电磁波传播问题的高性能数值算法研究
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摘要
现代技术的许多方面都与电磁场尤其是高频电磁场密切相关,对复杂工程电磁场问题的分析和计算成为现在技术发展的重要课题。本博士论文分别针对现代工程电磁场问题中的分层结构问题、多尺度问题、以及非线性问题在数值计算上存在的困难进行了研究探索。
分层结构分析的困难在于计算电磁学中传统数值法需要对整个分层结构离散,这有可能导致需要求解的系统矩阵方程规模十分巨大甚至不可接受。多尺度问题分析的困难主要有两方面,一是传统数值方法在空间离散上不灵活导致的未知量数目过多,二是时间步长大小受到限制,导致计算步数过多。这两方面导致多尺度问题分析的计算量非常大。非线性问题分析的困难在于传统的非线性数值计算方法都是基于迭代的方法对非线性方程进行求解,但在迭代过程中经常会遇到收敛速度慢或不稳定性的问题。
本文针对传统的数值算法在分层结构问题、多尺度问题、以及非线性问题分析中存在的主要困难,提出更加有效的数值算法,解决传统数值算法在现代电磁场工程中无法解决的困难。在频域问题方面,本文实现了对分层电磁结构的高精度高效率的数值分析,在时域问题方面,本文发展了一种求解瞬态多尺度电磁问题的高效时域混合算法,并将研究范围扩展到非线性电磁场问题,提出了一种基于参变量二次规划算法的时域有限元格式用于求解非线性麦克斯韦方程,具体研究内容如下：
针对分层结构分析中存在的困难,本文提出了基于精细积分的半解析有限谱单元算法。利用了分层结构沿纵向均匀的性质,仅需要在横截面进行网格离散。在Hamilton体系下,将力学中区段混合能的概念扩展到电磁波导问题中,利用能达到计算机精度的黎卡提方程的精细积分方法计算出每一个子结构的出口刚度阵,通过限谱单元离散子结构横截面,大幅降低了未知量数目,解决了目前传统有限元法在计算分层电磁结构问题过程中面临的计算效率过低问题。数值算例表明,在相同精度条件下,本文提出的算法效率比传统有限元提高了数个数量级。
针对瞬态多尺度问题分析中存在的困难,发展了一种基于间断Galerkin法的时域有限元／有限差分混合算法。首次提出了一种针对二维问题(TM模和TE模)的时域间断Galerkin有限元格式,并以时域间断Galerkin法为框架,将时域有限元法和有限差分法结合在一起,允许相邻各子区域在分界面上具有“非共形(non-conforming)"网格,极大提高了网格的灵活性,避免了这两种时域数值方法各自的劣势,充分发挥它们各自的优势,适合并行计算。解决了目前传统时域数值算法在计算瞬态多尺度电磁问题过程中存在的计算规模过大的问题。
针对非线性问题分析中存在的困难,发展了一种基于参变量二次规划算法的时域有限元算法。通过运用参变量二次规划的思想,将复杂非线性问题转化为一系列二次规划的线性互补问题,不需要迭代求解过程,因此避免了收敛速度慢或者发散的问题,相对于传统非线性计算方法具有更好的收敛性。为求解非线性麦克斯韦方程提供了新的思路和方法。
High frequency electromagnetics is closely related to many aspects of modern technology. It is a very important topic to solve the complex electromagnetic problems with numerical methods. With the increasing complexity of electromagnetic engineering problems, traditional numerical methods are facing more and more challenges.
Electromagnetic simulations of layered structures problem usually contains several parallel layers homogeneous along a specific direction. Due to the flexibility in geometric modeling, the conventional numerical method can be employed to perform full wave analysis, and thus to obtain the electrical properties of layered structures. However, as the number of layers and the complexity of each layer increase, directly using the conventional numerical method to discretize the whole structure may lead to a huge system of coupled equations, thus making the overall efficiency of conventional numerical method very low for the analysis of layered structures. Multiscale electromagnetic simulation is another type of problem with wide application but very challenging for conventional methods. For a multiscale structure, conventional methods use a single mesh/grid to discretize this kind of structure, which will lead to a large number of wasted unknowns. Temporal integration would be another difficulty. Small cells will lead to extremely small time steps and an unaffordable number of calculations in time integration for explicit schemes. The conventional nonlinear numerical methods for nonlinear Maxwell's equation usually require iteration, which maybe lead to a low convergence.
All in all, conventional numerical methods have not been satisfying the requirements of modern electromagnetic engineering. In order to meet the specific needs of practical engineering, some valuable studies for layered electromagnetic structure and multiscale electromagnetic problems are conducted in this thesis. We proposed a semi-analytical spectral element method for analysis of layered structures, a hybrid finite-element/finite-difference time domain technique for multiscale electromagnetic problems, and a novel finite element time domain method for nonlinear Maxwell's equations.
The main contributions of this thesis can be summarized as follows:
1. We proposed a semi-analytical method with high efficiency and high accuracy for frequency domain layered electromagnetic problems. A piecewise homogeneous3-D layered structure is divided into several substructures.2-D scalar and vector spectral elements are used to represent longitudinal and transverse unknowns on the cross section of each substructure, respectively. The semidiscretized system is then transformed from the Lagrangian system into the Hamiltonian system, where a Riccati equation-based high precision integration (HPI) method is utilized to perform integration along the longitudinal direction and to generate the stiffness matrix of a substructure. This method can be several orders more efficient and accurate than conventional FEM for layered structures.
2. We proposed a hybrid finite-element/finite-difference time domain technique for transient electromagnetic simulations. Based on the discontinuous Galerkin method we combined finite element time domain method with finite difference time domain method to take advantages of both the flexibility of the FETD method and the efficiency of the FDTD method. This hybrid scheme allows nonconforming meshes and multiple subdomains, thus makes it very flexible in geometric modeling. The proposed method is very competitive in solving multiscale electromagnetic problems.
3. We proposed a novel finite element time domain method to solve the nonlinear Maxwell's equations. Based on the quadratic programming method the nonlinear constitutive relations are treated as a series of linear complementary problems. Unlike conventional methods, the proposed does not require iteration during timestepping, which is favorable in simulating nonlinear electromagnetic phenomena.
分层结构分析的困难在于计算电磁学中传统数值法需要对整个分层结构离散,这有可能导致需要求解的系统矩阵方程规模十分巨大甚至不可接受。多尺度问题分析的困难主要有两方面,一是传统数值方法在空间离散上不灵活导致的未知量数目过多,二是时间步长大小受到限制,导致计算步数过多。这两方面导致多尺度问题分析的计算量非常大。非线性问题分析的困难在于传统的非线性数值计算方法都是基于迭代的方法对非线性方程进行求解,但在迭代过程中经常会遇到收敛速度慢或不稳定性的问题。
本文针对传统的数值算法在分层结构问题、多尺度问题、以及非线性问题分析中存在的主要困难,提出更加有效的数值算法,解决传统数值算法在现代电磁场工程中无法解决的困难。在频域问题方面,本文实现了对分层电磁结构的高精度高效率的数值分析,在时域问题方面,本文发展了一种求解瞬态多尺度电磁问题的高效时域混合算法,并将研究范围扩展到非线性电磁场问题,提出了一种基于参变量二次规划算法的时域有限元格式用于求解非线性麦克斯韦方程,具体研究内容如下：
针对分层结构分析中存在的困难,本文提出了基于精细积分的半解析有限谱单元算法。利用了分层结构沿纵向均匀的性质,仅需要在横截面进行网格离散。在Hamilton体系下,将力学中区段混合能的概念扩展到电磁波导问题中,利用能达到计算机精度的黎卡提方程的精细积分方法计算出每一个子结构的出口刚度阵,通过限谱单元离散子结构横截面,大幅降低了未知量数目,解决了目前传统有限元法在计算分层电磁结构问题过程中面临的计算效率过低问题。数值算例表明,在相同精度条件下,本文提出的算法效率比传统有限元提高了数个数量级。
针对瞬态多尺度问题分析中存在的困难,发展了一种基于间断Galerkin法的时域有限元／有限差分混合算法。首次提出了一种针对二维问题(TM模和TE模)的时域间断Galerkin有限元格式,并以时域间断Galerkin法为框架,将时域有限元法和有限差分法结合在一起,允许相邻各子区域在分界面上具有“非共形(non-conforming)"网格,极大提高了网格的灵活性,避免了这两种时域数值方法各自的劣势,充分发挥它们各自的优势,适合并行计算。解决了目前传统时域数值算法在计算瞬态多尺度电磁问题过程中存在的计算规模过大的问题。
针对非线性问题分析中存在的困难,发展了一种基于参变量二次规划算法的时域有限元算法。通过运用参变量二次规划的思想,将复杂非线性问题转化为一系列二次规划的线性互补问题,不需要迭代求解过程,因此避免了收敛速度慢或者发散的问题,相对于传统非线性计算方法具有更好的收敛性。为求解非线性麦克斯韦方程提供了新的思路和方法。
High frequency electromagnetics is closely related to many aspects of modern technology. It is a very important topic to solve the complex electromagnetic problems with numerical methods. With the increasing complexity of electromagnetic engineering problems, traditional numerical methods are facing more and more challenges.
Electromagnetic simulations of layered structures problem usually contains several parallel layers homogeneous along a specific direction. Due to the flexibility in geometric modeling, the conventional numerical method can be employed to perform full wave analysis, and thus to obtain the electrical properties of layered structures. However, as the number of layers and the complexity of each layer increase, directly using the conventional numerical method to discretize the whole structure may lead to a huge system of coupled equations, thus making the overall efficiency of conventional numerical method very low for the analysis of layered structures. Multiscale electromagnetic simulation is another type of problem with wide application but very challenging for conventional methods. For a multiscale structure, conventional methods use a single mesh/grid to discretize this kind of structure, which will lead to a large number of wasted unknowns. Temporal integration would be another difficulty. Small cells will lead to extremely small time steps and an unaffordable number of calculations in time integration for explicit schemes. The conventional nonlinear numerical methods for nonlinear Maxwell's equation usually require iteration, which maybe lead to a low convergence.
All in all, conventional numerical methods have not been satisfying the requirements of modern electromagnetic engineering. In order to meet the specific needs of practical engineering, some valuable studies for layered electromagnetic structure and multiscale electromagnetic problems are conducted in this thesis. We proposed a semi-analytical spectral element method for analysis of layered structures, a hybrid finite-element/finite-difference time domain technique for multiscale electromagnetic problems, and a novel finite element time domain method for nonlinear Maxwell's equations.
The main contributions of this thesis can be summarized as follows:
1. We proposed a semi-analytical method with high efficiency and high accuracy for frequency domain layered electromagnetic problems. A piecewise homogeneous3-D layered structure is divided into several substructures.2-D scalar and vector spectral elements are used to represent longitudinal and transverse unknowns on the cross section of each substructure, respectively. The semidiscretized system is then transformed from the Lagrangian system into the Hamiltonian system, where a Riccati equation-based high precision integration (HPI) method is utilized to perform integration along the longitudinal direction and to generate the stiffness matrix of a substructure. This method can be several orders more efficient and accurate than conventional FEM for layered structures.
2. We proposed a hybrid finite-element/finite-difference time domain technique for transient electromagnetic simulations. Based on the discontinuous Galerkin method we combined finite element time domain method with finite difference time domain method to take advantages of both the flexibility of the FETD method and the efficiency of the FDTD method. This hybrid scheme allows nonconforming meshes and multiple subdomains, thus makes it very flexible in geometric modeling. The proposed method is very competitive in solving multiscale electromagnetic problems.
3. We proposed a novel finite element time domain method to solve the nonlinear Maxwell's equations. Based on the quadratic programming method the nonlinear constitutive relations are treated as a series of linear complementary problems. Unlike conventional methods, the proposed does not require iteration during timestepping, which is favorable in simulating nonlinear electromagnetic phenomena.
引文
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