迭代表面离散化边界方程法及其应用
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摘要
本文基于现有的表面离散化边界方程(OS-DBE)法提出迭代表面离散化边界方程(IT-OS-DBE)法,并将该方法用于二维理想导体散射问题、三维静电问题及三维理想导体散射问题的分析。
对二维导体散射问题,原始OS-DBE方法以分域基函数和场点(或匹配点)仅覆盖导体表面部分区域的方式独立求解表面任意点的电流,非常适合并行计算。其内存消耗量低,矩阵阶数小于矩量法的相应矩阵阶数,且二者之比随着散射体电尺寸的增大而减小。该方法还可与快速多极子方法(FMA)或多层快速多极子方法(MLFMA)相结合,以进一步降低矩阵方程求解的计算复杂度。原始OS-DBE方法计算效率的瓶颈在于：对每个电流计算点均求解一次矩阵方程。本文提出的IT-OS-DBE由一系列交替进行的OS-DBE求解过程和修正过程构成。IT-OS-DBE的OS-DBE求解过程中的矩阵阶数可比原始OS-DBE低1至2个数量级。并且,对于二维导体散射问题和三维静电问题,IT-OS-DBE的OS-DBE求解过程可采用所谓的“一点求逆”技术：对二维导体散射问题,将导体表面某点的切向散射磁场与相关场点的切向入射电场之间的耦合系数向量用于整个散射体表面电流分布的计算；对三维静电问题,将导体或介质表面某点的相应的耦合系数向量用于整个表面电荷或极化电荷分布的计算,从而使OS-DBE矩阵方程的求解次数降低至仅一次,最大限度地降低了OS-DBE求解过程的时间消耗。IT-OS-DBE的修正过程可采用FMA或MLFMA,以降低计算复杂度。IT-OS-DBE具有可与所求解问题的尺度无关的快速收敛特点,迭代次数可控制在数次以内。鉴于IT-OS-DBE迭代次数少,其修正过程与FMA/MLFMA相结合时,可有多种不同的内存使用方案供选择。
本文对原始OS-DBE方法提出一种新的空域扫描计算技术,并将其应用于二维导体散射问题的分析,即将导体表面某点的切向散射磁场与相关场点的切向入射电场之间的耦合系数向量用于该点附近区域的电流分布计算,以减少原始OSDBE中矩阵方程的求解次数。扫描计算采用复频率跳变技术来实现自适应控制。与已有的空域渐近波形估计扫描计算技术相比,本文的空域扫描计算技术具有更小的计算复杂度、更大的扫描计算范围、方便与FMA/MLFMA相结合以及对不同的入射场不必重复计算耦合系数向量等优点。该方法保留了原始OS-DBE低内存消耗且非常适合并行计算的特点。
本文还对二维散射问题传统的FMA/MLFMA作精度和效率上的改进。精度上,提出远场作用的高阶解析近似,并对TE波入射时的电场积分方程及组合场积分方程的远场作用采用一种更节省内存的计算方式。效率上,主要是在转移矩阵元素计算时引入傅立叶变换及其逆变换,大幅降低了转移矩阵的计算时间以及其占总计算时间的比例。这些改进措施使得与FMA/MLFMA相结合的矩量法、OS-DBE及IT-OS-DBE具有更高的计算精度和效率。
Based on the existing on-surface discretized boundary equation (OS-DBE) method, an approach, called iterative OS-DBE (IT-OS-DBE), is proposed and applied to the analysis of two-dimensional (2D) and three-dimensional (3D) problems of scattering by perfect electric conducting (PEC) objects and3D electrostatic problems.
For the above2D problems, the original OS-DBE method allows independent determination of the electric surface current at arbitrary given point on the scatterer surface with subdomain basis functions and field points (matching points) covering just a part of the entire cylinder surface and centering around that given point. Hence, the method is very suitable for parallel computing. Furthermore, it has low memory requirement because a matrix of smaller order than that in the method of moments is used and the larger the problem is, the smaller the matrix order ratio is. The fast multipole algorithm (FMA) or the multilevel FMA (MLFMA) can be incorporated into the method to diminish the computational complexity. However, a matrix equation is solved repeatedly at each current calculation point in order to generate the whole current distribution and this is the efficiency bottleneck of the method. The present IT-OS-DBE method comprises a series of alternate OS-DBE solution and revision processes. Formally, an OS-DBE solution process seems to be the same as the original OS-DBE method. Within the present iterative scheme, however, it significantly decreases the OS-DBE matrix order by one to two orders of magnitude. The solution count of the matrix equation in obtaining the whole current distribution in the OS-DBE solution process can be reduced to just one for2D scattering problems and3D electrostatic problems using the proposed one matrix inversion technique, such that the computational burden for the OS-DBE solution process can be diminished to a minimum degree. For the2D problem of scattering by a PEC cylinder, the one matrix inversion technique utilizes the vector composed of coupling coefficients between the scattered tangential magnetic field at a rather arbitrarily chosen point on the cylinder surface and the incident tangential electric fields at the related field points all around the surface to produce the whole current distribution as if it were location independent. For3D electrostatic problems, the corresponding coupling coefficient vector obtained at one point on the surface of a conductor or dielectric can also be employed to determine the whole surface charge or surface bound charge density. The FMA/MLFMA can be implemented into the revision processes of the present IT-OS-DBE method as well to reduce the computational cost for concerned matrix vector multiplications. As the present method may converge fast and complete the solution in just a few iterations, independent of the problem scale, there are several optional forms regarding the memory usage in connection with the FMA/MLFMA adopted in the revision processes.
A new spatial sweep technique for the original OS-DBE is also proposed in this dissertation and applied to the analysis of scattering by2D PEC cylinders. It uses the aforementioned coupling coefficient vector generated at a given point to calculate the current distribution in its nearby area to reduce the solution count of the matrix equation in the original OS-DBE. The complex frequency hopping is employed for the automatic control of sweep calculations. Compared with the available spatial sweep technique based on the asymptotic waveform evaluation, the new technique has the advantages of lower computational complexity, wider sweep range, easier implementation of the FMA/MLFMA, and no necessity of recalculating coupling coefficient vectors when incident waves or exciting sources alter. The method has low memory requirement and is very suitable for parallel computing as the original OS-DBE method.
For2D scattering problems.efforts have also been made in this dissertation to improve the traditional FMA/MLFMA in both accuracy and efficiency. The higher order analytical approximations to far interactions in the FMA/MLFMA are derived. In addition.a scheme with more efficient memory usage is proposed for computing the far interactions related to the electric field integral equation and the combined field integral equation for the case of a TE wave incidence. The filling time of translation matrices and its portion within the total computation time are significantly diminished by the discrete Fourier transform and its inverse. These enhancements benefit the method of moments, the OS-DBE method.and the IT-OS-DBE method all in conjunction with the FMA/MLFMA in both accuracy and efficiency.
对二维导体散射问题,原始OS-DBE方法以分域基函数和场点(或匹配点)仅覆盖导体表面部分区域的方式独立求解表面任意点的电流,非常适合并行计算。其内存消耗量低,矩阵阶数小于矩量法的相应矩阵阶数,且二者之比随着散射体电尺寸的增大而减小。该方法还可与快速多极子方法(FMA)或多层快速多极子方法(MLFMA)相结合,以进一步降低矩阵方程求解的计算复杂度。原始OS-DBE方法计算效率的瓶颈在于：对每个电流计算点均求解一次矩阵方程。本文提出的IT-OS-DBE由一系列交替进行的OS-DBE求解过程和修正过程构成。IT-OS-DBE的OS-DBE求解过程中的矩阵阶数可比原始OS-DBE低1至2个数量级。并且,对于二维导体散射问题和三维静电问题,IT-OS-DBE的OS-DBE求解过程可采用所谓的“一点求逆”技术：对二维导体散射问题,将导体表面某点的切向散射磁场与相关场点的切向入射电场之间的耦合系数向量用于整个散射体表面电流分布的计算；对三维静电问题,将导体或介质表面某点的相应的耦合系数向量用于整个表面电荷或极化电荷分布的计算,从而使OS-DBE矩阵方程的求解次数降低至仅一次,最大限度地降低了OS-DBE求解过程的时间消耗。IT-OS-DBE的修正过程可采用FMA或MLFMA,以降低计算复杂度。IT-OS-DBE具有可与所求解问题的尺度无关的快速收敛特点,迭代次数可控制在数次以内。鉴于IT-OS-DBE迭代次数少,其修正过程与FMA/MLFMA相结合时,可有多种不同的内存使用方案供选择。
本文对原始OS-DBE方法提出一种新的空域扫描计算技术,并将其应用于二维导体散射问题的分析,即将导体表面某点的切向散射磁场与相关场点的切向入射电场之间的耦合系数向量用于该点附近区域的电流分布计算,以减少原始OSDBE中矩阵方程的求解次数。扫描计算采用复频率跳变技术来实现自适应控制。与已有的空域渐近波形估计扫描计算技术相比,本文的空域扫描计算技术具有更小的计算复杂度、更大的扫描计算范围、方便与FMA/MLFMA相结合以及对不同的入射场不必重复计算耦合系数向量等优点。该方法保留了原始OS-DBE低内存消耗且非常适合并行计算的特点。
本文还对二维散射问题传统的FMA/MLFMA作精度和效率上的改进。精度上,提出远场作用的高阶解析近似,并对TE波入射时的电场积分方程及组合场积分方程的远场作用采用一种更节省内存的计算方式。效率上,主要是在转移矩阵元素计算时引入傅立叶变换及其逆变换,大幅降低了转移矩阵的计算时间以及其占总计算时间的比例。这些改进措施使得与FMA/MLFMA相结合的矩量法、OS-DBE及IT-OS-DBE具有更高的计算精度和效率。
Based on the existing on-surface discretized boundary equation (OS-DBE) method, an approach, called iterative OS-DBE (IT-OS-DBE), is proposed and applied to the analysis of two-dimensional (2D) and three-dimensional (3D) problems of scattering by perfect electric conducting (PEC) objects and3D electrostatic problems.
For the above2D problems, the original OS-DBE method allows independent determination of the electric surface current at arbitrary given point on the scatterer surface with subdomain basis functions and field points (matching points) covering just a part of the entire cylinder surface and centering around that given point. Hence, the method is very suitable for parallel computing. Furthermore, it has low memory requirement because a matrix of smaller order than that in the method of moments is used and the larger the problem is, the smaller the matrix order ratio is. The fast multipole algorithm (FMA) or the multilevel FMA (MLFMA) can be incorporated into the method to diminish the computational complexity. However, a matrix equation is solved repeatedly at each current calculation point in order to generate the whole current distribution and this is the efficiency bottleneck of the method. The present IT-OS-DBE method comprises a series of alternate OS-DBE solution and revision processes. Formally, an OS-DBE solution process seems to be the same as the original OS-DBE method. Within the present iterative scheme, however, it significantly decreases the OS-DBE matrix order by one to two orders of magnitude. The solution count of the matrix equation in obtaining the whole current distribution in the OS-DBE solution process can be reduced to just one for2D scattering problems and3D electrostatic problems using the proposed one matrix inversion technique, such that the computational burden for the OS-DBE solution process can be diminished to a minimum degree. For the2D problem of scattering by a PEC cylinder, the one matrix inversion technique utilizes the vector composed of coupling coefficients between the scattered tangential magnetic field at a rather arbitrarily chosen point on the cylinder surface and the incident tangential electric fields at the related field points all around the surface to produce the whole current distribution as if it were location independent. For3D electrostatic problems, the corresponding coupling coefficient vector obtained at one point on the surface of a conductor or dielectric can also be employed to determine the whole surface charge or surface bound charge density. The FMA/MLFMA can be implemented into the revision processes of the present IT-OS-DBE method as well to reduce the computational cost for concerned matrix vector multiplications. As the present method may converge fast and complete the solution in just a few iterations, independent of the problem scale, there are several optional forms regarding the memory usage in connection with the FMA/MLFMA adopted in the revision processes.
A new spatial sweep technique for the original OS-DBE is also proposed in this dissertation and applied to the analysis of scattering by2D PEC cylinders. It uses the aforementioned coupling coefficient vector generated at a given point to calculate the current distribution in its nearby area to reduce the solution count of the matrix equation in the original OS-DBE. The complex frequency hopping is employed for the automatic control of sweep calculations. Compared with the available spatial sweep technique based on the asymptotic waveform evaluation, the new technique has the advantages of lower computational complexity, wider sweep range, easier implementation of the FMA/MLFMA, and no necessity of recalculating coupling coefficient vectors when incident waves or exciting sources alter. The method has low memory requirement and is very suitable for parallel computing as the original OS-DBE method.
For2D scattering problems.efforts have also been made in this dissertation to improve the traditional FMA/MLFMA in both accuracy and efficiency. The higher order analytical approximations to far interactions in the FMA/MLFMA are derived. In addition.a scheme with more efficient memory usage is proposed for computing the far interactions related to the electric field integral equation and the combined field integral equation for the case of a TE wave incidence. The filling time of translation matrices and its portion within the total computation time are significantly diminished by the discrete Fourier transform and its inverse. These enhancements benefit the method of moments, the OS-DBE method.and the IT-OS-DBE method all in conjunction with the FMA/MLFMA in both accuracy and efficiency.
引文
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