电磁散射分析中的快速方法
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摘要
军事民用技术的发展，迫切要求精确高效地分析电大尺寸复杂目标的散射特性。随着计算机科技的迅猛发展，以及全波快速分析方法的不断提出与改进，许多以前只能用传统高频近似方法求解的问题，现在能够在单台普及型计算机上进行高精度的分析。本论文研究的正是电磁领域最近发展起来的几种快速全波分析方法。
第一种快速方法是快速傅里叶变换(FFT)加速电磁分析算法。我们将它应用到分析介质体散射的离散偶极子近似方法及体积分方程的矩量法中，并且讨论了文献中报道的FFT技术结合迭代解法求解体积分方程几种不同实现方案的优缺点。
第二种快速方法是不等间距的快速傅里叶变换(NUFFT)加速电磁分析算法，它保留了等间距FFT方法的优点，将存储量减少到了O(N)，而计算复杂度降低到了O(NlogN)，但突破了等间距网格离散的要求，给不少实际问题的分析带来了方便。论文中详细讨论了不等间距FFT的正、逆变换的实现过程，并改进了逆变换的实现效率。还将这一算法结合弱形式的积分方程，应用到金属平板及一维、二维和三维介质体的散射分析。
第三种快速方法是多层快速多极子加速电磁分析算法，它应用在时谐电磁场问题时将存储量及计算复杂度都降低到O(NlogN)，并且适用于任意形状的网格离散，具有很强的灵活性，文中将这一加速算法应用到金属体、介质体、以及无限大底板的微带结构的电磁散射特性分析中。在分析介质体散射时，论文详细研究了体积积分方程中的无散度基函数的使用与实现。论文还讨论了将矢量有限元法结合快速多极子技术来分析电大尺寸的带有有耗介质涂层的金属体的散射，详细地讨论了吸收边界条件预条件矩阵的对称特性的形成和预条件矩阵方程的快速求解。
然而，上述技术并不能减少Krylov子空间迭代解法的总的迭代步数，它是由积分方程算子或离散线性系统矩阵的谱的分布特性决定的，本文还研究了在采用不同积分方程求解方法时，如何选择效率高的迭代算法以及采用高效的预条件技术来降低迭代步数。大量数值算例验证了不同方法中我们提出的迭代解法及预条件技术方法的准确性与高效性。
The accurate and efficient analyses of electromagnetic scattering of complex objectswith electrically large size are urgently required by the quick development of technologyfor military and civil use. Fortunately, the combination of the modern computer technologyand the fast algorithm for rigorous full-wave analysis make it possible to givehigh-performance researches for numerous practical problems in a popular personalcomputer, which traditionally can only be analyzed by use of high-frequency approximateapproaches.
The dissertation investigates three kinds of fast algorithms to achieve the low memorycost and low operation complexity in the electromagnetic analyses.
First is the fast Fourier transform technique. We apply this technique into theimplementation of discrete dipole approximation method, and the method of moment ofvolume integral equation. The FFT technique combined with iterative solvers reported inthe literatures is compared for the ordinary volume integral equation and weakform ones.
The second is a newly developed scheme called non-uniform FFT (NUFFT). Itinherits the efficiency of the FFT and removes the limitation of the uniform mesh grid. Theprocedure for the inverse transform is improved. The applications in the analyses ofelectricmagnetic scattering from conductor plates and dielectric objects are discussed. Thememory requirement and computational complexity is O(N) and O(NlogN), respectively.
The third fast algorithm investigated is multilevel fast multipole approach (MLFMA).It is powerful and flexible since the arbitrary mesh partition can be permitted. Thedissertation discusses in detail on the implementation of the algorithm, and applies it toanalyze the scattering from the conductor objects, dielectric objects, and microstrip patchantenna. The solenoidal basis functions are adopted in the scattering analysis of dielectricobjects, which lead to a less number of unknowns for the same mesh as in the traditionalSchaubert-Wilton-Glisson basis functions.
The vector-edge finite element method combined with boundary integral (FE-BI)technique with MLFMA accelerator is also investigated in the dissertation. This hybridapproach is able to analyze various complicated problems with the advantages of bothFEM and MLFMA. It has been considered as one of the most powerful schemes to theelectromagnetic problems.
All the above fast algorithms can only accelerate the speed of every iteration, however,they do not reduce the total iteration number. Among the various Krylov subspace iterativesolvers, different solver for the certain problems is discussed. Some preconditioningschemes are proposed to further reduce the computational time of solvers. A lot ofnumerical examples illustrate the efficiency and accuracy of the fast algorithms studied inthis dissertation.
第一种快速方法是快速傅里叶变换(FFT)加速电磁分析算法。我们将它应用到分析介质体散射的离散偶极子近似方法及体积分方程的矩量法中，并且讨论了文献中报道的FFT技术结合迭代解法求解体积分方程几种不同实现方案的优缺点。
第二种快速方法是不等间距的快速傅里叶变换(NUFFT)加速电磁分析算法，它保留了等间距FFT方法的优点，将存储量减少到了O(N)，而计算复杂度降低到了O(NlogN)，但突破了等间距网格离散的要求，给不少实际问题的分析带来了方便。论文中详细讨论了不等间距FFT的正、逆变换的实现过程，并改进了逆变换的实现效率。还将这一算法结合弱形式的积分方程，应用到金属平板及一维、二维和三维介质体的散射分析。
第三种快速方法是多层快速多极子加速电磁分析算法，它应用在时谐电磁场问题时将存储量及计算复杂度都降低到O(NlogN)，并且适用于任意形状的网格离散，具有很强的灵活性，文中将这一加速算法应用到金属体、介质体、以及无限大底板的微带结构的电磁散射特性分析中。在分析介质体散射时，论文详细研究了体积积分方程中的无散度基函数的使用与实现。论文还讨论了将矢量有限元法结合快速多极子技术来分析电大尺寸的带有有耗介质涂层的金属体的散射，详细地讨论了吸收边界条件预条件矩阵的对称特性的形成和预条件矩阵方程的快速求解。
然而，上述技术并不能减少Krylov子空间迭代解法的总的迭代步数，它是由积分方程算子或离散线性系统矩阵的谱的分布特性决定的，本文还研究了在采用不同积分方程求解方法时，如何选择效率高的迭代算法以及采用高效的预条件技术来降低迭代步数。大量数值算例验证了不同方法中我们提出的迭代解法及预条件技术方法的准确性与高效性。
The accurate and efficient analyses of electromagnetic scattering of complex objectswith electrically large size are urgently required by the quick development of technologyfor military and civil use. Fortunately, the combination of the modern computer technologyand the fast algorithm for rigorous full-wave analysis make it possible to givehigh-performance researches for numerous practical problems in a popular personalcomputer, which traditionally can only be analyzed by use of high-frequency approximateapproaches.
The dissertation investigates three kinds of fast algorithms to achieve the low memorycost and low operation complexity in the electromagnetic analyses.
First is the fast Fourier transform technique. We apply this technique into theimplementation of discrete dipole approximation method, and the method of moment ofvolume integral equation. The FFT technique combined with iterative solvers reported inthe literatures is compared for the ordinary volume integral equation and weakform ones.
The second is a newly developed scheme called non-uniform FFT (NUFFT). Itinherits the efficiency of the FFT and removes the limitation of the uniform mesh grid. Theprocedure for the inverse transform is improved. The applications in the analyses ofelectricmagnetic scattering from conductor plates and dielectric objects are discussed. Thememory requirement and computational complexity is O(N) and O(NlogN), respectively.
The third fast algorithm investigated is multilevel fast multipole approach (MLFMA).It is powerful and flexible since the arbitrary mesh partition can be permitted. Thedissertation discusses in detail on the implementation of the algorithm, and applies it toanalyze the scattering from the conductor objects, dielectric objects, and microstrip patchantenna. The solenoidal basis functions are adopted in the scattering analysis of dielectricobjects, which lead to a less number of unknowns for the same mesh as in the traditionalSchaubert-Wilton-Glisson basis functions.
The vector-edge finite element method combined with boundary integral (FE-BI)technique with MLFMA accelerator is also investigated in the dissertation. This hybridapproach is able to analyze various complicated problems with the advantages of bothFEM and MLFMA. It has been considered as one of the most powerful schemes to theelectromagnetic problems.
All the above fast algorithms can only accelerate the speed of every iteration, however,they do not reduce the total iteration number. Among the various Krylov subspace iterativesolvers, different solver for the certain problems is discussed. Some preconditioningschemes are proposed to further reduce the computational time of solvers. A lot ofnumerical examples illustrate the efficiency and accuracy of the fast algorithms studied inthis dissertation.
引文
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