结构动力方程的2种精细时程积分

Two Precise Time-Integration Methods for Structural Dynamic Analysis

分析了增维精细时程积分和扩展精细时程积分2种方法在求解结构动力方程中的异同.在演变随机激励为多项式、指数函数以及正弦/余弦(虚指数)函数组合的一般形式下,分别推导出了2种方法所对应的显式离散递推表达式.结果表明:2种积分方法所对应的显式递推格式最终都转化为积分步长的多项式函数,并且在相同泰勒级数展开项的条件下,扩展精细积分除包含增维精细积分的所有递推项外,还包含一些高阶小项,理论上具有更高的精度;忽略高阶小项,2种方法尽管算法实现不同,离散递推格式完全一致;工程实例计算表明,二者计算精度都可以达到10位以上有效数字,扩展精细积分计算时间较增维精细积分少一个数量级.

Similarities and differences in solving dynamic equations between precise time-integration method with augmented matrix（PTI-AM） and extended precise time-integration method（EPTI） were analyzed.The explicit,discrete and recursive expressions for both methods were deduced,respectively,with the evolutionary random excitations in a general combined form of polynomial,exponential,and sinusoid/cosine functions.Both recursive expressions can be transformed into polynomial functions corresponding to the integral steps.With the same number of terms in the Taylor series,the recursive expression for EPTI contains additional high-order terms besides all the terms in PTI-AM.Therefore,EPTI has higher precision than PTI-AM does.If those additional high-order terms are neglected,the two methods have an identical discrete and recursive expression.In this respect,the two methods are essentially the same despite of different programming realization.An engineering example was presented,showing that the computing precision of both methods was as high as 10-significant-figures,and the computing time of EPTI was over 1 order of magnitude less than that of PTI-AM.