In this paper, two second-order methods are proposed for reliability analysis. First, general random variables are transformed to standard normal random variables. Then, the limit-state function is additively decomposed into one-dimensional functions, which are then expanded at the mean-value point to second-order terms. The approximated limit-state function becomes the sum of independent variables following noncentral chi-square distributions or normal distributions. The first method computes the probability of failure by the saddle-point approximation. If a saddle-point does not exist, the second method is then used. The second method approximates the limit-state function by a quadratic function with independent variables following normal distributions with the same variances. This treatment leads to a quadratic function that follows a noncentral chi-square distribution. These methods generally produce more accurate reliability approximations than the first-order reliability method (FORM) with 2n + 1 function evaluations, where n is the dimension of the problem. The effectiveness of the proposed methods is demonstrated with three examples, and the proposed methods are compared with the first- and second-order reliability methods (SROMs).
Saddle Point and Second Order Optimality in Nondifferentiable Nonlinear Abstract Multiobjective Optimization