Advanced Dimension-Adaptive Sparse Grid Integration Method for Structural Reliability Analysis

Abstract

The conventional sparse grid (SG) integration method treats all considered random variables equally for the numerical integration of performance functions. The dimension-equal treatment inevitably causes a considerable demand of computation devoted to unimportant variables. To overcome this shortcoming, an advanced dimension-adaptive sparse grid (ADASG) integration method is proposed by introducing a numerical indicator quantifying the importance levels of tensor products of difference quadrature formulas. By achieving a target number of function evaluations, only several important tensor products of difference quadrature formulas are retained while the unimportant ones are removed. The proposed ADASG integration method is firstly employed to estimate the first four moments of performance functions. The estimated moments are then applied to structural reliability analysis by generating the probability density function of performance functions using the maximum entropy method. The advantage of the proposed ADASG integration method is demonstrated over four examples. The reliability index calculated by the proposed ADASG integration method shows favorable consistency to that obtained by the direct Monte Carlo simulation. Compared with conventional SG integration methods, the proposed ADASG integration method exhibits a better performance in terms of both accuracy and efficiency.