Study On Finite Element Method In Time Domain And Its Truncation Boundary Conditions

At the present time, finite element method (FEM) has been acknowledged to be one of the efficient approaches to solve the electromagnetic problems in frequency domain. Finite element method in time domain (TDFEM) is able to solve directly the electromagnetic problems in time domain, and the investigation was started in 1980's. TDFEM has been greatly developed in the following decades.In this dissertation, TDFEM will be investigated based on solving the second-order vector wave equation in time domain. The principles of FEM and TDFEM are introduced first, and then considering the eigen-frequency of the system, frequency sampling condition and the stability of the time discretization for a harmonic field, the stability condition of time discretization for TDFEM will be derived by a new approach.The major disadvantage of TDFEM is that a matrix equation needs to be solved in each time step, and it will result in a lower calculation efficiency of the whole system. Reduction of the complexity of the size or the form of the system matrix and development of the more efficient matrix solver are the two approaches to increase the calculation efficiency, and the first approach will be discussed only in this dissertation. The size of the system matrix mainly depends on the sizes of the computation domain and the grids. Besides the employment of the mesh generators with high performance, one can employ the efficient truncation boundary conditions in order to reduce the size of the computation domain.MEI (Measured Equation of Invariance) method was first used for finite-difference method as an absorbing boundary condition (ABC). Before long, the measured equation of invariance in time domain (TD-MEI) was innovated and applied to FDTD, and an accurate solution was obtained with a small computation domain. In this work, the measuring equation is derived by using the property of the basic function and the first order ABC, and the MEI-ABC is constructed. The analysis shows that MEI-ABC can maintain the sparsity of the system matrix.For the scalar electromagnetic problems, MEI-ABC can significantly reduce the size of the computation domain, and MEI-ABC is greatly superior to the first order ABC in the accuracy of the solution. The numerical results show that MEI-ABC generates the number of the unknowns being less than that of the first order ABC with the same accuracy.However, for the vector electromagnetic problems, MEI-ABC has the superiority no longer, and it will be unstable. A modified MEI-ABC is presented based on the invariance of MEI method, the surface impedance boundary condition in time domain (TDSIBC) is used, and it increases the accuracy significantly. However, the cost of the computation will go up consequently.Compared with MEI-ABC, perfectly matched layer (PML) can obtain better accuracy while need more computation time. In this dissertation, the conformal perfectly matched layer (CPML) is investigated and applied to TDFEM.CPML can be considered as a generalized PML, and the analysis shows that the geometry of CPML and the local constitutive parameters of the media strongly depend on the smallest convex surface enclosing the scatterer. Compared with PML, CPML has less PML domain and computation domain, thus results in less requisition of the unknowns and reducation of the computation complexity in each time step, the computation time reduces consequently. The numerical results demonstrate that CPML can employ a smaller computation domain than that of PML with the same accuracy.Based on the results of this dissertation, the new truncation boundary conditions will be of benefit to the future development of TDFEM.