力学学报 ›› 2017, Vol. 49 ›› Issue (6): 1223-1230.doi: 10.6052/0459-1879-17-292

所属专题: 力学学报创刊60周年专栏(2017年)

• 创刊60周年专栏 • 上一篇    下一篇

固体结构大规模矩阵正定性判定的快速算法

徐辉, 刘彬   

  1. 清华大学航天航空学院工程力学系应用力学教育部重点实验室, 北京 100084
  • 收稿日期:2017-08-26 出版日期:2017-11-15 发布日期:2017-12-02
  • 作者简介:刘彬,教授,主要研究方向:大规模多尺度多物理场计算、复合材料力学和断裂力学.E-mail:liubin@tsinghua.edu.cn
  • 基金资助:

    国家自然科学基金资助项目(51232004,11372158,11425208).

FAST ALGORITHMS FOR DETERMINING POSITIVE DEFINITENESS OF LARGE SCALE MATRICES IN SOLID STRUCTURE ANALYSIS

Xu Hui, Liu Bin   

  1. AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
  • Received:2017-08-26 Online:2017-11-15 Published:2017-12-02

摘要:

对于结构稳定性分析中超大规模矩阵正定性判定,必须采用并行计算方法,传统方法如计算特征值、主子式行列式及LDLT等直接方法难以实现.本文提出了一些可适用于并行的迭代判定算法.借鉴力学系统中能量下降的思想,发展了一种判定矩阵正定性的新思路,即将矩阵的正定性判定问题转化为一个优化问题,并基于优化算法来判定矩阵的正定性.提出了基于最速下降法和共轭梯度法来进行矩阵正定性判定的算法.然后考虑到力学系统刚度矩阵的稀疏性和结构刚失稳状态的弱非正定性,提出可以先截超平面后解方程求驻值点的方法来判定弱非正定矩阵的正定性.为了保证对强非正定矩阵判定的准确性,本文提出可以高效混杂使用截平面法和共轭梯度法.数值实验结果表明,本文提出的算法具有准确性和高效性.对于强非正定矩阵而言,共轭梯度算法更加高效;而对于弱非正定矩阵,则是截平面法和混杂算法更加高效.这些算法都容易在机群上实现并行计算,能够快速判定大规模矩阵的正定性.

关键词:

矩阵|正定性|共轭梯度法|截平面

Abstract:

In order to determine the positive definiteness of the super-large-scale matrix in the structural stability analysis, the parallel computing method must be adopted. However, the traditional direct methods such as the eigenvalue's analysis, the master subordinate determinant's computation and LDLT decomposition are difficult to realize in parallel computing. In this paper, some iterative algorithms which are suitable for parallel computing are proposed. A new approach is developed that determining the positive definiteness of a matrix can be transformed into an optimization problem, which is solved by various optimization algorithms. The algorithms based on the steepest descent method and the conjugate gradient method are proposed. Considering the sparseness of the stiffness matrix of the mechanical system and the weakly non-positive definite property of at the critical buckling state, we propose a method via calculating the stationary point on a cutting plane to determine the non-positive definite matrices. In order to ensure the accuracy of the determination of strong non-positive definite matrices, a hybrid method combing the cutting plane method and the conjugate gradient method is developed. The numerical results show that the proposed algorithms are accurate and efficient. The conjugate gradient algorithm is more efficient for strongly non-positive definite matrices while the hybrid method is more efficient for the weak non-positive definite matrices. These algorithms are easy to realize in parallel computing on the cluster and can quickly determine the positive definiteness of large-scale matrices.

Key words:

matrix|positive definite|conjugate gradient method|cutting plane

中图分类号: 

  • O346.1