指导下进行的研究工作 及取得的研究成果。据我所
3、知，除了文中特别加以标注和致谢的地方 外，论文中不包含其他人已经发表或撰写过的研究成果，也不包含为 获得电子科技大学或其它教育机构的学位或证书而使用过的材料。与 我一同工作的同志对本研究所做的任何贡献均已在论文中作了明确的 说明并表示谢意。 作者签名： 日期： 年 月 日 论文使用授权 本学位论文作者完全了解电子科技大学有关保留、使用学位论文 的规定，有权保留并向国家有关部门或机构送交论文的复印件和磁盘， 允许论文被查阅和借阅。本人授权电子科技大学可以将学位论文的全 部或部分内容编入有关数据库进行检索，可以采用影印、缩印或扫描 等复制手段保存、汇编学位论文。 （保密的学位论文在解密后应遵守此规
4、定） 作者签名： 导师签名： 日期： 年 月 日 摘要 I 摘 要 本文基于电磁理论中的 Caldern 关系与 Caldern 恒等式所揭示的不同积分算 子之间的关系，系统地研究了 Caldern 预条件技术及其在计算电磁学积分方程方 法中的应用。研究内容全面覆盖了求解理想导电体目标和均匀或分层均匀介质目 标电磁散射与辐射问题的积分方程中的 Caldern 预条件技术。在导体积分方程方 面，研究了电场积分方程在中频，低频，以及高频区的 Caldern 预处理方法。在 介质积分方程方面，则研究了 PMCHWT 积分方程的 Caldern 预处理方法，和 N- Mller 积分方程的 Calde
5、rn 技术。本文对金属问题中的第二类 Fredholm 积分方 程和介质问题中的第二类 Fredholm 积分方的精度改善进行了深入详尽的研究。 关键词：电磁散射，面积分方程方法，Caldern 预条件方法，数值计算精度，第 二类 Fredholm 积分方程 Abstract II ABSTRACT Revealed by the Caldern relation and the Caldern identities in electromagnetic theory, the properties and relation of different integral operators in
6、 the computational electromagnetics (CEM) are utilized to construct the Caldern preconditioning techniques, which are applied in the integral-equation-based methods in this thesis. A thorough and systematic research has been accomplished to cover the Caldern preconditioning techniques for the perfec
7、t electric conductor (PEC) and the dielectric cases. For the PEC case, the Caldern preconditioning for the electric-field integral equation (EFIE) at mid, low, and high frequencies are constructed and studied. For the dielectric cases, the Caldern preconditioning for the Poggio-Miller-Chang-Harringt
8、on- Wu-Tsai (PMCHWT) integral equation are investigated, and the Caldern technique for the N-Mller integral equation is developed. Moreover, the accuracy improving technique for the second-kind Fredholm integral equation for both PEC and dielectric cases is also studied in this thesis. Keywords: Ele
9、ctromagnetic scattering and radiation, surface-integral-equation-based Methods, Caldern preconditioning methods, numerical accuracy, Fred- holm integral equations of the second kind Contents III Contents Chapter 1 Introduction .1 1.1 Research Background and Significance .1 1.2 State of Arts1 1.3 Con
10、tents and Innovations of the Thesis2 1.4 Outline of the Thesis 2 Chapter 2 Theoretical Basics3 2.1 Integral Equations in Electromagnetics3 2.2 270 MHz Plan Wave Excitation.3 2.3 The Solution of Integral Equations in Electromagnetics4 2.3.1 General Principle of the Method of Moments4 2.3.2 Geometrica
11、l Modeling and Discretization of Object4 2.3.2.1 Planar Triangular Model.4 2.3.2.2 Curvilinear Triangular Model.4 2.3.3 The Choice of Basis Functions.5 2.3.3.1 Planar RWG Basis Functions .5 2.3.3.2 Curvilinear RWG Basis Functions .6 2.3.4 The Solution of Matrix Equations 6 2.3.4.1 Direct Algorithms.
12、6 2.3.4.2 Iterative Algorithms6 2.4 Conclusion6 Chapter 3 Caldern Preconditioner at Mid Frequencies 7 3.1 Introduction 7 3.2 Caldern Relation and Caldern Identities.7 3.3 Caldern Preconditioner at Mid Frequencies .7 3.4 Numerical Examples 7 3.5 Conclusion7 Chapter 4 Caldern Preconditioning Technique
13、 for N-Mller 8 4.1 Introduction 8 4.2 N-Mller Integral Equations.8 4.3 The Derivation of N-Mller Equations 8 Contents IV 4.4 The Discretization of N-Mller Equations .8 4.5 Numerical Examples 8 4.6 Conclusion8 Chapter 5 Conclusions 9 5.1 Concluding Remarks 9 5.2 Future Work9 Acknowledgements10 Refere
14、nces 11 Research Results Obtained During the Study for Master Degree12 Chapter 1 Introduction 1 Chapter 1 Introduction 1.1 Research Background and Significance Integral-equation-based numerical methods combined with fast algorithms are capable of solving electromagnetic problems of complex structure
15、s and material properties with a good accuracy and a high efficiency. They are widely used in a variety of engineering applications, such as the efficient analysis of three dimensional radar scattering problems, the simulation of the input impedance and the radiation properties of antenna systems, t
16、he calculation of the input response and the transmission efficiency of microwave circuits, the evaluation of the electromagnetic interference (EMI) between complex electromagnetic systems, and the computer aided electromagnetic compatibility (EMC) designs. The versatility, capability, accuracy and
17、efficiency of the integral-equation-based methods have made them an important and cost effective approach in the analysis and design of electromagnetic problems and applications. 1.2 State of Arts From the 1960s, the numerical methods of electromagnetic analysis have been fast developed because of t
18、heir versatility and flexibility. Many well-known numerical methods have been introduced during that time, including the finite element method (FEM) 1 and the finite difference time domain method (FDTD) 2,3, which are based on the solution to the Maxwells equations in differential form, and the meth
19、od of moments (MoM) 2, 4-6, which is based on the solution to the Maxwells equations in integral form. Especially from 1990s, with the fast developments of high performance computing systems, the theories and methods of computational electromagnetics have been advanced dramatically. The increases of
20、 the clock speed and the memory size of computer systems and the developments of highly efficient electromagnetic computing algorithms make the numerical methods capable of solving electromagnetic engineering problems.1 1 Master Thesis of University of Electronic Science and Technology of China 2 1.
21、3 Contents and Innovations of the Thesis Based on the Caldern relation and the Caldern identities, this thesis has developed several Caldern preconditioning techniques and investigated their applications in the integral-equation-based computational electromagnetic methods. The research content has c
22、overed the Caldern preconditioning techniques for the perfect electric conductor (PEC) and dielectric cases. For the PEC1 case, the Caldern preconditions at mid, low, and high frequencies are investigated. For the dielectric case, the Caldern preconditioning techniques for the PMCHWT and N-Mller int
23、egral equations are developed. The numerical accuracy of the second-kind Fredholm integral equations are investigated and improved in this thesis. 1.4 Outline of the Thesis This thesis is organized as follows. 1 Chapter 2 Theoretical Basics 3 Chapter 2 Theoretical Basics In this chapter, the general
24、 methods of constructing the commonly used integral equations in electromagnetics are introduced based on the surface equivalence principle and the volume equivalence principle. 2.1 Integral Equations in Electromagnetics In the integral-equation-based computational electromagnetic methods, the unkno
25、wn functions in the electromagnetic problems such as the scattering or radiation fields are modeled in terms of the equivalent surface or volume electric/magnetic sources by applying the surface or volume equivalence principles, respectively. 2.2 270 MHz Plan Wave Excitation In order to investigate
26、the its performance in handling electrically very large problems with over one million unknowns, the same numerical example is repeated again by increasing the frequency to 270 MHz, and keeping the incident angle and polarization of the plane wave unchanged. To have a better insight, the memory cons
27、umption and CPU time requirements of the EFIE, the CP-CFIE(0.8), and the CP- AEFIE algorithms are given in Table 2-1. Table 2-1 Comparison of Computational Data of Different Algorithms CPU Time Solution Time Total Memory (Mb) Setup (h) Iter. (m) Tol. (h) EFIE 3215.84 1.14 3.18 63 CP-CFIE(0.8) 6386.1
28、2 7.84 7.04 27.69 CP-AEFIE 5750.43 6.71 7.47 19.05 All the calculations are carried out on a HP Z400 workstation with a Fedora 10 operating system. Master Thesis of University of Electronic Science and Technology of China 4 2.3 The Solution of Integral Equations in Electromagnetics 2.3.1 General Pri
29、nciple of the Method of Moments The integral equations constructed in the preceding section can be solved with adequate numerical methods. One of the most commonly used methods in solving integral equations is the method of moments (MoM) introduced by R. F. Harrington in 19685. The general principle
30、 and key points of MoM will be reviewed in this section. 2.3.2 Geometrical Modeling and Discretization of Object From the description in the preceding section, it is clear that in order to solve for the unknown equivalent electromagnetic currents defined on the surface or in the volume of an obstruc
31、tion, the definition domain of the unknown currents, which is the geometry, needs to be described mathematically. This is the so-called geometrical modeling. In computational electromagnetics, geometrical modeling is the basic of electromagnetic modeling and numerical calculation, and its quality wi
32、ll affect the accuracy of the numerical solution directly. 2.3.2.1 Planar Triangular Model The simplest and most commonly used element in the geometrical modeling is the planar triangle, which is defined by its three vertices (nodes). 2.3.2.2 Curvilinear Triangular Model The curved surface of an obj
33、ect can be better modeled with curvilinear triangular elements which are the second-order curved surfaces. A curvilinear triangle can be defined by six nodes, three of which are the vertices of the triangle, the other three are the midpoints of three curved edges. Shown in Figure 2-1 is the sketch o
34、f a curvilinear triangular element. The curved surface of an object can be better modeled with curvilinear triangular elements which are the second-order curved surfaces. A curvilinear triangle can be defined by six nodes, three of which are the vertices of the triangle, the other three are the midp
35、oints of three curved edges. Shown in Figure 2-1 is the sketch of a curvilinear Chapter 2 Theoretical Basics 5 triangular element. Figure 2-1 The sketch of a curvilinear triangular element. (a)The curvilinear triangle in the coordinate system; (b) The curvilinear triangle in the coordinate system Us
36、ing the following coordinate transformation, the curvilinear triangle in the rectangular coordinate system, as shown in Figure 2-1(a), can be mapped onto the triangle defined in a parametric coordinate system, as shown in Figure 2-1(b) (2-1) 612123(,)(,)jjrr where denote the rectangular coordinates
37、of the six controlling nodes in Figure jr 2-1a, ， ， are the parametric coordinates varying from 0 to 1, and they satisfy 123 the relation (2-2)123 From (2-2), it is clear that only two variables out of these three are independent. 2.3.3 The Choice of Basis Functions After the geometrical discretizat
38、ion of the object surface using planar or curvilinear triangular elements, basis functions can be defined on these triangular elements to expand the unknown vector functions. 2.3.3.1 Planar RWG Basis Functions Introduced by Rao Wilton, and Glisson in 1982, the RWG basis function 6 is defined over tw
39、o adjacent triangular elements.1 Master Thesis of University of Electronic Science and Technology of China 6 2.3.3.2 Curvilinear RWG Basis Functions In order to give a better representation of curved surfaces, the curvilinear triangular elements can be used. Correspondingly, the curvilinear RWG basi
40、s functions 7 can be defined on the curvilinear triangular elements. 2.3.4 The Solution of Matrix Equations The matrix equation can be solved with two types of algorithms, the direct algorithms and the iterative algorithms. They will be introduced briefly in this subsection 8. 2.3.4.1 Direct Algorit
41、hms The commonly used direct algorithms include the Gaussian elimination, the LU decomposition, and the singular value decomposition (SVD) 9-10. 2.3.4.2 Iterative Algorithms When the dimension of the impedance matrix is very large, the direct solution becomes very expensive. 2.4 Conclusion 1 Chapter
42、 3 Caldern Preconditioner at Mid Frequencies 7 Chapter 3 Caldern Preconditioner at Mid Frequencies 3.1 Introduction The integral equations (IEs) are used to model the electromagnetic scattering, 3.2 Caldern Relation and Caldern Identities In a scattering problem, according to the surface equivalence
43、 principle, 3.3 Caldern Preconditioner at Mid Frequencies Based on the discussion in the preceding section, 3.4 Numerical Examples Two simple examples are given to demonstrate the fast convergence of the Caldern preconditioner at mid frequencies. 3.5 Conclusion The Caldern preconditioner for the EFI
44、E at mid frequencies is reviewed in this chapter. Master Thesis of University of Electronic Science and Technology of China 8 Chapter 4 Caldern Preconditioning Technique for N-Mller 4.1 Introduction Analysis of low-frequency electromagnetic problems has received more attention, 4.2 N-Mller Integral
45、Equations Consider the problem of electromagnetic wave scattering by a conducting surface, Theorem 1 Proof: Consider the problem of electromagnetic wave scattering by a conducting surface, the problem is proved. 4.3 The Derivation of N-Mller Equations The derivation begins from the preconditioning o
46、f the EFIE, 4.4 The Discretization of N-Mller Equations The derivation begins from the preconditioning of the EFIE, 4.5 Numerical Examples In this section, the performance of the N-Muller equations is investigated. 4.6 Conclusion In this chapter, Chapter 5 Conclusions 9 Chapter 5 Conclusions 5.1 Con
47、cluding Remarks The accurate and efficient numerical solutions of the Maxwells equations have important significance to the analysis of electromagnetic scattering and radiation problems. 5.2 Future Work The researches reported in this dissertation have covered most important areas including the conv
48、ergence acceleration of the first-kind integral equations and the accuracy improvement of the second-kind integral equations for both the PEC and the dielectric cases. Nevertheless, due to the time limitation, there are still spaces for the future development of the Calderon-technique-related methods. Master Thesis of University of Electronic Science and Technology of China 10 Acknowledgements On the completion of this thesis, References 11 References 1 W. C. Chew, J. M. Jin, E. Michielssen, et al.
