基于微分求积法及V-变换的大规模动力系统快速数值计算方法

PDF(2472 KB)

振动与冲击 ›› 2016, Vol. 35 ›› Issue (3) : 73-78.

论文

汪芳宗，廖小兵

作者信息 +

Fast numerical calculation method for large-scale dynamic systems based on differential quadrature method and V-transformation#br#

WANG Fang-zong LIAO Xiao-bing

Author information +

文章历史 +

摘要

针对大规模动力系统动态响应的数值计算，传统的微分求积法通常在时间域上逐步离散、整体求解，存在“维数灾”问题。在多级高阶时域微分求积法的基础上，提出了基于V-变换的大规模动力系统动态响应的快速数值计算方法。利用微分求积法的加权系数矩阵满足V-变换这一重要特性，将离散后的雅可比矩阵方程进行解耦分块，推导形成了多级分块递推计算方法。数值算例表明，即使采用相当于Newmark方法2s倍的步长，微分求积法的计算精度仍比Newmark方法要高出2~3个数量级。进一步对3个不同规模的算例系统进行了测试，结果表明：相对于传统的数值计算方法，多级分块递推计算方法可以获得较大的加速比，能够显著提高大规模动力系统动态响应的计算效率。

Abstract

For numerical simulation of large-scale dynamic systems’ response, the traditional differential quadrature method (DQM) usually adopts successive discrete and global solution in time domain, where there is the problem of “curse of dimensionality”. On the basis of the multi-stage high-order time domain differential quadrature method, a fast numerical calculation method for large-scale dynamic systems’ response based on V-transformation is proposed. Using the V-transformation possessed by the weighting coefficient matrix of DQM, the whole Jacobian matrix equations involved in the traditional approach of DQM can be decoupled into blocks, thus the multi-stage block recursive method is formed. Numerical examples show that, even using 2s times step size of the Newmark method, the calculation precision of the differential quadrature method is about 2~3 orders of magnitude higher than that of the Newmark method. Furthermore, three different scale systems are used for computational efficiency test and the results show that the multi-stage block recursive method can obtain high speedups compared with the traditional numerical methods, which can significantly improve the computational efficiency of large-scale dynamic systems’ response.

Key words: large-scale dynamic systems; fast calculation; differential quadrature method; V-transformation; multi-stage block recursive method

关键词

大规模动力系统 / 快速计算 / 微分求积法 / V-变换 / 多级分块递推方法

Key words

large-scale dynamic systems

/ fast calculation / differential quadrature method / V-transformation / multi-stage block recursive method

引用本文

EndNote

Ris (Procite)

Bibtex

导出引用

汪芳宗，廖小兵. 基于微分求积法及V-变换的大规模动力系统快速数值计算方法[J]. 振动与冲击, 2016, 35(3): 73-78

WANG Fang-zong LIAO Xiao-bing. Fast numerical calculation method for large-scale dynamic systems based on differential quadrature method and V-transformation#br#[J]. Journal of Vibration and Shock, 2016, 35(3): 73-78

参考文献

[1] Hairer E, Nørsett S P, Wanner G. Solving Ordinary Differ¬ential Equations I: Nonstiff Problems[M]. Berlin: Springer, 1987.
[2] Butcher J C. Numerical Methods for Ordinary Differential Equations[M]. New York: Wiley, 2008.
[3] Newmark N M. A method of computation for structural dynamics[J]. Journal of the Engineering Mechanics Division, ASCE, 1959, 85: 67-94.
[4] 李鸿晶, 王通, 廖旭. 关于Newmark-β法机理的一种解释[J]. 地震工程与工程振动, 2011, 31(2): 55-62.
LI Hong-jing, WANG Tong, LIAO Xu. An interpretation on Newmark-β methods in mechanism of numerical analysis[J]. Journal of Earthquake Engineering and Engineering Vibration, 2011, 31(2): 55-62.
[5] 邢誉峰, 郭静. 与结构动特性协同的自适应Newmark方法[J]. 力学学报, 2012, 44(5): 904-911.
XING Yu-feng, GUO Jing. A self-adaptive Newmark method with parameters dependent upon structural dynamic characteris¬tics[J]. Chinese Journal of Theoretical and Applied Me¬chanics, 2012, 44(5): 904-911.
[6] 方德平,王全凤. 加速度修正的Wilson-θ法的精度分析[J]. 振动与冲击, 2010, 29(6): 216-218+246.
FANG De-ping, WANG Quan-feng. Accuracy analysis of Wilson-θ method with modified acceleration[J]. Journal of Vibra¬tion and Shock, 2010, 29(6): 216-218+246.
[7] 张雄, 王天舒. 计算动力学[M]. 北京: 清华大学出版社, 2007.
ZHANG Xiong, WANG Tian-shu. Computational Dynamics [M]. Beijing: Tsinghua University Press, 2007.
[8] 钟万勰. 结构动力方程的精细时程积分法[J]. 大连理工大学学报, 1994, 34(2): 131-136.
ZHONG Wan-xie. On precise time-integration method for struc¬tural dynamics[J]. Journal of Dalian University of Technol¬ogy, 1994, 34(2): 131-136.
[9] 高强, 吴锋, 张洪武, 等. 大规模动力系统改进的快速精细积分方法[J]. 计算力学学报, 2011, 28(4): 493-498.
GAO Qiang, WU Feng, ZHANG Hong-wu, et al. A fast pre¬cise integration method for large-scale dynamic structures[J]. Chi¬nese Journal of Computational Mechanics, 2011, 28(4): 493-498.
[10] 吴泽艳, 王立峰, 武哲. 大规模动力系统高精度増维精细积分方法快速算法[J]. 振动与冲击, 2014, 33(2): 188-192.
WU Ze-yan, WANG Li-feng, WU Zhe. Fast algorithm for pre¬cise integration with high accuracy dimension expanding method with for large-scale dynamic systems[J]. Journal of Vibra¬tion and Shock, 2014, 33(2): 188-192.
[11] Bellman R E, Casti J. Differential quadrature and long term integration[J]. Journal of Mathematical Analysis and Applica¬tion, 1971, 34 (2): 235-238.
[12] Bellman R E, Kashef B G, Casti J. Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations[J]. Journal of Computational Physics, 1972, 10, 40-52.
[13] Shu C. Differential Quadrature and Its Application in Engineer¬ing[M]. Berlin: Springer, 2000.
[14] 王鑫伟. 微分求积法在结构力学中的应用[J]. 力学进展, 1995, 25(2): 232-240.
WANG Xin-wei. Differential quadrature in the analysis of struc¬tural components[J]. Advances in Mechanics, 1995, 25(2): 232-240.
[15] 程昌钧, 朱正佑. 微分求积法及其在力学应用中的若干新进展[J]. 上海大学学报(自然科学版), 2009, 15(6): 551-559.
CHENG Chang-Jun, ZHU Zheng-you. Recent advances in differ¬ential quadrature method with application to mechan¬ics[J]. Journal of Shanghai University (Natural Science Edi¬tion), 2009, 15(6): 551-559.
[16] Fung T C. Solving initial value problems by differential quad¬rature method-Part 1: first order equations[J]. International Jour¬nal for Numerical Methods in Engineering, 2001, 50(6): 1411-1427.
[17] Fung T C. Solving initial value problems by differential quad¬rature method-Part 2: second and higher order equations[J]. Inter¬national Journal for Numerical Methods in Engineering, 2001, 50(6): 1429-1454.
[18] Fung T C. Stability and accuracy of differential quadrature method in solving dynamic problems[J]. Computer Methods in Applied Mechanics and Engineering, 2002, 191(13-14): 1311-1331.
[19] 刘剑, 王鑫伟. 基于微分求积的逐步积分法与常用时间积分法的比较[J]. 力学季刊, 2008, 29(2): 304-309.
LIU Jian, WANG Xin-wei. Comparisons of successive integra¬tion method based on differential quadrature with com¬monly used time integration schemes[J]. Chinese Quarterly of Me¬chanics, 2008, 29(2): 304-309.
[20] Wang F Z, Liao X B, Xie X. Characteristics of the differential quadrature method and its improvement[J]. Mathematical Problems in Engineering, 2015. doi:10.1155/2015/657494.