传统时域有限差分（Finite-Difference Time-Domain，FDTD）方法在求解含精细结构目标电磁问题时会受到Courant-Friedrichs-Lewy（CFL）条件的限制使其劣势开始显现。一种基于空间滤模的显式无条件稳定FDTD（Explicit and unconditionally stable FDTD method based on spatial modes filtering, SMF-FDTD）方法能够在保证计算精度的前提下，从根本上消除当时间步长不满足CFL条件时发生的不稳定。SMF-FDTD方法提出这种不稳定是由于数值系统中的高阶模式，也称作不稳定模式导致的，将这些模式从数值系统中去除后就可以得到稳定的仿真结果。然而，这些不稳定模式需通过求解系统矩阵的特征值问题获得，当计算域中未知量数目过多时，求解特征值的开销巨大，严重降低了计算效率，限制了应用场景。此外，该算法目前应用范围也相对较窄，且若将现有的SMF-FDTD方法用于圆柱类目标电磁问题的分析，会产生台阶化误差从而导致计算结果不准确。

本文阐述了SMF-FDTD方法的基本原理框架，在现有SMF-FDTD方法的基础上，针对特征值求解效率低、应用场景较窄以及分析圆柱类目标时计算精度较差等问题给出了对应的解决方法。本文工作主要分为以下四个部分：

（1）发展了局部不稳定模式提取的快速SMF-FDTD方法（Fast SMF-FDTD method for local unstable modes extraction, FSMF-FDTD）。该方法的总体思路是通过降低待求解矩阵维度，从而降低特征值的求解时间。文中定量地分析了特征值与网格尺寸的关系，根据是否包含不稳定模式将计算域分为两个子域，通过求解包含不稳定模式子域对应小矩阵的特征值问题获得所有不稳定模式，达到从局部区域提取全域不稳定模式的目的。文中列出了两个实现FSMF-FDTD方法的关键问题，并给出了解决这两个关键问题的不同实施方案，均能有效提高计算效率。

（2）发展了周期结构SMF-FDTD方法。为了对周期结构进行简化，文中根据Floquet定理，阐述了当入射波垂直入射时将周期边界条件（Periodic Boundary Condition, PBC）引入SMF-FDTD方法的过程。这一过程是通过修改网格矩阵中未知量的耦合关系实现的。之后，文中对周期系统矩阵的特征值求解进行了说明。文中通过分析无限大介质板、耶路撒冷十字频选表面以及光子晶体等典型周期结构的电磁问题，说明了周期SMF-FDTD方法的准确性和高效性。针对光子晶体的特殊排布方式，结合SMF-FDTD方法的自身特点，文中还给出了一种更高效获取不稳定模式的方法。此外，研究了不同形状介质柱的几何参数对光子晶体禁带（photonic band gap, PBG）的影响。这增强了SMF-FDTD方法的工程实用性，扩大了其应用范围。

（3）在原有SMF-FDTD方法的基础上，提出了柱坐标SMF-FDTD（Cylindrical SMF-FDTD, CSMF-FDTD）方法用以快速分析圆柱类结构的电磁问题。文中介绍了传统柱坐标FDTD（Cylindrical FDTD, C-FDTD）方法的理论框架，之后系统地阐述了所提柱坐标SMF-FDTD方法中网格的剖分、系统矩阵的建立等基本理论。由于柱网格尺寸在φ方向上渐变，导致系统矩阵出现了不对称的情况，文中介绍了一种对称化处理方案，保证了所提方法显式中心差分格式的稳定性。为了更好的模拟一些精细的物理结构，文中还讨论了含细网格情形下的CSMF-FDTD方法。与直角坐标系类似，CSMF-FDTD方法也会出现全域特征值计算效率低的问题，本章在第三章基础上发展了快速CSMF-FDTD（Fast CSMF-FDTD for local unstable modes extraction, FCSMF-FDTD）方法，且该方法是一种通用方法。FCSMF-FDTD方法在CSMF-FDTD方法降低迭代时间的基础上，进一步提高了特征值分析的效率。

（4）发展了混合SMF-FDTD与传统FDTD方法。SMF-FDTD方法虽能明显降低显式时间步进的时长，但其代价是需要求解矩阵特征值问题。而传统FDTD方法无需进行矩阵运算，计算效率较高。当区域中没有不稳定模式时，传统FDTD方法和SMF-FDTD方法等价，二者的时间步长均由粗网格尺寸确定。为了最大程度的保留传统方法优点的同时又能实现无条件稳定，采用将SMF-FDTD方法和传统FDTD方法相结合的方式。文中分别介绍了直角坐标系和柱坐标系下的混合方法，说明了算法区域的划分以及场值交互的过程。数值结果说明了该方法的准确性和高效性。

Traditional finite-difference time-domain (FDTD) method is limited by Courant- Friedrichs-Lewy (CFL) condition when solving electromagnetic problems involving fine structures. The disadvantage of the traditional FDTD method shows up. An explicit and unconditionally stable FDTD method based on spatial modes filtering (SMF-FDTD) can eliminate the instability caused by the time step beyond the CFL condition fundamentally without compromising accuracy. The SMF-FDTD method claims that the instability is caused by a few high-order modes, which are also called unstable modes. By removing these modes from numerical system, a stable simulation result can be obtained. However, these unstable modes are found by solving eigenvalue problem of the system matrix. When there are large number of unknowns in the computational domain, the cost of solving eigenvalue is expensive, reducing the computational efficiency seriously and limits application scenarios. Besides, the range of applications of the existing SMF-FDTD method is relatively narrow. In addition, the step error is caused when analyzing cylindrical structures, which may leads to inaccurate results.

 

This work illustrates the framework of the SMF-FDTD method. Based on the existing SMF-FDTD method, the corresponding solutions are provided to address the issues such as low efficiency in solving eigenvalues, narrow application scenarios, and poor accuracy when analyzing cylindrical structures. This work is mainly divided into the following four parts:

 

(1) A fast SMF-FDTD method for local unstable modes extraction (FSMF-FDTD) is developed. The core of this method is to reduce the size of the system matrix, thereby the time for eigenvalue solution can be reduced. The relationship between the eigenvalues and grid size is quantificationally analyzed, and then the domain can be divided into two subdomains according to whether the unstable modes exist. By solving the eigenvalue problem of the small matrix corresponding to the subdomain contains unstable modes, all unstable modes can be obtained. It achieves obtaining global unstable modes in local area. Two key issues in FSMF-FDTD methods are listed and the corresponding implementations are given, which all improve the calculation efficiency.

 

(2) The SMF-FDTD method for periodic structures is proposed. First, in order to simplify the periodic structure, the periodic boundary condition (PBC) is introduced into SMF-FDTD method based on the Floquet theorem on the condition of vertical incidence of the plane wave. It is achieved by modifying the coupling relationship of unknowns in the grid matrix. Then the way to solve eigenvalues of the periodic system matrix is indicated. The accuracy and efficiency of the periodic SMF-FDTD method are demonstrated by analyzing the electromagnetic problems of several typical periodic structures, such as the infinite dielectric plate, the Jerusalem cross frequency selective surface and photonic crystals. For the special arrangement of photonic crystals, a more efficient way to obtain unstable modes is given. Further, the factors affecting the photonic band gap (PBG) are also discussed. It enhances the engineering practicality of the SMF-FDTD method and expands its application scope.

 

(3) Based on SMF-FDTD method, the cylindrical SMF-FDTD (CSMF-FDTD) method is proposed to analyze electromagnetic problems of cylindrical structures efficiently. Firstly, the framework of the traditional cylindrical FDTD (C-FDTD) method is introduced. Then the grid and the establishment of system matrix in CSMF-FDTD method are systematically elaborated. Due to the grid size in φ axis changes gradually, the system matrix is asymmetric. To ensure the stability of the explicit central difference scheme, a symmetrization scheme is introduced. In order to simulate fine structures, the CSMF-FDTD method involving fine grids is also discussed in this work. Similar to the problem exists in Cartesian coordinate system, the CSMF-FDTD method also suffers from low efficiency in solving global eigenvalues. Thus, a fast CSMF-FDTD (FCSMF-FDTD) method based on Chapter 3 is developed. In addition, it is a general method. The FCSMF-FDTD method not only reduces the iteration time compared to C-FDTD method, but also improves the efficiency of solving eigenvalue problem compared to CSMF-FDTD.

 

(4) A hybrid SMF-FDTD and traditional FDTD method is developed. Although the SMF-FDTD method could reduce the time for explicit time marching, it needs to solve the eigenvalue problem. The traditional FDTD method is a matrix-free method so it has relatively high computational efficiency. When there exists no unstable mode in the domain, the traditional FDTD method and the SMF-FDTD method are equivalent, and the time step are both determined by the size of the coarse grid. In order to preserve the advantages of traditional method while achieving unconditional stability, a combination of SMF-FDTD method and traditional FDTD method is adopted. The hybrid method are introduced in Cartesian and cylindrical coordinate systems. The domain decomposition and the correctness of boundary fields are illustrated. Numerical results demonstrate the accuracy and efficiency of this method.

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