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配点型区间有限元法

Collocation interval finite element method

  • 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.

     

    Abstract: Based on shortcoming analysis of `point approximation'interval finite element method with Taylor expansion, collocation intervalfinite element method based on the first Chebyshev polynomials which canapproach objective function in global domain is proposed in this paper. Themethod does not require the sensitivities of the objective function withrespect to uncertain variables and the assumption of narrow interval is alsonot needed. The method is suitable for solving the case that the objectivefunction is strongly nonlinear with respect to the uncertain variables. Theorthogonal expansion coefficients of the objective function are obtainedfrom Gauss-Chebyshev quadrature formula. So Gauss integration points arecollocated in the intervals of uncertain variables. The main computationaleffort is to calculate the values of objective function at Gaussianintegration points. When the number of the uncertain variables is m and theten-point Gauss integral method is introduced, it is needed to analyze thesystem with 12m times. Examples show that the collocation interval finiteelement method can still obtain almost exact interval bounds in the casethat other interval finite element methods are invalid.

     

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