First passage probability of elasto-plastic systems by importance sampling with adapted process

A new importance sampling method is presented for computing the first passage probability of elasto-plastic systems under white noise excitations. The importance sampling distribution corresponds to shifting the mean of the excitation to an 'adapted' ('predictable') stochastic process whose future is determined based on information only up to the present. Choosing the adapted process involves designing an adaptive control force algorithm in a stochastic environment that targets to drive the response to first passage failure based on information up to the present. Algorithms for single-degree-of-freedom linear and elasto-plastic systems are proposed and their resulting computational efficiency investigated. Numerical results show that the use of adapted process is particularly useful for nonlinear hysteretic systems where hysteretic effects undermine the effectiveness of conventional importance sampling method based on fixed design points.