电磁本征值问题的理论与数值计算方法研究

Electromagnetic eigenvalue problems are a very important research topic in the area of microwave techniques, the design of waveguides and resonant cavities is close relation with the solutions of electromagnetic eigenvalue problems. Firstly, this project will propose a sufficient and necessary condition for the existence of the pure TE and TM modes in the closed waveguide filled with a homogeneous, fully anisotropic and lossless medium, and prove the orthogonality of TE and TM modes in the anisotropic waveguide. The sufficient and necessary condition is the relation between the permittivity and permeability tensors, and this theory is the extension of the classic electromagnetic waveguide theory. Next, we use mixed finite element method and mixed spectral element method to solve the closed waveguide problems filled with the most general anisotropic media, and then analyze the propagation and dispersion characteristics of the propagation modes in the closed waveguide. In order to verify the correctness and reasonability of the above proposed sufficient and necessary condition, these numerical methods are applied to simulate the closed waveguide problem with a homogeneous, fully anisotropic and lossless medium. Finally, several numerical computational methods which can be used to solve 3-D resonant cavity problems with lossy media are proposed. All the numerical computational methods in the project will enforce the Gauss's law of electromagnetic field, such that these numerical computational methods can not introduce any spurious modes in solving electromagnetic eigenvalue problems.

电磁本征值问题是微波技术领域中十分重要的研究课题，波导与谐振腔的设计与电磁本征值问题的求解密切相关。首先，本项目将建立填充均匀、完全各向异性、无损耗介质的闭波导存在纯净TE模和TM模的一个充分必要条件，并证明各向异性波导中TE模和TM模的正交性。此充分必要条件与介质的介电张量与磁导率张量有关，这个理论是经典电磁波导理论的推广。接着我们使用混合有限元法和混合谱元法去求解填充最一般各向异性介质的闭波导问题，并分析这些闭波导中传播模式的传输特性与色散特性。为了验证上面所提充分必要条件的正确性与合理性，应用这些数值方法去模拟填充均匀、完全各向异性、无损耗介质的闭波导问题。最后设计一些可用于求解填充耗散介质的三维谐振腔问题的数值计算方法。本项目中所有的数值计算方法将约束电磁场的高斯定律，使得这些数值计算方法在求解电磁本征值问题中不会引入任何伪模式。

电磁本征值问题是微波技术领域中十分重要的研究课题，波导与谐振腔的设计与电磁本征值问题的求解密切相关。本项目建立了填充均匀、完全各向异性、无损耗介质的闭波导存在纯净TE模和TM模的一个充分必要条件，此充分必要条件与介质的介电张量与磁导率张量有关，这个理论是经典电磁波导理论的推广。这个理论的建立将有助于辨识各项异性波导中传播模式类型，存在一定的应用前景。此外，数值实验验证了该理论的正确性。本项目设计了一些可用于求解三维线性谐振腔问题的数值计算方法，在数值模拟过程中由于约束了电磁场的高斯定律，从而使得这些数值计算方法在求解三维线性谐振腔特征值问题中不会引入任何伪模式，包括伪零模式。特别的，当谐振腔中的介质是无磁损耗介质时，本项目中提出的增广方法可以十分高效地求解这种的谐振腔问题，而且增广方法没有破坏矩阵的稀疏性，因此可以利用高效的迭代算法来求解广义矩阵特征值问题，从而可以快速获得三维谐振腔的谐振模式。以上这些数值算法可以快速求解三维谐振腔的工作模式，这有助于微波谐振腔的工业应用设计。