Symplectic Analysis And Dual Finite Element Method For Electromagnetic Waveguide

This thesis presents some theoretical analyses and numerical methods for electromagnetic waveguides, in the frame of Hamiltonian system.By treating the transverse electric and magnetic componets as dual variables to each other, the dual variables variational principle is presented, and the governing equations for electromagnetic waveguides are derived to Hamiltonian system formulation and symplectic geometric form. The whole set of methodology, which includs method of separation of variables, the adjoint symplectic orthonormality relationship, and the symplectic eigen-function expansion method, etc., can be used in waveguide problems. Symplectic analysis can be superior to conventional analytical method for some complex problems, such as field singularities near conducting and dielectric edges. Moreover, symplectic analysis for electromagnetic waveguides provides the theoretical basis for constructing dual finite element method (FEM).The main content of the thesis is dual FEM computation for electromagnetic waveguides. When applied to computational electromagnetics, conventional node-based FEM has three fundamental difficulties, they are the occurrence of nonphysical solutions for the lack of enforcement of the divergence condition, the inconvenience of imposing boundary conditions at interfaces between different materials, and the difficulty in presenting singular electromagnetic field near conducting and dielectric edges and corners. Based on dual variables variational principle for electromagnetic waveguides, we present dual node-based FEM, in conjunction with singular value decompsiton, sub-region analysis, and singular analytical element method to solve these fundamental difficulties for node-based FEM.Vector-based FEM is a sort of popular numerical method for computational electromagnetics for their special interpolating functions. On the basis of dual variables variational principle for electromagnetic waveguides, dual edge element is presented in this thesis. Dual edge element can easily treat anisotropic media and solve different problems in uniform formulation, and has merits in numerical precision and computation for high order modes when compared with conventional edge element.For waveguide discontinuity problems, waveguide sections which are homogeneous in the longitudinal direction are treated as substructures and semi-analytically discretized by 2-dimensional dual edge element, the stiffness matrices can be calculated by Riccatiequations-based high precision integral. The whole waveguide discontinuities problems can be solved by a combination of substructures and conventional 3-dimensional edge element for inhomogeneous sections. This method has high efficiency and can be implemented in optimazation for microwave devices.