一种改进的2n因子乘积形式的逆矩阵求解算法

A GENERALIZED 2n-FACTOR FORM OF THE INVERSE OF A MATRIX

复杂结构的有限元分析需要耗费巨大的内存和计算机时间 .在此过程中 ,有限元线性方程组的求解时间占很大比例 ,因此 ,发展高效的线性方程组求解算法是提高有限元分析效率的关键 .针对矩阵求逆问题 ,该文将有限元方程组的系数矩阵视为等分块矩阵 ,并基于等分块矩阵的概念推广了传统的逆矩阵的n因子和 2n因子乘积形式 .推广后得到的这组乘积形式在适当条件下又分别可以退化到传统的逆矩阵的n因子和 2n因子乘积形式 .应用基于推广后得到的这组乘积形式的求逆算法来求解板材冲压成形有限元数值模拟中的大型有限元线性方程组 ,结果表明 ,该算法可以显著地提高大型有限元线性方程组的求解效率 .

The Finite Element Analyses (FEA) of complex constructions need huge memory and consume much time. Generally it takes a large portion of time to solve the large-scale linear simultaneous equations (LLSE) involved in the FEA process. Therefore the key to improve the efficiency of that process is to shorten the solving time of that equations. The coefficient matrix of the LLSE involved in the process is treated as a uniform block matrix in this paper. To get the inverse of the matrix, the traditional n-factor form and 2n-factor form of an inverse matrix are generalized to provide two new forms based on the concept of uniform block matrix, which can degenerate into the traditional forms under some given conditions, specifically. The algorithms based on either the traditional form or the new form are coded to solve the LLSE involved in the numerical simulation of several sheet metals stamping processes. The results indicate that the algorithm based on the new form can greatly improve the efficiency of solving the LLSE involved in the FEA of complex constructions.