ADVANCED TIME-DOMAIN NUMERICAL METHODS FOR ELECTROMAGNETIC MODELING AND SIMULATION
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In this thesis, systematic investigations on divergence property of the divergence preserved alternatively-direction-implicit finite-difference time-domain (ADI-FDTD) method and the meshless method are carried out. It is found that divergence preserved ADI-FDTD method maintains the unconditional stability and the same numerical dispersion as that of the traditional ADI-FDTD method, while preserving electromagnetic divergence properties. To further improve its efficiency, an efficiency improved version is proposed. Theoretical proof of both the unconditional stability and the divergence property is provided. Almost 41.7% less count of floating-point operations than the original one is obtained. Investigations on the meshless method lead to the following results. (1) A meshless method for the wave equation is proposed to improve the efficiency based on the mathematical equivalence of the Maxwell’s equations and wave equation. Since the proposed method only requires to solve electrical field, computational efficiency of the proposed method is largely improved. (2) A divergence preserved meshless method based on the vector radial basis function (RBF) is proposed to solve the Maxwell’s equations. The conventional meshless method using Gaussian RBF cannot preserve the divergence property of electric and magnetic fields. The proposed method is theoretically proven to be divergence free and will not introduce the artificial charges in its numerical solutions. Numerical results show that the proposed method can accurately model the charge distribution in the simulations. (3) A stable meshless method based on QR-decomposition method is proposed to overcome the spatial stability issue of the conventional meshless method. The source of the spatial instability is removed from the Gaussian RBF through the QR-decomposition method. A new stable basis function, which share the same function space as that of the Gaussian RBF, is obtained. (4) The relationship between the meshless method and the FDTD method is theoretically investigated in terms of numerical dispersion. When node distribution and field component location is the same as that of the FDTD method, numerical dispersion of the meshless method can become the same as that of the FDTD method. It implies that the meshless method is a general method which includes the FDTD method as its special case.