A unified formalism of the GE-GDEE for generic continuous responses and first-passage reliability analysis of multi-dimensional nonlinear systems subjected to non-white-noise excitations

The stochastic response analysis and first-passage reliability evaluation of multi-dimensional nonlinear systems subjected to non-white-noise engineering dynamic excitations have long been challenging problems. The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE) provides a promising tool and is extended in the present paper for these purposes. Firstly, a more general derivation of the GE-GDEE is given for generic Markov or non-Markov path-continuous stochastic processes. In this unified formalism, the GE-GDEE can be established with respect to one- or two-dimensional path-continuous response(s) of interest in a multi-dimensional system subjected to non-white-noise excitations. Further, for the first-passage reliability evaluation, a new process absorbed at the boundary (ABP) corresponding to failure criterion is constructed, and the GE-GDEE of ABP can be established as a one- or two-dimensional partial differential equation (PDE). This equation can then be solved to obtain the probability density in the safety domain, and the first-passage reliability can thereby be obtained via an integral. Meanwhile, the eligibility of imposing an absorbing boundary condition for the first-passage reliability is rigorously proved. Numerical algorithms are elaborated. Several examples of first-passage reliability analysis of multi-dimensional linear/nonlinear systems subjected to white/non-white noise are illustrated, demonstrating the efficiency and accuracy of the proposed method.