时域有限元法及其截断边界条件的研究
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摘要
当前,有限元法(Finite Element Method,简称为FEM)已成为求解电磁场问题的有效的频域数值方法之一。时域有限元法(Finite Element Methodin Time Domain,简称TDFEM)能够直接在时域内求解电磁场问题。时域有限元法的研究始于20世纪80年代,并在随后的十几年中得到了快速的发展。
本文研究基于求解时域二阶矢量波动方程的时域有限元法。文中首先阐述有限元法及时域有限元法的一般原理,然后结合系统的本征频率、频率采样条件以及时谐场对时间离散间隔的稳定性要求等,从一种新的途径导出时域有限元法的时间离散稳定性条件。
时域有限元法的主要缺点是在每个时间步内都必需求解矩阵方程,因而系统的计算效率较低。为了提高系统的计算效率,可以进行两方面努力:一方面是减小系统矩阵的规模或改变其形式:另一方面是寻求更快的矩阵求解算法,本文讨论前一种措施。系统矩阵的规模主要取决于计算区域的大小和网格的粗细。为了减小系统矩阵的规模,除了使用高质量的网格剖分工具外,还可使用高效的截断边界条件。
MEI(Measured Equation of Invariance,简称为MEI)方法最初作为吸收边界条件用于有限差分法。很快人们提出了时域MEI方法(MeasuredEquation of Invariance in Time Domain,简称TD-MEI),并成功地应用于FDTD,能够以较小的计算区域取得精确的解。本文将TD-MEI方法应用于时域有限元法,结合基函数的性质,从一阶吸收边界条件(Absorbing BoundaryCondition,简称ABC)反推出MEI测试方程,并据此构建时域测度不变方程吸收边界条件,或称为MEI-ABC。分析发现MEI-ABC能够保持系统矩阵的稀疏性。
对于标量电磁问题的计算,MEI-ABC能够有效地减小计算区域。与一阶吸收边界条件相比,MEI-ABC在精度上具有明显的优势。数值算例证明,在相同的精度要求下,MEI-ABC的未知量更少。
但是,对于矢量电磁问题的计算,MEI-ABC相对一阶吸收边界条件的优势有限,而且稳定性差。为此,本文基于MEI不变性,对MEI-ABC作了改进,构建一种时域阻抗边界条件(Surface Impedance Boundary Condition intime domain,简称TDSIBC),使精度有了很大的提高,但是计算代价也相应增加。
相比MEI-ABC的快速优势,完全匹配层(Perfectly Matched Layer,简称PML)边界条件的计算速度较慢,但是精度更高。本文研究共形完全匹配层(Conformal Perfectly Matched Layer,简称CPML)技术,并应用于时域有限元法,取得了良好的结果。
CPML可理解为广义PML,分析发现CPML的形状和介质参数与包围散射体的最小凸曲面有关。数值算例证明,与常规PML相比,使用CPML时,PML区域及整个计算区域更小,因而未知量减少,同时简化了每个时间步内的计算步骤,因而节省计算时间,且不降低精度。
本文研究的截断边界条件为今后的时域有限元分析提供了更多的选择余地。
At the present time, finite element method (FEM) has been acknowledged to be one of the efficient approaches to solve the electromagnetic problems in frequency domain. Finite element method in time domain (TDFEM) is able to solve directly the electromagnetic problems in time domain, and the investigation was started in 1980's. TDFEM has been greatly developed in the following decades.
In this dissertation, TDFEM will be investigated based on solving the second-order vector wave equation in time domain. The principles of FEM and TDFEM are introduced first, and then considering the eigen-frequency of the system, frequency sampling condition and the stability of the time discretization for a harmonic field, the stability condition of time discretization for TDFEM will be derived by a new approach.
The major disadvantage of TDFEM is that a matrix equation needs to be solved in each time step, and it will result in a lower calculation efficiency of the whole system. Reduction of the complexity of the size or the form of the system matrix and development of the more efficient matrix solver are the two approaches to increase the calculation efficiency, and the first approach will be discussed only in this dissertation. The size of the system matrix mainly depends on the sizes of the computation domain and the grids. Besides the employment of the mesh generators with high performance, one can employ the efficient truncation boundary conditions in order to reduce the size of the computation domain.
MEI (Measured Equation of Invariance) method was first used for finite-difference method as an absorbing boundary condition (ABC). Before long, the measured equation of invariance in time domain (TD-MEI) was innovated and applied to FDTD, and an accurate solution was obtained with a small computation domain. In this work, the measuring equation is derived by using the property of the basic function and the first order ABC, and the MEI-ABC is constructed. The analysis shows that MEI-ABC can maintain the sparsity of the system matrix.
For the scalar electromagnetic problems, MEI-ABC can significantly reduce the size of the computation domain, and MEI-ABC is greatly superior to the first order ABC in the accuracy of the solution. The numerical results show that MEI-ABC generates the number of the unknowns being less than that of the first order ABC with the same accuracy.
However, for the vector electromagnetic problems, MEI-ABC has the superiority no longer, and it will be unstable. A modified MEI-ABC is presented based on the invariance of MEI method, the surface impedance boundary condition in time domain (TDSIBC) is used, and it increases the accuracy significantly. However, the cost of the computation will go up consequently.
Compared with MEI-ABC, perfectly matched layer (PML) can obtain better accuracy while need more computation time. In this dissertation, the conformal perfectly matched layer (CPML) is investigated and applied to TDFEM.
CPML can be considered as a generalized PML, and the analysis shows that the geometry of CPML and the local constitutive parameters of the media strongly depend on the smallest convex surface enclosing the scatterer. Compared with PML, CPML has less PML domain and computation domain, thus results in less requisition of the unknowns and reducation of the computation complexity in each time step, the computation time reduces consequently. The numerical results demonstrate that CPML can employ a smaller computation domain than that of PML with the same accuracy.
Based on the results of this dissertation, the new truncation boundary conditions will be of benefit to the future development of TDFEM.
本文研究基于求解时域二阶矢量波动方程的时域有限元法。文中首先阐述有限元法及时域有限元法的一般原理,然后结合系统的本征频率、频率采样条件以及时谐场对时间离散间隔的稳定性要求等,从一种新的途径导出时域有限元法的时间离散稳定性条件。
时域有限元法的主要缺点是在每个时间步内都必需求解矩阵方程,因而系统的计算效率较低。为了提高系统的计算效率,可以进行两方面努力:一方面是减小系统矩阵的规模或改变其形式:另一方面是寻求更快的矩阵求解算法,本文讨论前一种措施。系统矩阵的规模主要取决于计算区域的大小和网格的粗细。为了减小系统矩阵的规模,除了使用高质量的网格剖分工具外,还可使用高效的截断边界条件。
MEI(Measured Equation of Invariance,简称为MEI)方法最初作为吸收边界条件用于有限差分法。很快人们提出了时域MEI方法(MeasuredEquation of Invariance in Time Domain,简称TD-MEI),并成功地应用于FDTD,能够以较小的计算区域取得精确的解。本文将TD-MEI方法应用于时域有限元法,结合基函数的性质,从一阶吸收边界条件(Absorbing BoundaryCondition,简称ABC)反推出MEI测试方程,并据此构建时域测度不变方程吸收边界条件,或称为MEI-ABC。分析发现MEI-ABC能够保持系统矩阵的稀疏性。
对于标量电磁问题的计算,MEI-ABC能够有效地减小计算区域。与一阶吸收边界条件相比,MEI-ABC在精度上具有明显的优势。数值算例证明,在相同的精度要求下,MEI-ABC的未知量更少。
但是,对于矢量电磁问题的计算,MEI-ABC相对一阶吸收边界条件的优势有限,而且稳定性差。为此,本文基于MEI不变性,对MEI-ABC作了改进,构建一种时域阻抗边界条件(Surface Impedance Boundary Condition intime domain,简称TDSIBC),使精度有了很大的提高,但是计算代价也相应增加。
相比MEI-ABC的快速优势,完全匹配层(Perfectly Matched Layer,简称PML)边界条件的计算速度较慢,但是精度更高。本文研究共形完全匹配层(Conformal Perfectly Matched Layer,简称CPML)技术,并应用于时域有限元法,取得了良好的结果。
CPML可理解为广义PML,分析发现CPML的形状和介质参数与包围散射体的最小凸曲面有关。数值算例证明,与常规PML相比,使用CPML时,PML区域及整个计算区域更小,因而未知量减少,同时简化了每个时间步内的计算步骤,因而节省计算时间,且不降低精度。
本文研究的截断边界条件为今后的时域有限元分析提供了更多的选择余地。
At the present time, finite element method (FEM) has been acknowledged to be one of the efficient approaches to solve the electromagnetic problems in frequency domain. Finite element method in time domain (TDFEM) is able to solve directly the electromagnetic problems in time domain, and the investigation was started in 1980's. TDFEM has been greatly developed in the following decades.
In this dissertation, TDFEM will be investigated based on solving the second-order vector wave equation in time domain. The principles of FEM and TDFEM are introduced first, and then considering the eigen-frequency of the system, frequency sampling condition and the stability of the time discretization for a harmonic field, the stability condition of time discretization for TDFEM will be derived by a new approach.
The major disadvantage of TDFEM is that a matrix equation needs to be solved in each time step, and it will result in a lower calculation efficiency of the whole system. Reduction of the complexity of the size or the form of the system matrix and development of the more efficient matrix solver are the two approaches to increase the calculation efficiency, and the first approach will be discussed only in this dissertation. The size of the system matrix mainly depends on the sizes of the computation domain and the grids. Besides the employment of the mesh generators with high performance, one can employ the efficient truncation boundary conditions in order to reduce the size of the computation domain.
MEI (Measured Equation of Invariance) method was first used for finite-difference method as an absorbing boundary condition (ABC). Before long, the measured equation of invariance in time domain (TD-MEI) was innovated and applied to FDTD, and an accurate solution was obtained with a small computation domain. In this work, the measuring equation is derived by using the property of the basic function and the first order ABC, and the MEI-ABC is constructed. The analysis shows that MEI-ABC can maintain the sparsity of the system matrix.
For the scalar electromagnetic problems, MEI-ABC can significantly reduce the size of the computation domain, and MEI-ABC is greatly superior to the first order ABC in the accuracy of the solution. The numerical results show that MEI-ABC generates the number of the unknowns being less than that of the first order ABC with the same accuracy.
However, for the vector electromagnetic problems, MEI-ABC has the superiority no longer, and it will be unstable. A modified MEI-ABC is presented based on the invariance of MEI method, the surface impedance boundary condition in time domain (TDSIBC) is used, and it increases the accuracy significantly. However, the cost of the computation will go up consequently.
Compared with MEI-ABC, perfectly matched layer (PML) can obtain better accuracy while need more computation time. In this dissertation, the conformal perfectly matched layer (CPML) is investigated and applied to TDFEM.
CPML can be considered as a generalized PML, and the analysis shows that the geometry of CPML and the local constitutive parameters of the media strongly depend on the smallest convex surface enclosing the scatterer. Compared with PML, CPML has less PML domain and computation domain, thus results in less requisition of the unknowns and reducation of the computation complexity in each time step, the computation time reduces consequently. The numerical results demonstrate that CPML can employ a smaller computation domain than that of PML with the same accuracy.
Based on the results of this dissertation, the new truncation boundary conditions will be of benefit to the future development of TDFEM.
引文
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