The Study And Application Of Discontinuous Galerkin Time-domain Method Of Computational Electromagnetics

Compared with traditional finite element methods, Discontinuous Galerkin Time-domain method (DGTD) is on the basis of discontinuous basis functions and discontinuous testing functions, it takes the first-order vector Maxwell’s equations as the governing equation and adopts Galerkin methods to find solutions. Therefore, the DGTD is a special way of finite element methods. Its essence is to weakly enforced the continuity across mesh interface and solve the integration by employing numerical fluxes. Due to the versatile choices of spatial discretization and temporal integration, DGTD can be very promising in simulating complex electromagnetic problems like transient multi-scale problems. Moreover, when dealing with transient problems, DGTD deduces block-diagonal mass matrices. Using explicit time marching can obtain explicit scheme and only the small matrix in each element needs to be solved. It avoids the difficulties of solving a huge matrix system as in FETD schemes in every time step. This advantage of DGTD methods can save a large amount of memory during time marching, and furthermore, it makes parallel computation straightforward for a DGTD system. This paper, consisting of four parts, is mainly about several key steps of DGTD and its preliminary application in electromagnetic field.In the first section, some kinds of numerical methods are reviewed, as the differential form of FDTD, FETD and the integral form of TDIE. By focusing more on FDTD, a comparatively analysis of the merits, demerits, and application areas of the methods mentioned above is made in detail.The second section pays attention to the theory and numerical implementation of FEM, which is a continuous Galerkin method, providing solid basis to further researches of DGTD.In the third section, after investigating different versions of DGTD, a generalization of four main procedures of general format this method is presented. To be more specific, the first procedure is to determine governing equation, then choosing element shape and corresponding basis functions for the spatial discretization of each subdomain. After that applying numerical fluxes onto interfaces to stitch all subdomains together, and finally selecting a time scheme based on properties of a discretized system.Boundary conditions are extremely essential in transient problems. So section four deals with three issues. In the beginning, the realization of PEC condition with fictitious element is given, and an UPML formulation suited for DGTD is proposed for solving the electromagnetic problems in infinite space. In addition, an attempt of new time stepping scheme named as Newmark technique and applying a fast algorithm named "Casting Box" to accelerate finding the adjacent elements are developed. At last, the validity and efficiency of this method are proved by the numerical results.