时域边界积分方程及其快速算法的研究与应用
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摘要
时域积分方程(TDIE)方法在计算瞬态、宽带电磁问题中表现出特别的优势,现已成为计算电磁学领域的研究热点之一。然而TDIE的数值解法之一——时间步进算法(MOT)在实现中会面临两方面的问题:数值解的后期振荡和较高的计算量与存储量,为了促进TDIE在工程实践中的广泛应用,论文主要围绕MOT的这两方面问题展开研究。
论文首先研究了求解任意线、面导体目标的MOT算法,建立了时域电场积分方程(TDEFIE)、时域磁场积分方程(TDMFIE)和时域混合场积分方程(TDCFIE)的MOT矩阵线性方程组,比较了三种TDIE在计算一般电磁目标瞬态特性时的稳定性和精度问题,研究了TDCFIE抑制因目标内谐振而引起的MOT后期振荡的能力,并通过数值算例确定了TDCFIE中比例系数的安全取值范围。论文建立了微分形式TDEFIE求解任意细线结构的MOT矩阵方程组,提出了一种精确计算细线阻抗矩阵元素的半数值半解析方法,并对半径渐变的细线结构、分支结构、对数周期结构进行了精确的建模和仿真。论文针对TDMFIE中立体角的通用近似处理方法,引入了一种更加精确的等效立体角,改进了TDMFIE在求解具有尖锐棱边电小尺寸目标的数值精度。
为了降低MOT算法的高计算量和存储量,论文重点研究了时域积分方程MOT的加速算法。首先论文提出了一种基于电流的物理光学(PO)耦合TDIE的混合算法,通过将目标表面分成PO区与IE区,实现了TDIE精确计算与PO高效计算的有机结合,极大地降低了经典MOT算法的计算量和存储量。其次,基于窄脉冲信号与目标作用的物理机理,通过分析目标表面电流的瞬态分布并引入时域PO电流,论文实现了一种能够显著减少MOT算法计算量的近似方法。论文还深入研究了一种高精度、高效率的TDIE加速算法——时域平面波算法(PWTD)的理论基础、数值实现的关键技术,以及两层、多层PWTD加速MOT的实现方案、程序流程以及数值性能。综合上述研究,最后论文提出了一种PWTD加速的PO-TDIE混合算法,该算法能够进一步提高PO-TDIE算法的计算效率。
为了进一步提高TDIE的计算效率、降低存储需求,论文探索了大型稀疏矩阵压缩存储技术在MOT中的应用问题,实践证明采用按行/列、按坐标压缩存储阻抗矩阵,能够显著降低对计算机的内存需求;作为TDIE数值计算的一个必需环节,论文还研究了迭代算法结合预条件技术在求解大型稀疏矩阵线性方程组中的效率和精度问题。
Time domain integral equation (TDIE), providing an appealing avenue for analyzing transient and broadband electromagnetics problems, has been of considerable interest in the computational electromagnetic (CEM) community. Unfortunately, marching-on-in-time (MOT)-based TDIE solvers have been suffered from numerically late-time oscillation and high computational complexities and large memory requirements. The purpose of this thesis is to develop a stable and fast TDIE solver for their applications to practical, real-world problems.
Derivations of MOT matrix equations for time domain EFIE, MFIE and CFIE to obtain the transient responses from arbitrarily shaped wire and surface conducting objects are presented firstly in this thesis. The stability and precision of this three TDIEs are studied, when applied to transient analysis from general EM problems. The validity of TDCFIE to eliminate the late-time oscillation for the frequency components corresponding to the internal resonance of the structure is identified, and the effective parameter of linear combination is confirmed by numerical examples. A half-numerical and half-analytical method is proposed for the accurate computation of resistance matrix elements, when the differential form of TDEFIE is applied to transient analysis of wire structure. And the exact modeling and simulation for the wire structure with stepped-radius, wire junction and log-period structure are presented in this thesis. A more accurate equivalent solid angle value in TDMFIE is introduced to correcte the general assumption of solid angle, which allows the accurate analysis of electrically small sharp-edged objects as TDEFIE does.
Significant efforts have been expended on the development of fast algorithms of MOT-based TDIE solvers for reducing the high computation complexities and large memory requirements in classical MOT. Firstly, a novel current-based hybrid method of PO coupled TDIE is proposed for analysis of large bodies with significant features that are not electrically large. Dividing the structure into PO region and IE region, the proposed method employs the accuracy of TDIE and the efficiency of PO completely, that dramatically reduces the high computation complexities and memory requirements in classical MOT. Secondly, an approximate technique is implemented to cut down the computational complexity of MOT, according to the distribution of transient current generated during the object is elluminated by a short pulse. As a fast TDIE solver with high accuracy and efficiency, plane wave time domain (PWTD) algorithm is studied deeply in this thesis. The implementations of two-level and multilevel PWTD-enhanced MOT schemes are described and the numerical efficiency is presented. Furthermore, a PWTD-enhanced PO-TDIE hybrid scheme is proposed, which can improve the computational efficiency of aforementioned hybrid method.
Compressed storage technique for large sparse matrix is applied in fast TDIE solver for the purpose of further decreasing memory requirements. The precision and efficiency of iterative method, along with preconditioner, are explored in solving the large sparse matrix linear equation which is an indispensable part of the TDIE solver.
论文首先研究了求解任意线、面导体目标的MOT算法,建立了时域电场积分方程(TDEFIE)、时域磁场积分方程(TDMFIE)和时域混合场积分方程(TDCFIE)的MOT矩阵线性方程组,比较了三种TDIE在计算一般电磁目标瞬态特性时的稳定性和精度问题,研究了TDCFIE抑制因目标内谐振而引起的MOT后期振荡的能力,并通过数值算例确定了TDCFIE中比例系数的安全取值范围。论文建立了微分形式TDEFIE求解任意细线结构的MOT矩阵方程组,提出了一种精确计算细线阻抗矩阵元素的半数值半解析方法,并对半径渐变的细线结构、分支结构、对数周期结构进行了精确的建模和仿真。论文针对TDMFIE中立体角的通用近似处理方法,引入了一种更加精确的等效立体角,改进了TDMFIE在求解具有尖锐棱边电小尺寸目标的数值精度。
为了降低MOT算法的高计算量和存储量,论文重点研究了时域积分方程MOT的加速算法。首先论文提出了一种基于电流的物理光学(PO)耦合TDIE的混合算法,通过将目标表面分成PO区与IE区,实现了TDIE精确计算与PO高效计算的有机结合,极大地降低了经典MOT算法的计算量和存储量。其次,基于窄脉冲信号与目标作用的物理机理,通过分析目标表面电流的瞬态分布并引入时域PO电流,论文实现了一种能够显著减少MOT算法计算量的近似方法。论文还深入研究了一种高精度、高效率的TDIE加速算法——时域平面波算法(PWTD)的理论基础、数值实现的关键技术,以及两层、多层PWTD加速MOT的实现方案、程序流程以及数值性能。综合上述研究,最后论文提出了一种PWTD加速的PO-TDIE混合算法,该算法能够进一步提高PO-TDIE算法的计算效率。
为了进一步提高TDIE的计算效率、降低存储需求,论文探索了大型稀疏矩阵压缩存储技术在MOT中的应用问题,实践证明采用按行/列、按坐标压缩存储阻抗矩阵,能够显著降低对计算机的内存需求;作为TDIE数值计算的一个必需环节,论文还研究了迭代算法结合预条件技术在求解大型稀疏矩阵线性方程组中的效率和精度问题。
Time domain integral equation (TDIE), providing an appealing avenue for analyzing transient and broadband electromagnetics problems, has been of considerable interest in the computational electromagnetic (CEM) community. Unfortunately, marching-on-in-time (MOT)-based TDIE solvers have been suffered from numerically late-time oscillation and high computational complexities and large memory requirements. The purpose of this thesis is to develop a stable and fast TDIE solver for their applications to practical, real-world problems.
Derivations of MOT matrix equations for time domain EFIE, MFIE and CFIE to obtain the transient responses from arbitrarily shaped wire and surface conducting objects are presented firstly in this thesis. The stability and precision of this three TDIEs are studied, when applied to transient analysis from general EM problems. The validity of TDCFIE to eliminate the late-time oscillation for the frequency components corresponding to the internal resonance of the structure is identified, and the effective parameter of linear combination is confirmed by numerical examples. A half-numerical and half-analytical method is proposed for the accurate computation of resistance matrix elements, when the differential form of TDEFIE is applied to transient analysis of wire structure. And the exact modeling and simulation for the wire structure with stepped-radius, wire junction and log-period structure are presented in this thesis. A more accurate equivalent solid angle value in TDMFIE is introduced to correcte the general assumption of solid angle, which allows the accurate analysis of electrically small sharp-edged objects as TDEFIE does.
Significant efforts have been expended on the development of fast algorithms of MOT-based TDIE solvers for reducing the high computation complexities and large memory requirements in classical MOT. Firstly, a novel current-based hybrid method of PO coupled TDIE is proposed for analysis of large bodies with significant features that are not electrically large. Dividing the structure into PO region and IE region, the proposed method employs the accuracy of TDIE and the efficiency of PO completely, that dramatically reduces the high computation complexities and memory requirements in classical MOT. Secondly, an approximate technique is implemented to cut down the computational complexity of MOT, according to the distribution of transient current generated during the object is elluminated by a short pulse. As a fast TDIE solver with high accuracy and efficiency, plane wave time domain (PWTD) algorithm is studied deeply in this thesis. The implementations of two-level and multilevel PWTD-enhanced MOT schemes are described and the numerical efficiency is presented. Furthermore, a PWTD-enhanced PO-TDIE hybrid scheme is proposed, which can improve the computational efficiency of aforementioned hybrid method.
Compressed storage technique for large sparse matrix is applied in fast TDIE solver for the purpose of further decreasing memory requirements. The precision and efficiency of iterative method, along with preconditioner, are explored in solving the large sparse matrix linear equation which is an indispensable part of the TDIE solver.
引文
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