基于同伦分析方法的随机结构静力响应求解  

SOLUTION FOR STATIC RESPONSE OF STRUCTURE WITH RANDOM PARAMETERS BASED ON HOMOTOPY ANALYSIS METHOD

作　　者：张衡 王鑫[1,2] 陈辉 黄斌[1] ZHANG Heng;WANG Xin;CHEN Hui;HUANG Bin(School of Civil Engineering&Architecture,Wuhan University of Technology,Wuhan 430070,China;Wuchang Shouyi University,Wuhan 430064,China;College of Post and Telecommunication,Wuhan Institute of Technology,Wuhan 430073,China)

机构地区：[1]武汉理工大学土木工程与建筑学院,武汉430070 [2]武昌首义学院,武汉430064 [3]武汉工程大学邮电与信息工程学院,武汉430073

出　　处：《工程力学》2019年第11期27-33,61,共8页Engineering Mechanics

基　　金：国家自然科学基金项目(51578431)

摘　　要：该文提出了一种基于同伦分析方法的求解含随机参数结构的静力响应的新方法。该方法将随机静力平衡方程重新进行同伦构造,利用含随机变量和趋近函数的同伦级数展式来表示结构的随机静力位移响应,该同伦级数的各阶确定性系数和趋近函数可通过对一系列的变形方程求解得到。由于趋近函数的引入,该同伦级数解相较于传统的摄动法有更大的收敛范围,对于含较大变异性随机参数的结构也能获得不错的求解精度。同时,该文提出了一种降维策略来提高该方法的计算效率。数值算例表明,与目前广泛应用的广义正交多项式展开法(GPC)相比,从计算精度上看,该文方法的3阶展开与GPC2阶展开相当,该文方法的6阶展开与GPC4阶展开相当,而计算时间上前者均明显少于后者。此外,该文方法也可以方便地应用到随机结构的几何非线性分析当中,并具有较好的计算精度和计算效率。A homotopy-based method for calculating the stochastic responses of a structure under static loads involving random parameters is proposed. In this method, the static equilibrium equation is reconstructed based on the homotopy analysis method, the stochastic responses are represented by an infinite multivariate homotopy series of the random variables and approaching functions, and all the deterministic coefficients in the multivariate series are determined through solving a series of various order of deformation equations. This homotopy series solution obtained has a relatively large convergence domain due to the approach function compared with the Taylor series, which makes the series solution independent of random parameters with small fluctuation. Further, a dimension reduction strategy is proposed to improve the computational efficiency of the solution. The numerical examples show that: when considering the computational accuracy, compared with the recently widely used generalized polynomial chaos method(GPC), a third-order expansion of the proposed method is comparable with the second-order expansion of GPC, a sixth-order expansion of the proposed method is comparable with the fourth-order expansion of GPC. However, the computational effort of the former is significantly less than the latter. In addition, the presented method are also suitable for solving geometric nonlinearity problems.

关 键 词：随机静力响应分析 同伦分析方法 摄动法 随机有限元方法 几何非线性分析 

分 类 号：TU311.4[建筑科学—结构工程]