A sampling-variability-free dimension-reduced probability density evolution equation method for high-dimensional nonlinear stochastic dynamic analysis

Abstract

Effective stochastic dynamic analysis of high-dimensional nonlinear structural systems is essential for ensuring structural safety. However, two key challenges persist: (1) accurately capturing physical system response characteristics with limited samples while avoiding sampling-induced variability, and (2) effectively extracting probabilistic information from stochastic response samples. To address these issues, a sampling-variability-free Dimension-Reduced Probability Density Evolution Equation (DR-PDEE) method is proposed by integrating a deterministic sampling method, i.e., the New Generating Vectors-based Number-Theoretic Method (NGV-NTM), with the DR-PDEE framework. In this method, the NGV-NTM is first performed to efficiently generate deterministic high-dimensional point sets with excellent space-filling property, enabling variability-free representation of stochastic input samples. These samples are used to compute dynamic responses, from which intrinsic drift functions are subsequently estimated, as required by the DR-PDEE. The DR-PDEE method then yields one- or two-dimensional partial differential equations governing the evolution of the response probability density function, which can be solved for effective stochastic dynamic response analysis. In this work, the theoretical foundations of the proposed method are first established via number theory and the Kramers–Moyal expansion, followed by a three-step numerical implementation strategy. Numerical examples demonstrate that the proposed method achieves superior accuracy and efficiency while eliminating sampling variability, outperforming both the Monte Carlo simulation and the random-sampling-based DR-PDEE solution.