A Time-Dependent Reliability Analysis Method using a Niching Genetic Algorithm

A reliability analysis method is presented for time-dependent systems under uncertainty. A level-crossing problem is considered where the system fails if its maximum response exceeds a specified threshold. The proposed method uses a double-loop optimization algorithm. The inner loop calculates the maximum response in time for a given set of random variables, and transforms a time-dependent problem into a time-independent one. A time integration method is used to calculate the response at discrete times. For each sample function of the response random process, the maximum response is found using a global-local search method consisting of a genetic algorithm (GA), and a gradient-based optimizer. This dynamic response usually exhibits multiple peaks and crosses the allowable response level to form a set of complex limit states, which lead to multiple most probable points (MPPs). The outer loop of the proposed method identifies all potential significant MPPs, which have the highest contribution to the system probability of failure. Approximate MPPs are first calculated at a reasonable computational cost using a Niching GA, which can robustly identify the multiple solutions of a multimodal problem. The location of each MPP is subsequently, refined using a gradient-based optimizer. Among all MPPs, the significant ones are identified using a simple correlation analysis. Approximate limit states are built at the identified MPPs, and the system failure probability is estimated using bi-modal bounds or a Monte Carlo simulation using an approximated failure domain. An importance sampling technique may also used to calculate the probability of failure for comparison purposes. The vibration response of a cantilever plate under random oscillating pressure load and a point load is used as an example. A finite-element model is used to calculate the dynamic response.