电磁场数值求解中迭代方法与预条件技术研究
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摘要
在工程应用领域,随着电子计算机与相关科学技术的发展,高效的数值分析方法已越来越具有重要性。电磁领域问题的数值分析方法中,有限元法和矩量法是两种重要的且广泛应用的方法,两种方法分析的结果都将产生一个大型复矩阵线性方程组,对该方程组的求解用时在整个仿真用时中占据很大比重。因此,各种高效率、低存储的快速求解算法的研究成为迫切需要。本文研究主要基于:矢量有限元法分析高频电磁场Helmholtz方程边值问题所生成的大型复对称且高度非正定的稀疏线性方程组的快速求解算法,以及电场积分方程矩量法求解所生成的大型复对称非正定的稠密线性方程组的快速求解算法,研究主要致力于高效迭代算法和预条件算法。
首先采用对称化的迭代方法如:COCG、SQMR、以及简化的LBCG(SLBCG),并详细研究了其对复对称稠密线性系统以及复对称稀疏线性系统求解的效率,同时与几种常用迭代法相比较。提出了组合迭代的方法,通过组合不同迭代法的残差收敛机制来充分消除残差,这样既能避免COCG在高精度迭代的等方性崩溃,又能在一些问题的求解中获得比单独迭代法更高的收敛效率。
详细研究了几种常用的预条件算法在求解不同类型的矩阵方程组以及与不同迭代法结合时的功效。提出了一种针对高度非正定的矢量有限元系数矩阵的改进的AINV(MAINV)预条件算法,该算法通过在分解过程中引入动态主元加强技术来获得更加稳定且高效的预条件子;在此基础上,引入了复偏移Laplace算子方案来尽可能减少预条件子的构造代价,并进一步加强预条件子的健壮性。提出了一种主元补偿的松弛IC分解(RIC)预条件算法,该算法利用待丢弃的小元对相应的对角主元进行加强,以获得更加稳定高效的预条件子,同时也研究了其在与不同迭代法结合时的功效。常规预条件子的产生通常基于系数矩阵A,本文针对特定电磁问题的求解提出了一种基于A的实部矩阵分解的预条件算法,并比较了与不同迭代法结合进行求解时的效率。该预条件子不仅获得了比常规方法更好的求解性能,还能节约相当一部分的存储量。
此外,研究了对不完全分解和稀疏近似逆分解预条件算法影响较大的前处理技术,主要包括缩放和重排序技术。采用了一种易于实施的缩放技术来降低矩阵条件数,并采用重排序技术来同时获得分解的高效率和低存储。对采用前处理与不采用前处理所得到的预条件子的性能进行详细比较,以说明前者的优越性。
With the development of the computer science and techniques, effective numerical methods become more and more important. in many engineering fields. The finite element method and the method of moment are two important and widely used numerical methods to analyze the electromagnetic problems. However, the application of these two methods usually yield large complex and symmetric linear systems, and the time for solving this kind of linear system is the main part of which for whole numerical simulation process. Therefore the researches on the algorithms of high efficiency and low memory needs are necessary. In this dissertation, the work is mainly based on the effective algorithm for solving the large complex symmetric and highly indefinite linear systems arising from the FEM analyzing electromagnetic field Helmholtz equations, and the MOM analyzing EFIE equations. And the work is devoted to iterative and preconditioning algorithms.
We first apply the symmetrical iterative methods such as COCG, SQMR, and the simplified linear BCG(LBCG) for solving complex symmetric dense and sparse linear systems, and detailedly investigate the convergence behaviour by compared with several conventional iterative methods. We then proposed a co-iterative method which can avoid the isotropic breakdowns and outperform the conventional iterative methods for solving several electromagnetic problems.
We apply several conventional preconditioners to solve complex symmetric dense and sparse linear systems, and detailedly investigate their performance when combined with different iterative methods. For highly indefinite FEM linear systems, we propose a modified AINV(MAINV) preconditioner. It is designed by adding pivots compensation strategy to the basic AINV process, thus the preconditioner can be more stable and effective. Based on this algorithm, we employed a complex shifted Laplace operater scheme to reduce the computational cost and get a more robust preconditioner. We also propose a modified relaxed IC (RIC) algorithm and investigate its convergence beheavior when applied to different iterative methods. It uses the small elements to compensate the correspond diagonal pivots to get a more stable preconditioner. Conventional preconditioner is usually constructed based on the coefficient matrix A, in this dissertation we develop an novel preconditioner, which is based on the real part of matrix A, for solving specific electromagnetic problems. Compared with the conventional ones, this preconditioner is not only more efficient but also can reduce the memory needs.
Besides, we investigate the preprocessing techniques, including scaling and reordering, which is very useful for the incomplete factorization and the factorized sparse approximate inverse algorithms. We use a scaling technique to decrease the condition number of the coefficient matrix. We also use the reordering techniques to achieve the efficient factorization and low memory needs. And the preconditioner is more effective when compared with which resulting from the factorization without preprocessing.
首先采用对称化的迭代方法如:COCG、SQMR、以及简化的LBCG(SLBCG),并详细研究了其对复对称稠密线性系统以及复对称稀疏线性系统求解的效率,同时与几种常用迭代法相比较。提出了组合迭代的方法,通过组合不同迭代法的残差收敛机制来充分消除残差,这样既能避免COCG在高精度迭代的等方性崩溃,又能在一些问题的求解中获得比单独迭代法更高的收敛效率。
详细研究了几种常用的预条件算法在求解不同类型的矩阵方程组以及与不同迭代法结合时的功效。提出了一种针对高度非正定的矢量有限元系数矩阵的改进的AINV(MAINV)预条件算法,该算法通过在分解过程中引入动态主元加强技术来获得更加稳定且高效的预条件子;在此基础上,引入了复偏移Laplace算子方案来尽可能减少预条件子的构造代价,并进一步加强预条件子的健壮性。提出了一种主元补偿的松弛IC分解(RIC)预条件算法,该算法利用待丢弃的小元对相应的对角主元进行加强,以获得更加稳定高效的预条件子,同时也研究了其在与不同迭代法结合时的功效。常规预条件子的产生通常基于系数矩阵A,本文针对特定电磁问题的求解提出了一种基于A的实部矩阵分解的预条件算法,并比较了与不同迭代法结合进行求解时的效率。该预条件子不仅获得了比常规方法更好的求解性能,还能节约相当一部分的存储量。
此外,研究了对不完全分解和稀疏近似逆分解预条件算法影响较大的前处理技术,主要包括缩放和重排序技术。采用了一种易于实施的缩放技术来降低矩阵条件数,并采用重排序技术来同时获得分解的高效率和低存储。对采用前处理与不采用前处理所得到的预条件子的性能进行详细比较,以说明前者的优越性。
With the development of the computer science and techniques, effective numerical methods become more and more important. in many engineering fields. The finite element method and the method of moment are two important and widely used numerical methods to analyze the electromagnetic problems. However, the application of these two methods usually yield large complex and symmetric linear systems, and the time for solving this kind of linear system is the main part of which for whole numerical simulation process. Therefore the researches on the algorithms of high efficiency and low memory needs are necessary. In this dissertation, the work is mainly based on the effective algorithm for solving the large complex symmetric and highly indefinite linear systems arising from the FEM analyzing electromagnetic field Helmholtz equations, and the MOM analyzing EFIE equations. And the work is devoted to iterative and preconditioning algorithms.
We first apply the symmetrical iterative methods such as COCG, SQMR, and the simplified linear BCG(LBCG) for solving complex symmetric dense and sparse linear systems, and detailedly investigate the convergence behaviour by compared with several conventional iterative methods. We then proposed a co-iterative method which can avoid the isotropic breakdowns and outperform the conventional iterative methods for solving several electromagnetic problems.
We apply several conventional preconditioners to solve complex symmetric dense and sparse linear systems, and detailedly investigate their performance when combined with different iterative methods. For highly indefinite FEM linear systems, we propose a modified AINV(MAINV) preconditioner. It is designed by adding pivots compensation strategy to the basic AINV process, thus the preconditioner can be more stable and effective. Based on this algorithm, we employed a complex shifted Laplace operater scheme to reduce the computational cost and get a more robust preconditioner. We also propose a modified relaxed IC (RIC) algorithm and investigate its convergence beheavior when applied to different iterative methods. It uses the small elements to compensate the correspond diagonal pivots to get a more stable preconditioner. Conventional preconditioner is usually constructed based on the coefficient matrix A, in this dissertation we develop an novel preconditioner, which is based on the real part of matrix A, for solving specific electromagnetic problems. Compared with the conventional ones, this preconditioner is not only more efficient but also can reduce the memory needs.
Besides, we investigate the preprocessing techniques, including scaling and reordering, which is very useful for the incomplete factorization and the factorized sparse approximate inverse algorithms. We use a scaling technique to decrease the condition number of the coefficient matrix. We also use the reordering techniques to achieve the efficient factorization and low memory needs. And the preconditioner is more effective when compared with which resulting from the factorization without preprocessing.
引文
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