Efficiency evaluation of structural nonlinear analysis method based on the Woodbury formula
Efficiency evaluation of structural nonlinear analysis method based on the Woodbury formula
复制标题
基于Woodbury公式的结构非线性分析方法效率评估
DOI：
10.1108/ec-09-2018-0393
复制
发表时间：
2019-05-13
影响因子：
1.6
通讯作者：
Li, Hong-Nan
中科院分区：
文献类型：
--
作者：
Li, Gang;Jia, Shuo;Li, Hong-Nan
目的
本文的目的是从线性方程组在每一增量步的求解效率和所选用的迭代算法两个方面对一种基于伍德伯里公式的非线性分析方法进行理论上的综合效率评价。
设计/方法/方式
首先，本研究利用时间复杂度理论来定量比较伍德伯里公式与求解线性方程组常用的LDLT因式分解方法的效率。此外，迭代算法的性能也显着影响分析的效率。为此，采用收敛阶数为8的三点法求解基于伍德伯里公式的非线性分析方法的平衡方程，以改善牛顿-拉夫森（N-R）法的迭代性能。
结果
结果表明，当非弹性自由度远小于自由度时，伍德伯里公式的渐近时间复杂度远低于LDLT分解方法，表明伍德伯里公式对局部非线性问题更有效.通过对N-R方法和三点方法时间复杂度的比较，表明对于大尺度结构或IDOFs数与DOFs数之比较大的局部非线性问题，三点方法比N-R方法更有效。
独创性/价值
本文从理论上评价了基于伍德伯里公式的非线性分析方法的有效性，并定量地说明了比较方法的适用条件。比较结果为不同非线性问题的算法选择提供了理论依据。
Purpose
The purpose of this paper is to make a theoretical comprehensive efficiency evaluation of a nonlinear analysis method based on the Woodbury formula from the efficiency of the solution of linear equations in each incremental step and the selected iterative algorithms.
Design/methodology/approach
First, this study employs the time complexity theory to quantitatively compare the efficiency of the Woodbury formula and the LDLT factorization method which is a commonly used method to solve linear equations. Moreover, the performance of iterative algorithms also significantly effects the efficiency of the analysis. Thus, the three-point method with a convergence order of eight is employed to solve the equilibrium equations of the nonlinear analysis method based on the Woodbury formula, aiming to improve the iterative performance of the Newton–Raphson (N–R) method.
Findings
First, the result shows that the asymptotic time complexity of the Woodbury formula is much lower than that of the LDLT factorization method when the number of inelastic degrees of freedom (IDOFs) is much less than that of DOFs, indicating that the Woodbury formula is more efficient for local nonlinear problems. Moreover, the time complexity comparison of the N–R method and the three-point method indicates that the three-point method is more efficient than the N–R method for local nonlinear problems with large-scale structures or a larger ratio of IDOFs number to the DOFs number.
Originality/value
This study theoretically evaluates the efficiency of nonlinear analysis method based on the Woodbury formula, and quantitatively shows the application condition of the comparative methods. The comparison result provides a theoretical basis for the selection of algorithms for different nonlinear problems.