| High frequency electromagnetics is closely related to many aspects of modern technology. It is a very important topic to solve the complex electromagnetic problems with numerical methods. With the increasing complexity of electromagnetic engineering problems, traditional numerical methods are facing more and more challenges.Electromagnetic simulations of layered structures problem usually contains several parallel layers homogeneous along a specific direction. Due to the flexibility in geometric modeling, the conventional numerical method can be employed to perform full wave analysis, and thus to obtain the electrical properties of layered structures. However, as the number of layers and the complexity of each layer increase, directly using the conventional numerical method to discretize the whole structure may lead to a huge system of coupled equations, thus making the overall efficiency of conventional numerical method very low for the analysis of layered structures. Multiscale electromagnetic simulation is another type of problem with wide application but very challenging for conventional methods. For a multiscale structure, conventional methods use a single mesh/grid to discretize this kind of structure, which will lead to a large number of wasted unknowns. Temporal integration would be another difficulty. Small cells will lead to extremely small time steps and an unaffordable number of calculations in time integration for explicit schemes. The conventional nonlinear numerical methods for nonlinear Maxwell’s equation usually require iteration, which maybe lead to a low convergence.All in all, conventional numerical methods have not been satisfying the requirements of modern electromagnetic engineering. In order to meet the specific needs of practical engineering, some valuable studies for layered electromagnetic structure and multiscale electromagnetic problems are conducted in this thesis. We proposed a semi-analytical spectral element method for analysis of layered structures, a hybrid finite-element/finite-difference time domain technique for multiscale electromagnetic problems, and a novel finite element time domain method for nonlinear Maxwell’s equations.The main contributions of this thesis can be summarized as follows:1. We proposed a semi-analytical method with high efficiency and high accuracy for frequency domain layered electromagnetic problems. A piecewise homogeneous3-D layered structure is divided into several substructures.2-D scalar and vector spectral elements are used to represent longitudinal and transverse unknowns on the cross section of each substructure, respectively. The semidiscretized system is then transformed from the Lagrangian system into the Hamiltonian system, where a Riccati equation-based high precision integration (HPI) method is utilized to perform integration along the longitudinal direction and to generate the stiffness matrix of a substructure. This method can be several orders more efficient and accurate than conventional FEM for layered structures.2. We proposed a hybrid finite-element/finite-difference time domain technique for transient electromagnetic simulations. Based on the discontinuous Galerkin method we combined finite element time domain method with finite difference time domain method to take advantages of both the flexibility of the FETD method and the efficiency of the FDTD method. This hybrid scheme allows nonconforming meshes and multiple subdomains, thus makes it very flexible in geometric modeling. The proposed method is very competitive in solving multiscale electromagnetic problems.3. We proposed a novel finite element time domain method to solve the nonlinear Maxwell’s equations. Based on the quadratic programming method the nonlinear constitutive relations are treated as a series of linear complementary problems. Unlike conventional methods, the proposed does not require iteration during timestepping, which is favorable in simulating nonlinear electromagnetic phenomena. |
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