Domain Decomposition Method for Calculating the Failure Probability of Linear Dynamic Systems Subjected to Gaussian Stochastic Loads

In this paper the problem of calculating the probability of failure of linear dynamic systems subjected to random vibrations is considered. This is a very important and challenging problem in structural reliability. The failure domain in this case can be described as a union of linear failure domains whose boundaries are hyperplanes. Each linear limit state function can be completely described by its own design point, which can be analytically determined, allowing for an exact analytical calculation of the corresponding failure probability. The difficulty in calculating the overall failure probability arises from the overlapping of the different linear failure domains, the degree of which is unknown and needs to be determined. A novel robust reliability methodology, referred to as the domain decomposition method (DDM), is proposed to calculate the probability that the response of a linear system exceeds specified target thresholds. It exploits the special structure of the failure domain, given by the union of a large number of linear failure regions, to obtain an extremely efficient and highly accurate estimate of the failure probability. The number of dynamic analyses to be performed in order to determine the failure probability is as low as the number of independent random excitations driving the system. Furthermore, calculating the reliability of the same structure under different performance objectives does not require any additional dynamic analyses. Two numerical examples are given demonstrating the proposed method, both of which show that the method offers dramatic improvement over standard Monte Carlo simulations, while a comparison with the ISEE algorithm shows that the DDM is at least as efficient as the ISEE.