First excursion probability of dynamical systems: A review on computational methods

The theory of dynamic reliability, predicated on the first excursion failure criterion, holds significant importance in the domains of seismic and wind resistance of structures, as well as in the assessment of the reliability of machinery and airplanes. This theoretical framework offers a mathematical description of failure probabilities, which serve as critical indicators for the safety evaluations of dynamic systems. However, dynamical systems such as large structures, machines or airplanes are composed of numerous members and nodes that may be influenced by uncertainties related to loads, geometric imperfections, and material properties. The inherent high-dimensional randomness and pronounced nonlinear coupling effects contribute to the complexity and implicit nature of the system failure modes in these systems. Consequently, the computation of the first excursion probability for complex dynamical systems presents a formidable challenge that necessitates comprehensive investigation. To summarize the current methodologies, this paper delineates a state-of-the-art review of dynamic reliability theory, with a particular emphasis on its potential to address the first excursion probability in dynamical systems.