Second-order time-variant reliability analysis method based on Hessian matrix approximation

First-order time-variant reliability analysis(TRA) methods have been widely studied in the literature. However, these methods have low accuracy when solving time-variant reliability analysis problems with high nonlinear performance functions. To solve this problem, this paper proposes a second-order TRA method based on Hessian matrix approximation. First, the symmetric rank-one based quasi-Newton method is introduced to calculate the most probable point(MPP) and obtain approximated Hessian matrix of instantaneous performance function. Then, the instantaneous performance function is approximated by a second-order parabolic model, and therefore the instantaneous response is transformed into the weight sum of a normal variable and a chi-square variable. Correspondingly, the time-variant response is transformed into a non-stationary non-Gaussian stochastic process(NNP), which improves the accuracy of TRA. The Kriging models of the MPP trajectory and the mean curvature radius function are constructed, and the second-order time-variant reliability is estimated by sampling in the original random space to avoid the difficulty of modeling and simulation of the NPP in the response space. Finally, the results of two simulation examples show that the proposed method can significantly improve the accuracy of TRA without increasing the computational cost compared with first-order TRA method.