配点型区间有限元法  

COLLOCATION INTERVAL FINITE ELEMENT METHOD

作　　者：邱志平[1] 祁武超[1] 

机构地区：[1]北京航空航天大学固体力学研究所,北京100191

出　　处：《力学学报》2011年第3期496-504,共9页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金(10872017);国家自然科学基金重大研究计划(90816024)资助项目~~

摘　　要：在分析Taylor展开"点逼近"区间有限元法不足的基础上,提出了基于Chebyshev第一类正交多项式全局逼近目标函数的配点型区间有限元法.该方法不需要计算目标函数对不确定性变量的灵敏度,不要求不确定性变量的变化范围为小区间,并适合求解目标函数为不确定变量非线性函数的情形.目标函数正交展开式的系数采用Gauss-Chebyshev求积公式得到,故需要在不确定性变量所在区间内配置高斯积分点.计算目标函数在高斯点的取值是该方法的主要工作量,当不确定性变量数为m,并选用高斯十点法进行积分时,需要对系统进行12m次分析.算例表明,在其他区间有限元法失效的情况下,配点型区间有限元法依然能够得到几乎精确的区间界限.Based on shortcoming analysis of ＇point approximation＇ interval finite element method with Taylor expansion,collocation interval finite element method based on the first Chebyshev polynomials which can approach objective function in global domain is proposed in this paper.The method does not require the sensitivities of the objective function with respect to uncertain variables and the assumption of narrow interval is also not needed.The method is suitable for solving the case that the objective function is strongly nonlinear with respect to the uncertain variables.The orthogonal expansion coefficients of the objective function are obtained from Gauss-Chebyshev quadrature formula.So Gauss integration points are collocated in the intervals of uncertain variables.The main computational effort is to calculate the values of objective function at Gaussian integration points.When the number of the uncertain variables is m and the ten-point Gauss integral method is introduced,it is needed to analyze the system with 12m times.Examples show that the collocation interval finite element method can still obtain almost exact interval bounds in the case that other interval finite element methods are invalid.

关 键 词：区间有限元 Chebyshev第一类正交多项式 GAUSS积分 TAYLOR展开 函数逼近 

分 类 号：O174.1[理学—数学] O242.2[理学—基础数学]