Multisphere-based importance sampling for structural reliability

An innovative Importance Sampling (IS) in calculating reliability for a structural engineering problem using multiple spheres is proposed. Radial-based Importance Sampling (RBIS) builds a single sphere with its center at the origin and a radius of beta(adistance from the most probable point to the origin) to recognize the safety samples located inside the sphere. Such samples are excluded for function evaluation to reduce the computational cost. Adaptive radial-based importance sampling (ARBIS) extended RBIS with an adaptive scheme to determine the optimal radius beta. To maximize the number of safety samples, multiple spheres with various centers and radii are recommended in current study. Two types of spheres are introduced: the "origin" and "non-origin spheres". It is shown that in addition to "origin sphere", the "non-origin spheres" can exclude more safety samples. As a results, computational efficiency is significantly enhanced. Similar to RBIS, samples outside the "origin sphere" are generated in the proposed method. However, only part of these samples is evaluated by the limit state function. A simple but robust line search is adopted to determine the radius of each sphere. Effects of sphere number and locations are discussed. Robustness and efficiency of the proposed method are demonstrated by various benchmark problems. Results show that for most cases, the proposed method greatly reduces the number of function evaluation with similar accuracy and uncertainty levels compared to those of Monte Carlo Simulation (MCS), RBIS and ARBIS.