基于电磁场体积分方程的区域分解方法研究
发布时间：2018-08-11 13:37
【摘要】：现代信息技术的许多领域都涉及求解复杂的电磁场问题,这促使许多高效、实用的电磁算法发展起来。但是,对于三维电大尺寸、高媒质参数、高对比度复杂介质结构电磁散射特性的快速、有效分析仍然面临挑战。为应对这样的挑战,本文研究基于电磁场体积分方程的区域分解方法,并研究在区域分解方法基础上进一步减少存储需求,提高计算效率,增强内迭代收敛性的方法。主要贡献如下：1.提出了基于体积分方程的矩量法(VIE-MoM)矩阵的快速填充方法。该方法对剖分单元进行统筹安排,消除了冗余计算,使矩阵填充时间减少了约80%。2.提出了基于体积分方程的拟合Green函数快速Fourier变换法(VIE-FG-FFT)。该方法通过拟合Green函数到均匀笛卡尔网格节点上,使得矩阵-向量积可被FFT加速,其计算复杂度为O(N log N),存储复杂度为O(N),其中N是未知量个数。与先前出现的基于FFT的方法比较,VIE-FG-FFT具有精度高、预处理时间少的优点。特别地,利用拟合Green函数步骤与媒质参数无关的特点,进一步将VIE-FG-FFT的应用范围扩展到了电各向异性介质目标。3.提出了基于体积分方程的重叠型区域分解方法(VIE-ODDM)及其严格的数学建模步骤。该方法将一个电大尺寸介质目标的VIE-MoM模型的全局求解问题方式转化为许多子域问题进行局部迭代求解,大幅度降低了内存需求,能够有效地求解那些其快速算法模型对用户计算机太大的介质体电磁散射问题。特别地,从理论和数值的角度研究了VIE-ODDM的外迭代格式的收敛性,证明是非常好的,并进一步将VIE-ODDM的应用范围扩展到电各向异性介质目标。4.提出了基于体积分方程的重叠型区域分解方法与拟合Green函数快速Fourier变换法的混合的方法(VIE-ODDM-FG-FFT)。该方法保持了VIE-ODDM的优点,进一步降低了存储需求,大幅度提高了计算效率,增强了内迭代的收敛性。与采用多层快速多极子算法(MLFMA)加速的方法比较,这里没有“次波长中断”问题。此外,由于引入了嵌套均匀笛卡尔网格方案,即使在媒质参数分布很不均匀的情形(如高对比度结构),该方法的计算效率也不会受到重大影响。5.提出了基于体积分方程的非重叠型区域分解方法(VIE-NDDM)并与FG-FFT形成混合方法。与VIE-ODDM不同,VIE-NDDM采用显式边界条件来形成各相邻子域之间的信息耦合,从而省去了构造缓冲区的工作量,减少了算法的预处理时间。类似于、TE-ODDM-FG-FFT那样,FG-FFT的引入进一步降低了存储需求,大幅度提高了计算效率。
[Abstract]:Many fields of modern information technology involve in solving complex electromagnetic problems, which promote the development of many efficient and practical electromagnetic algorithms. However, the fast and effective analysis of electromagnetic scattering characteristics of three-dimensional electrically large size, high medium parameters and high contrast complex dielectric structures still faces a challenge. In order to meet this challenge, this paper studies the domain decomposition method based on the volume fraction equation of electromagnetic field, and further reduces the storage requirement, improves the computational efficiency and enhances the convergence of the internal iteration based on the domain decomposition method. The main contributions are as follows: 1. A fast filling method for the method of moments (VIE-MoM) matrix based on the volume fraction equation is proposed. This method arranges the subdivision unit as a whole, eliminates the redundant calculation, and reduces the filling time of the matrix by about 80%. 2. A fast Fourier transform method (VIE-FG-FFT) for fitting Green function based on volume fraction equation is proposed. By fitting the Green function to the uniform Cartesian grid node, the matrix vector product can be accelerated by FFT, and its computational complexity is that the O (N log N), storage complexity is O (N), where N is the number of unknown variables. Compared with the previous methods based on FFT, VIE-FG-FFT has the advantages of high precision and less pretreatment time. In particular, by using the feature of fitting the Green function step independent of the medium parameters, the application of VIE-FG-FFT is further extended to the target of electrically anisotropic medium .3. An overlapping domain decomposition method (VIE-ODDM) based on volume fraction equation and its strict mathematical modeling steps are proposed. In this method, the global solution of a VIE-MoM model for an electrically large dielectric target is transformed into a number of subdomain problems for local iterative solution, which greatly reduces the memory requirement. It can effectively solve the electromagnetic scattering problem of dielectric bodies whose fast algorithm models are too large for user computers. In particular, the convergence of VIE-ODDM 's external iteration scheme is studied theoretically and numerically, which is proved to be very good. Furthermore, the application of VIE-ODDM is extended to the target of electrically anisotropic media .4. An overlap domain decomposition method based on volume fraction equation and a hybrid method (VIE-ODDM-FG-FFT) for fast Fourier transform of fitting Green function are proposed. This method preserves the advantages of VIE-ODDM, further reduces the storage requirements, greatly improves the computational efficiency and enhances the convergence of the inner iteration. Compared with the multilayer fast multipole algorithm (MLFMA) acceleration, there is no subwavelength interruption problem. In addition, due to the introduction of a nested uniform Cartesian mesh scheme, the computational efficiency of the method will not be significantly affected even in the case of very uneven distribution of medium parameters (such as high contrast structure). A nonoverlapping domain decomposition method (VIE-NDDM) based on volume fraction equation is proposed and mixed with FG-FFT. Unlike VIE-ODDM, VIE-NDDM uses explicit boundary conditions to form information coupling between adjacent subdomains, which saves the workload of constructing buffer zones and reduces the preprocessing time of the algorithm. The introduction of FG-FFT similar to that of TE-ODDM-FG-FFT further reduces the storage requirement and greatly improves the computational efficiency.
【学位授予单位】：东南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O441.4
[Abstract]:Many fields of modern information technology involve in solving complex electromagnetic problems, which promote the development of many efficient and practical electromagnetic algorithms. However, the fast and effective analysis of electromagnetic scattering characteristics of three-dimensional electrically large size, high medium parameters and high contrast complex dielectric structures still faces a challenge. In order to meet this challenge, this paper studies the domain decomposition method based on the volume fraction equation of electromagnetic field, and further reduces the storage requirement, improves the computational efficiency and enhances the convergence of the internal iteration based on the domain decomposition method. The main contributions are as follows: 1. A fast filling method for the method of moments (VIE-MoM) matrix based on the volume fraction equation is proposed. This method arranges the subdivision unit as a whole, eliminates the redundant calculation, and reduces the filling time of the matrix by about 80%. 2. A fast Fourier transform method (VIE-FG-FFT) for fitting Green function based on volume fraction equation is proposed. By fitting the Green function to the uniform Cartesian grid node, the matrix vector product can be accelerated by FFT, and its computational complexity is that the O (N log N), storage complexity is O (N), where N is the number of unknown variables. Compared with the previous methods based on FFT, VIE-FG-FFT has the advantages of high precision and less pretreatment time. In particular, by using the feature of fitting the Green function step independent of the medium parameters, the application of VIE-FG-FFT is further extended to the target of electrically anisotropic medium .3. An overlapping domain decomposition method (VIE-ODDM) based on volume fraction equation and its strict mathematical modeling steps are proposed. In this method, the global solution of a VIE-MoM model for an electrically large dielectric target is transformed into a number of subdomain problems for local iterative solution, which greatly reduces the memory requirement. It can effectively solve the electromagnetic scattering problem of dielectric bodies whose fast algorithm models are too large for user computers. In particular, the convergence of VIE-ODDM 's external iteration scheme is studied theoretically and numerically, which is proved to be very good. Furthermore, the application of VIE-ODDM is extended to the target of electrically anisotropic media .4. An overlap domain decomposition method based on volume fraction equation and a hybrid method (VIE-ODDM-FG-FFT) for fast Fourier transform of fitting Green function are proposed. This method preserves the advantages of VIE-ODDM, further reduces the storage requirements, greatly improves the computational efficiency and enhances the convergence of the inner iteration. Compared with the multilayer fast multipole algorithm (MLFMA) acceleration, there is no subwavelength interruption problem. In addition, due to the introduction of a nested uniform Cartesian mesh scheme, the computational efficiency of the method will not be significantly affected even in the case of very uneven distribution of medium parameters (such as high contrast structure). A nonoverlapping domain decomposition method (VIE-NDDM) based on volume fraction equation is proposed and mixed with FG-FFT. Unlike VIE-ODDM, VIE-NDDM uses explicit boundary conditions to form information coupling between adjacent subdomains, which saves the workload of constructing buffer zones and reduces the preprocessing time of the algorithm. The introduction of FG-FFT similar to that of TE-ODDM-FG-FFT further reduces the storage requirement and greatly improves the computational efficiency.
【学位授予单位】：东南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O441.4
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