非连续变形分析(DDA)线性方程组的高效求解算法

High efficient algorithms for solving linear equations in discontinuous deformation analysis

非连续变形分析(DDA)方法对大规模工程问题的数值模拟耗时太长,其中线性方程组求解耗时可占总计算时间的70%以上,因此,高效的线性方程组解法是重要研究课题。首先,阐述了适用于DDA方法的基于块的行压缩法和基于试验-误差迭代格式的非0位置记录;然后,针对DDA的子矩阵技术,将块雅可比迭代法(BJ)、预处理的块共轭梯度法(PCG,包括Jacobi-PCG、SSOR-PCG)引入DDA方法,重点研究了线性方程组求解过程中的关键运算;最后,通过两个洞室开挖算例,分析了各线性方程组求解算法在DDA中的计算效率。研究表明:与迭代法相比,直解法无法满足大规模工程计算需要;BJ迭代法与块超松弛迭代法(BSOR)的效率差别不大,但明显不如PCG迭代法。因此,建议采用PCG迭代法求解DDA线性方程组,特别是SSOR-PCG值得推广;如果开展并行计算研究,Jacobi-PCG是较好的选择,当刚度矩阵惯性优势明显时,BJ迭代法同样有效。

Simulating large-scale engineering problems with discontinuous deformation analysis（DDA） is extremely time-consuming. The solving process of linear equations normally costs more than 70% of the total computing time, and thus the computing efficiency of algorithms for linear equations is a significant research topic. Firstly, two contents of non-zero storage in the DDA have been described. One is the block compressed sparse row method, and the other is the iterative scheme of non-zero position recording based on the trial-error approach. Secondly, in view of the sub-matrix technology, the block Jacobi（BJ） iteration method and pre-processing block conjugate gradient（PCG, including Jacobi and symmetric successive over relaxation（SSOR）pre-processing） iteration method have been introduced into DDA, and then the key operations of solving linear equations have been analysed. Last, the calculation efficiency of various algorithms for solving linear equations are investigated through two examples of tunnelling excavation. The results show that the direct solution cannot meet the requirements of large-scale engineering computing compared with the iterative method. Although there are few differences of computing efficiency between BJ and BSOR iteration methods, both of them are obviously not as well as the PCG method. Therefore, the PCG method, in particular SSOR-PCG method is highly recommended. Jacobi-PCG is the best method to perform parallel computing, however BJ iteration is also an acceptable choice when there is an apparent inertial advantage of the stiffness matrix.