数值保角变换及其在电磁理论中的应用
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摘要
保角变换在现代技术的许多领域如在电磁理论、热传输、流体力学、力学、声学等方面有着广泛的应用,具有强大的生命力。本文主要研究保角变换的数值方法,并讨论其在电磁理论中的应用。
第一章概述研究保角变换在电磁理论中应用的意义,并简要回顾保角变换的研究概况。
第二章简单介绍保角变换的基本理论及其基本方法。归纳常用解析变换的特点及各种变换的单叶性区域。
第三章介绍数值保角变换的各种方法。内容包括级数展开法,积分方程法,变分法。在级数展开法中,介绍Kantorovich法和快速傅立叶变换法。在积分方程法中,介绍Lichtenstein法、Theodorsen法和Symm法。在变分法中,介绍基于面积最小化和周长最小化的数值保角变换法。讨论许瓦兹—克里斯托夫变换的数值求解问题。并归纳双连通区域的数值变换。
第四章介绍电磁问题的数学模型。内容包括平面平行矢量场的复数表示,梯度、散度和旋度的复数表示,静电和静磁问题的复数位、复数场。本章还介绍拉普拉斯方程、泊松方程的保角变换求解及其本征值问题的保角变换解法。
第五章介绍保角变换法在静电和静磁问题中的应用,详细讨论在平面均匀电场作用下,不同导体边界下的静电位和场的保角变换解法,也讨论静磁问题的求解。
第六章介绍保角变换法在传输线特征阻抗方面的应用。将复杂截面的传输线用解析或者数值的方法变换为圆环形区域,提出平面分数阶多极子并用于计算特征阻抗,讨论分数阶多极子的选取原则及方法。
第七章介绍保角变换法在均匀波导截止频率计算中的应用。将复杂截面的波导用解析或者数值的方法变换为圆形区域,由于在圆形区域内边界形状简单,从而可以比较方便地选取全域基函数,这样用矩量法计算复杂截面波导截止频率时在编程处理时可以统一考虑和处理。通过数值例子验证方法的正确性和灵活性。本章还讨论保角变换在波导不连续性方面的应用。
第八章简要归纳本文的研究重点。讨论用保角变换和其它数值方法结合求解边值问题。
Conformal mapping is a powerful method of analysis with many successful applications in modern technology. Uses conformal mapping, we can solved a wide range of problem in electromagnetics, heat flow, fluid flow, mechanics, and acoustics. This thesis discussed the numerical method of conformal mapping and its applications in the theory of electromagnetics.
In chapter 1, a brief survey of study of conformal mapping and its applications in the theory of electromagnetics is presented.
In chapter 2, the classical methods of conformal mapping are briefly introduced firstly, and then a review of the basic properties of analytic functions is presented, those functions of a complex variable which can transform orthogonal grids in one plane to orthogonal grids in another plane. The condition of univalent of some elementary functions is given.
In chapter 3, the numerical methods of conformal mapping are presented, which contain the method of approximation, method of variational, method of integral equation. In the method of approximation, we describe the method of Kantorovich and method of Fourier transform, in the method of integral equation, we describe the method of Lichtenstein, method of Theodorsen, and method of Symm, in the method of variational, we describe methods based on minimum circumference principle and minimum area principle. Then discussed the numerical determination of the Schwarz-Christoffel transformation and the numerical transformation of doubly connected regions.
In chapter 4, we present the mathematical models of the electromagnetic theory. First we describe how the Laplace equation, Poisson equation and equation for wave is transformed during conformal mapping. Then we develop the relationship between the potential function in the physical plane and the corresponding potential function in the model plane. Next we present the complex gradient, complex divergence and complex rotation for the planar field. Finally, complex potential function for the electrostatics and magnetostatics problem is discussed.
In chapter 5, we describe the applications of conformal mapping in electrostatics and magnetostatics problem such as electric fields at points of high intensity, and magnetic fields of stationary structures.
In chapter 6, we address the use of conformal mapping in the analysis of transmission lines. This chapter presents a fractional multipole model for the calculation of the transmission line characteristic impedance. The model has a higher accuracy in analyzing the potential of a point close to a sharp conducting edge. The validity of this model is conformed by numerical results.
In chapter 7, we describe the applications of conformal mapping for the calculation of the cutoff frequencies in uniform waveguide firstly, a conformal mapping is applied to transforms the waveguides section onto the circle, and the variational equation is solved by Galerkin’s method. Comparisons with numerical results found in the literature validate the presented method. Then we discuss the problem of waveguides with discontinuities by conformal mapping.
Chapter 8 is concluding. We presentation a discussion of the use of conformal mapping in conjunction with other methods of solving boundary value problems.
第一章概述研究保角变换在电磁理论中应用的意义,并简要回顾保角变换的研究概况。
第二章简单介绍保角变换的基本理论及其基本方法。归纳常用解析变换的特点及各种变换的单叶性区域。
第三章介绍数值保角变换的各种方法。内容包括级数展开法,积分方程法,变分法。在级数展开法中,介绍Kantorovich法和快速傅立叶变换法。在积分方程法中,介绍Lichtenstein法、Theodorsen法和Symm法。在变分法中,介绍基于面积最小化和周长最小化的数值保角变换法。讨论许瓦兹—克里斯托夫变换的数值求解问题。并归纳双连通区域的数值变换。
第四章介绍电磁问题的数学模型。内容包括平面平行矢量场的复数表示,梯度、散度和旋度的复数表示,静电和静磁问题的复数位、复数场。本章还介绍拉普拉斯方程、泊松方程的保角变换求解及其本征值问题的保角变换解法。
第五章介绍保角变换法在静电和静磁问题中的应用,详细讨论在平面均匀电场作用下,不同导体边界下的静电位和场的保角变换解法,也讨论静磁问题的求解。
第六章介绍保角变换法在传输线特征阻抗方面的应用。将复杂截面的传输线用解析或者数值的方法变换为圆环形区域,提出平面分数阶多极子并用于计算特征阻抗,讨论分数阶多极子的选取原则及方法。
第七章介绍保角变换法在均匀波导截止频率计算中的应用。将复杂截面的波导用解析或者数值的方法变换为圆形区域,由于在圆形区域内边界形状简单,从而可以比较方便地选取全域基函数,这样用矩量法计算复杂截面波导截止频率时在编程处理时可以统一考虑和处理。通过数值例子验证方法的正确性和灵活性。本章还讨论保角变换在波导不连续性方面的应用。
第八章简要归纳本文的研究重点。讨论用保角变换和其它数值方法结合求解边值问题。
Conformal mapping is a powerful method of analysis with many successful applications in modern technology. Uses conformal mapping, we can solved a wide range of problem in electromagnetics, heat flow, fluid flow, mechanics, and acoustics. This thesis discussed the numerical method of conformal mapping and its applications in the theory of electromagnetics.
In chapter 1, a brief survey of study of conformal mapping and its applications in the theory of electromagnetics is presented.
In chapter 2, the classical methods of conformal mapping are briefly introduced firstly, and then a review of the basic properties of analytic functions is presented, those functions of a complex variable which can transform orthogonal grids in one plane to orthogonal grids in another plane. The condition of univalent of some elementary functions is given.
In chapter 3, the numerical methods of conformal mapping are presented, which contain the method of approximation, method of variational, method of integral equation. In the method of approximation, we describe the method of Kantorovich and method of Fourier transform, in the method of integral equation, we describe the method of Lichtenstein, method of Theodorsen, and method of Symm, in the method of variational, we describe methods based on minimum circumference principle and minimum area principle. Then discussed the numerical determination of the Schwarz-Christoffel transformation and the numerical transformation of doubly connected regions.
In chapter 4, we present the mathematical models of the electromagnetic theory. First we describe how the Laplace equation, Poisson equation and equation for wave is transformed during conformal mapping. Then we develop the relationship between the potential function in the physical plane and the corresponding potential function in the model plane. Next we present the complex gradient, complex divergence and complex rotation for the planar field. Finally, complex potential function for the electrostatics and magnetostatics problem is discussed.
In chapter 5, we describe the applications of conformal mapping in electrostatics and magnetostatics problem such as electric fields at points of high intensity, and magnetic fields of stationary structures.
In chapter 6, we address the use of conformal mapping in the analysis of transmission lines. This chapter presents a fractional multipole model for the calculation of the transmission line characteristic impedance. The model has a higher accuracy in analyzing the potential of a point close to a sharp conducting edge. The validity of this model is conformed by numerical results.
In chapter 7, we describe the applications of conformal mapping for the calculation of the cutoff frequencies in uniform waveguide firstly, a conformal mapping is applied to transforms the waveguides section onto the circle, and the variational equation is solved by Galerkin’s method. Comparisons with numerical results found in the literature validate the presented method. Then we discuss the problem of waveguides with discontinuities by conformal mapping.
Chapter 8 is concluding. We presentation a discussion of the use of conformal mapping in conjunction with other methods of solving boundary value problems.
引文
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