【作者】 王俊；

【导师】 史小卫；

【作者基本信息】 西安电子科技大学 ， 电磁场与微波技术， 2015， 硕士

【摘要】 本文对时域不连续伽辽金法(Time-domain Discontinuous Galerkin Method,DGTD)进行了基本的理论研究,并与麦克斯韦方程相结合,研究了一维和二维DGTD方法。主要包括DGTD方法的基本理论和关键技术——数值流,以及不同时间积分策略和各种电磁边界条件在DGTD中的施加方法,最后还给出了二维形式的Debye色散媒质的建模方法。论文主要工作由以下几个部分组成:1.介绍了DGTD的基本理论,并同其他电磁场数值方法做了比较。通过麦克斯韦方程的守恒形式,推出DGTD方法下的麦克斯韦方程的弱形式。其中给出了基于Legendre-Gauss-Lobatto节点的高阶节点型基函数的基本空间分布,并且详细介绍和比较了DGTD的技术核心——数值流(包括中值流和迎风流)——的不同的数值特性,并给出了基于不同数值流的DGTD离散方程形式。在得到离散方程之后,给出了两种不同的数值时间积分策略来求解常微分方程组,即龙格库塔方法和蛙跳方法。最后,在一维情况下,给出算例,并分析了DGTD的空间划分形式和插值阶数对计算精度的影响。2.给出了二维形式下的DGTD一般形式,包括单元节点的数学表达式和空间分布,弱形式方程的基本形式和二维数值流的实现过程。接着,重点讨论了不同的电磁边界条件在DGTD方法下的施加方法,如PEC,PMC等,并引入了总场散射场边界和一阶Silver-Müller吸收边界,以便计算散射问题和辐射问题。但是,为了弥补和克服一阶Silver-Müller吸收边界的性能问题,引入了单轴各向异性介质完全匹配层(Uniaxial Perfectly Matched Laye,UPML)。为了在DGTD中有效地实现UPML,使用一种叫做辅助方程法(Auxiliary Equation Method,ADE)的方法,引入辅助量去得到解和UPML的参数。进一步地给出了Debye色散介质的基本电磁特性方程,并同样使用ADE方法,建模了DGTD形式下的Debye色散介质方程,并针对Debye的离散方程,改进了时间积分策略。在数值算例中,与解析解、FEM或FDTD做比较,验证了算法的正确性。更多还原

【Abstract】 In this paper, we studied the fundamental theory of Time-Domain Discontinuous Galerkin Method(DGTD), and combined it with Maxwell Equation in 1-D and 2-D. It mainly includes three parts: 1) the discussion of the fundamental theory of DGTD and its key idea — numerical fluxes; 2) two different temporal integration schemes and the implementation of various electromagnetic boundary conditions; 3) numerical model of the Debye dispersive medium with DGTD in 2-D.The thesis mainly consists of the following ideas:1. We introduced the DGTD theory and compared it with other electromagnetic numerical methods. We gave the weak form of DGTD system based on the conservation form of Maxwell Equations. In the introduction of DGTD weak form, the spatial distribution of the higher order nodal basis function is presented which is based on the Legendre-Gauss-Lobatto points. Then we presented and discussed the numerical properties of two different numerical fluxes, including central flux and upwind flux, and gave the discrete linear system of DGTD based on the two fluxes. After obtaining the discrete system, different temporal integration schemes—the Runge-Kutta method and Leap Frog method—are introduced and discussed to solve the ordinary differential equation system. Finally, the numerical example shows the hp-adaptivity of the DGTD method.2. The DGTD system in 2-D is presented, including the equation of spatial distribution of the nodal basis function, the weak form and the numerical fluxes in 2D. Then we discussed the implementation of various electromagnetic boundary conditions, e.g. PEC, PMC, etc. Moreover, the total-field/scatter field boundary condition and the 1st order Silver-Müller absorbing condition are adopted to simulate the scattering and radiation problems. However, in order to get a good absorbing effect of electromagnetic waves, we incorporated the Uniaxial Perfectly Matched Layer(UPML) with DGTD. The Auxiliary Equation Method(ADE) is used to model the UPML in DGTD. Similarly, the same technique is employed to obtain the discrete system of Debye medium in DGTD. An improved temporal integration scheme based on Runge-Kutta method is adopted to solve the Debye equations. The numerical examples which are compared with the analytical results including FEM and FDTD demonstrated the validity of the method.更多还原