改进的预处理共轭斜量法及其在工程有限元分析中的应用

Improved Preconditoned Conjugate Gradient Method and Its Application in F. E. A. for Engineering

本文就预处理共轭斜量法(PCCG法)给出了两个具有理论和实际意义的定理,它们分别讨论了迭代解的定性性质和迭代矩阵的构造原则.作者提出了新的非M-矩阵的不完全LU分解技术和迭代矩阵的构造方法.用此改进的PCCG法,对病态问题和大型三维有限元问题进行了计算并与其他方法作了对比,分析了PCCG法在求解病态方程组时的反常现象.计算结果表明本文建议的方法是求解大型有限元方程组和病态方程组的一种十分有效的方法.

In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient'; method (PCCG). The theorems discuss respectively the qualitative property of the iterative solution and the,construction principle of the iterative matrix. The authors put forward a new incompletely LU factorizing technique for aon-M-matrix and the method of constructing the iterative matrix. This improved PCCG is used -to calculate the ill-conditioned problems and large-scale three-dimensional finite element problems, and simultaneously contrasted with other methods. The abnormal phenomenon is analyzed when PCCG is used to solve the system of ill-conditioned equations. It is shown that the method proposed in this paper is quite effective in solving the system of large-scale finite element equations and the system of ill-conditioned equations.