First excursion probabilities for linear systems by very efficient importance sampling

• Published in: Probabilistic engineering mechanics Vol. 16; no. 3; pp. 193 - 207
• Main Authors: Au, S.K., Beck, J.L.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.07.2001; Elsevier Science
• Subjects: Exact sciences and technology; First excursion problem; First passage problem; Fracture mechanics (crack, fatigue, damage...); Fracture mechanics, fatigue and cracks; Fundamental areas of phenomenology (including applications); Importance sampling; Linear systems; Monte Carlo simulation; Physics; Reliability; Solid mechanics; Structural and continuum mechanics; Linear systems; First passage problem; Reliability; First excursion problem; Monte Carlo simulation; Importance sampling; Level crossing; Monte Carlo method; Probabilistic approach; Random excitation; Rupture; Modeling; Linear model; Gaussian noise; First passage time; Random vibration; Structural analysis
• ISSN: 0266-8920; 1878-4275
• DOI: 10.1016/S0266-8920(01)00002-9

An analytical study of the failure region of the first excursion reliability problem for linear dynamical systems subjected to Gaussian white noise excitation is carried out with a view to constructing a suitable importance sampling density for computing the first excursion failure probability. Central to the study are ‘elementary failure regions’, which are defined as the failure region in the load space corresponding to the failure of a particular output response at a particular instant. Each elementary failure region is completely characterized by its design point, which can be computed readily using impulse response functions of the system. It is noted that the complexity of the first excursion problem stems from the structure of the union of the elementary failure regions. One important consequence of this union structure is that, in addition to the global design point, a large number of neighboring design points are important in accounting for the failure probability. Using information from the analytical study, an importance sampling density is proposed. Numerical examples are presented, which demonstrate that the efficiency of using the proposed importance sampling density to calculate system reliability is remarkable.