Importance Sampling Method for Discrete Critical State Space

For performance functions with characteristics of non-normality, small failure probability and nonlinearity, it is difficult to obtain satisfactory accuracy of structural failure probability at low overhead by using traditional methods. For the structural reliability problem with consideration of complex performance functions, the UGF-importance sampling method focusing on the discrete critical state space is proposed by organically combining the universal generating function with the importance sampling. The continuous random variables are discretized in low density, and the universal generating function of the structure can be obtained by means of combination operations, thus the random space is divided into failure domain, critical state domain and reliability domain. The failure probability of the failure domain is directly figured out by the universal generation function, whereas the failure probability of the critical state domain is mainly obtained by employing the important sampling for the hot focal elements. The sum of the failure probabilities of the two domains is the estimate of the structural failure probability. The numerical examples verify the rationality and feasibility of the proposed method. The new method not only gives full play to the universality of the universal generating function technique for arbitrarily distributed random variables and nonlinear performance functions, but avoids combination explosion with the help of importance sampling as well, so as to give a feasible path for the rapid and accurate calculation of structural reliability involving complex performance functions.