Partition of the probability-assigned space in probability density evolution analysis of nonlinear stochastic structures

Based on a partition of probability-assigned space, a strategy for determining the representative point set and the associated weights for use in the probability density evolution method (PDEM) is developed. The PDEM, which is capable of capturing the instantaneous probability density function of responses of linear and nonlinear stochastic systems, was developed in the past few years. The determination of the representative point set and the assigned probabilities is of paramount importance in this approach. In the present paper, a partition of probability-assigned space related to the representative points and the assigned probabilities are first examined. The error in the resulting probability density function of the stochastic responses is then analyzed, leading to two criteria on strategies for determining the representative points and a set of indices in terms of discrepancy of the point sets. A two-step algorithm is proposed, in which an initial uniformly scattered point set is mapped to an optimal set. The implementation of the algorithm is elaborated. Two methods for generating the initial point set are outlined. These are the lattice point sets and the Number-Theoretical nets. A density-related transformation yielding the final point set is then analyzed. Numerical examples are investigated, where the results are compared to those obtained from the standard Monte Carlo simulation and the Latin hyper-cube sampling, demonstrating the accuracy and efficiency of the proposed approach.