Efficient importance analysis methods for structures with distribution parameter uncertainty based on cubature formula

The importance analysis of a structural system with distribution parameter uncertainty can identify key parameters that significantly affect its output performance, thus providing importance guidance for its design and optimization. However, the traditional importance analysis method requires the three-loop Monte Carlo sampling to estimate the importance measurement index of a distribution parameter with such output characteristic values as mean and variance, whose computational cost is too large. To solve this problem, two efficient cubature formula methods based on the surrogate sampling probability density function (SSPDF) for the importance analysis of distribution parameters are proposed: ①the double-loop cubature formula based on the surrogate sampling probability density function (S-DLCF); ②the single-loop cubature formula based on the surrogate sampling probability density function (S-SLCF). The two methods use cubature formulas to efficiently compute the nested mean and variance in the importance measurement index of a distribution parameter, thus solving the problem that the computational effort of propagating parameter uncertainty to output characteristic values depends on parameter dimensionality because of SSPDF. The S-DLCF makes full use of the efficiency and accuracy of the cubature formula to estimate output statistical moments; the S-SLCF simplifies the integral to calculate output moments by expanding the dimensionality of the distribution parameter. The numerical and engineering examples verify the efficiency and accuracy of the two methods for the importance analysis of distribution parameters.