随机激励下高维强非线性系统瞬态响应联合概率密度的一种高效求解方法

Engineering structures may fail under strong nonlinear random vibrations, yet existing theories and methods still fall from the need in practical engineering, especially for the high-dimension strongly nonlinear systems. Moreover, systems often undergo a prolonged transient phase before reaching a steady state, during which failures may occur. Therefore, conducting transient response analysis for high-dimension strongly nonlinear systems is of significant importance. To date, the classical theory for describing the probabilistic response of such systems remains the Fokker-Planck-Kolmogorov (FPK) equation. However, as dimensionality increases, conventional mesh-dependent numerical methods like finite difference and finite element methods suffer from the curse of dimensionality, leading to rapidly declining efficiency or even computational infeasibility. This paper proposes a novel meshless solution technique combined with dimension-reduced FPK equation strategy to efficiently solve multidimensional transient FPK equations (up to 5-dimensional in our case studies). Specifically, the subspace decomposition method is employed to reduce the dimension of the high-dimensional FPK equation. Subsequently, by integrating physics-based and data-driven strategies with a small amount of prior data from deterministic analysis, a continuous-time radial basis function neural network regression model is employed to estimate the complex-shaped equivalent drift and diffusion coefficients. Singular initial conditions are handled via a short-time Gaussian assumption. Finally, a meshless physics-informed radial basis function neural network combined with a hypersphere sampling strategy is used to solve the reduced transient FPK equation, obtaining the non-Gaussian joint probability density function (PDF) of the interested quantities. Two numerical examples of high-dimensional nonlinear systems validate the accuracy and efficiency of the proposed method against Monte Carlo simulations (MCS). Results show that with only 400 deterministic analyses, the method accurately captures the joint PDFs of the reduced system responses at all time instants, achieving a tail accuracy of 10⁻⁴ and a computational efficiency up to 3700 times that of MCS.