A body-of-revolution discontinuous Galerkin time domain method based on curved mesh for Maxwell equations

Numerical simulation of body-of-revolution (BoR) or axisymmetric structures for scattering and radiation problems has a broad spectrum of technology applications in microwave and optical engineering. In this work, we propose a high-order BoR discontinuous Galerkin time-domain method (BoR-DGTD) for solving Maxwell equations in the cylindrical coordinate system. The presented approach is based on $ \sin (m\varphi ) $ sin⁡(mφ) or $ \cos (m\varphi ) $ cos⁡(mφ) (m is the mode number and φ is the azimuth) expansions for different field components and thus yields a real-valued final system of equations. It saves half of the unknowns as compared to the existing method in which all the field components are assumed to have the same φ variation, i.e. $ e^{im\varphi } $ eimφ. In contrast to the common practice of deriving variational formulation for the BoR Maxwell equations on two-dimensional grids in the meridian plane, we build it on three-dimensional coaxial toroidal subdomains. As a result, the singularity difficulty induced by the cylindrical coordinates is eliminated naturally. In addition, for BoR with curved material interfaces in the angular cross-section, a solution using curved grids in conjunction with the low-storage weight-adjusted approximation scheme is provided, which significantly enhances the simulation quality. Particularly, to achieve a set of optimized interpolation shape functions for the isoparametric transformation, an essential assistant in calculating the coefficient matrices, a heuristic procedure is developed to properly position the geometric control points on the curved grids. Some numerical experiments are conducted to demonstrate the accuracy and performance of our proposed BoR-DGTD method.