Numerical Conformal Mapping And Its Applications In The Electromagnetic Theory

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.