一种结构动力时程分析的积分求微方法

李鸿晶; 梅雨辰; 任永亮

doi:10.6052/0459-1879-19-105

一种结构动力时程分析的积分求微方法

doi: 10.6052/0459-1879-19-105

李鸿晶^*2), ,,
梅雨辰^*,
任永亮^†

* 南京工业大学土木工程学院,南京 211816
? 江苏省第二建筑设计研究院有限责任公司,江苏盐城 224005

基金项目: 1)国家自然科学基金项目(51478222);高等学校博士学科点专项科研基金项目助(20123221110011)

详细信息

通讯作者:
李鸿晶

中图分类号: O325
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出版历程
- 收稿日期: 2019-04-26
- 刊出日期: 2019-09-18

AN INTEGRAL DIFFERENTIATION PROCEDURE FOR DYNAMIC TIME-HISTORY RESPONSE ANALYSIS OF STRUCTURES

* College of Civil Engineering, Nanjing Tech University, Nanjing 211816, China
? Jiang Su Second Architectural Design & Research Institute Corporation Limited, Yancheng 224005, Jiangsu, China

摘要

摘要: 传统采用微分求积(differential quadrature,DQ)法求解动力问题时都是以位移响应作为基本未知量,而将速度响应和加速度响应表示为位移响应的加权和的形式.如此做法需要处理线性方程组或者矩阵方程(Sylvester方程)才能求得动力响应,导出的算法一般为有条件稳定算法.本文利用动力响应的Duhamel积分解,逆用DQ原理,提出了一种计算卷积的高精度显式算法.该算法可以逐时段地求解出动力时程响应,当各时段内DQ节点分布完全一致时,仅须进行一次Vandermonde矩阵求逆计算即可应用于各个时段,一次性获得时段内多个时刻的位移响应值,因而具有计算效率高的优点.通过分析动力方程积分格式,证明本文动力算法传递矩阵的谱半径恒等于1,因而该算法具有无条件稳定特性,且计算过程中不会产生数值耗散. 本文算法的数值精度取决于分析时段内布置的DQ节点数量$N$,具有$N-1$阶代数精度.实际操作时可以取10个甚至更多的DQ节点数,从而获得比较高的数值精度.
- 动力分析 /
- 卷积计算 /
- 微分求积 /
- 显式算法 /
- 无条件稳定
Abstract: Traditionally, the response of displacement is selected to be the basic unknown and the responses of velocity and acceleration are usually expressed by linear weighted sum of the displacement when the differential quadrature (DQ) method is applied to the solution of dynamic problems. Either the linear equations or matrix equations (Sylvester equation) has to be processed in such procedure for dynamic solutions and the derived algorithm is conditionally stable in general. In this paper,the DQ principle is used in the inverse way to implement a high-accuracy explicit algorithm for the operation of convolution, and the algorithm is applied to dynamic analysis via the solution of Duhamel's integral. The dynamic response can be solved over a finite time interval according to this procedure, so that the total time-history of response could be obtained step by step. The inverse of Vandermonde matrix is required only once if the distribution of DQ nodes are completely consistent in each time interval and the response at several time instants during the interval can be obtained simultaneously. Hence, the procedure for dynamic solutions numerically achieves a high computational efficiency. It is proved that the spectral radius of the transfer matrix in the dynamic algorithm is always equal to 1, so the algorithm has unconditional stability and no numerical dissipation occurs during the calculation. The numerical accuracy of this algorithm depends on $N$, the number of DQ nodes within the analyzing time interval, and an algebraic accuracy with the order of $N-1$can be achieved. In practice, 10 and even more DQ nodes of $N$are suggested in order to gain high accuracy for dynamic problems.
- dynamic analysis /
- convolution operation /
- differential quadrature /
- explicit algorithm /
- unconditional stability

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参考文献(1)

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