Efficient Algorithm for Second‐Order Reliability Analysis

Abstract

In the second‐order reliability method the principal curvatures of the limit‐state surface at the design point are used to construct a paraboloid approximation of the surface, which is then used to compute a second‐order estimate of the failure probability. The principal curvatures are the eigenvalues of the Hessian of the surface. In this paper an efficient algorithm is developed to determine the principal curvatures without computing the Hessian. The curvatures are computed in an iterative manner using the gradient of the limit‐state function, and are obtained in the decreasing order of their absolute magnitudes, which is also the order of their importance in reliability analysis. The computation can be terminated when the last curvature obtained is sufficiently small. The method is efficient for problems with large numbers of random variables, especially when an efficient algorithm for computing the gradient is available. Several numerical examples, including a finite‐element application involving 99 random variables, demonstrate the accuracy and efficiency of the method.