Theory And Application Of Discontinuous Galerkin Method In Transient Electromagnetic Problems

Discontinuous Galerkin methods(DG methods) are a class of finite element methods using completely discontinuous basis functions to solve vector Maxwell’s equations in first-order form. The DG methods lead to block-diagonal mass matrices and therefore yield fully explicit scheme when coupled with explicit time stepping. Consequently, these methods possess high computational efficiency, and are inherently suitable for parallel computing. By the use of numerical fluxes, continuity is weakly enforced across mesh interfaces to recover the global solutions of the electromagnetic initial-boundary value problem. Furthermore, the DG methods can employ both structured and unstructured grids that conform to complex boundary geometry, and therefore are suitable for modelling complex electromagnetic problems. This dissertation focuses on investigating the DG methods and their applications for transient electromagnetic problems, and the main contents are listed as follows:Due to existence of different versions of DG methods in the literatures(such as in weak and strong form, for vector and scalar basis), a general theory of DG methods is firstly summarized. Based on the general theory, the DG methods with tetrahedrons and hexahedrons are studied exhaustively and systematically. Using the energy approach, the L2-stability of the corresponding DG methods is proved, and a sufficient condition on the time step is also given explicitly.In order to simulate the electromagnetic transient processes, different boundary conditions are deeply studied in the DG methods. Coupled with the numerical fluxes, the implentations of PEC boundary, analytic absorbing boundary, and the plane-wave source on the surface of perfect conductor is proposed. Furthermore, an unsplit PML formulation suited for DG methods is introduced to build better simulations of reflectionless propagation of electromagnetic waves. At last, numerical examples validate the accuracy and efficiency of different boundary conditions.Spurious modes generation in DG methods is deeply investigated in this dissertation. At the same time, two interior penalty schemes are proposed based on different time integration methods, and the stabilities are respectively proved by the energy approach. Fourier analysis shows that both schemes can not only decrease numerical dispersion errors, but they can also eliminate spurious modes propagation. Thus, the mechanism of suppression of the spurious modes for the DG methods with penalty terms is revealed.To mitigate the problem of declining computational efficiency caused by strong cell size disparities, two forms of local time-stepping methods(respectively based on the leapfrog and the Verlet methods) are studied systematically. Moreover, an improved local time-stepping algorithm is proposed based on a pre-existing work. This method has lower numerical dispersive errors and the capability of suppressing spurious modes. A correction procedure is presented to greatly reduce the computation time over a single time-step.Finally, pre-treatment processes of the DG methods are introduced in detail, including geometry modeling and clean-up, the discretization of computational domain by tetrahedral and hexahedral grids. This has practical meanings for pushing the electromagnetic numerical methods to engineering to achieve the transfer between academic significance and industial values. The results of transient scattering from complex objects are in good agreement with that of other numerical methods, which illustrates the accuracy and efficiency of the DG methods.