Hyper-spherical extrapolation method (HEM) for general high dimensional reliability problems

•An extrapolation method is developed to estimate the failure probability.•The method employs hyper-spherical formulations using geometric insights.•The method works well for general high dimensional reliability problems.•Performance is not sensitive to probability level, dimension or nonlinearity.•The method can estimate the first-passage probability of nonlinear random vibrations. To tackle challenges in low-probability, high-dimensional reliability analysis, this paper proposes hyper-spherical extrapolation method to estimate the failure probabilities efficiently and accurately. The extrapolation method employs hyper-spherical formulations of reliability problems developed based on geometric insights of high dimensional standard normal space. The proposed method can extrapolate the low probability region of interest using failure probabilities obtained from high/median probability region. Owing to the generality of the formulation, the proposed method is expected to work for general, component and system reliability problems defined in high-dimensional space of random variables. Using different presumptions on extrapolation, two slightly different versions of the extrapolation method are developed. Numerical examples with analytical limit-state functions and those concerning nonlinear random vibration of a hysteretic oscillator are investigated to test and demonstrate the performance of the proposed method. Finally, to facilitate an in-depth understanding of the proposed method and further developments, insights gained from the development of the method are also provided. The supporting source code and data are available for download at https://github.com/ziqidwang/Hyper-spherical-extrapolation-method.