This research thesis presents a novel higher-order spectral element method (SEM) formulated in cylindrical coordinates for analyzing electromagnetic fields in waveguides filled with complex anisotropic media. In this study, we consider a large class of cylindrical waveguides: radially-bounded and radially-unbounded domains; homogeneous and inhomogeneous waveguides; concentric and non-concentric geometries; Hermitian and non-Hermitian anisotropic media tensors. This work explores different wave equation formulations for one-layer eccentric and multilayer cylindrical waveguides. For the first case, we can define a new normalized scalar Helmholtz equation for decoupling TM and TE modes, and for the second, a vectorial Helmholtz equation for hybrid modes in multilayered anisotropic structures. Additionally, we formulate a transformation optics (TO) framework to include non-symmetric and non-Hermitian media tensors for non-concentric multilayer waveguides. Lastly, we model excitation sources for logging sensors applied in geophysical problems using the fields obtained by SEM. We validate the proposed approach against analytical solutions, perturbation-based and mode-matching-based methods, finite-elements, and finite-integration numerical methods. Our technique obtains accurate results with fewer elements and degrees of freedom (DoF) than Cartesian-based SEM and ordinary finite-element approaches. To this end, we use higher-order two-dimensional basis functions associated with the zeros of the completed Lobatto polynomial to model the fields in each reference element. The convergence analysis demonstrates the absence of the Runge effect as the expansion order increases. Numerical results show that our formulation is efficient and accurate for modeling cylindrical waveguided geometries filled with complex media.

翻译：本研究提出了一种新颖的高阶谱元方法，该方法在柱坐标系中构建，用于分析填充复杂各向异性介质的波导中的电磁场。在本研究中，我们考虑了一大类柱面波导：径向有界和径向无界域；均匀和非均匀波导；同心和非同心几何结构；厄米特和非厄米特各向异性介质张量。本工作探讨了单层偏心及多层圆柱波导的不同波动方程表述。对于第一种情况，我们可以定义一个新的归一化标量亥姆霍兹方程来解耦TM和TE模式；对于第二种情况，则定义了一个矢量亥姆霍兹方程用于分析多层各向异性结构中的混合模式。此外，我们构建了一个变换光学框架，以将非对称和非厄米特介质张量纳入非同心多层波导的建模中。最后，我们利用谱元法计算得到的场，为应用于地球物理问题的测井传感器建立了激励源模型。我们通过解析解、基于微扰和模式匹配的方法、有限元法以及有限积分数值方法对所提方法进行了验证。与基于笛卡尔坐标的谱元法及普通有限元方法相比，我们的技术能以更少的单元和自由度获得精确结果。为此，我们使用与完整洛巴托多项式零点相关的高阶二维基函数来建模每个参考单元中的场。收敛性分析表明，随着展开阶数的增加，龙格效应并未出现。数值结果表明，我们的公式对于填充复杂介质的圆柱波导几何结构的建模是高效且精确的。