Generalized distribution reconstruction based on the inversion of characteristic function curve for structural reliability analysis

Recovering the probability distribution of the performance function with an arbitrary shape is still a difficult task for structural reliability analysis. Since working with the characteristic function curve provides an alternative and frequently much simpler route than working directly with the probability distribution, this paper proposes a generalized density reconstruction method based on the inversion of characteristic function, which is estimated based on the complex fractional moments. First, the complex fractional moments of the limit state function are numerically evaluated by using the sample estimator. Then, the characteristic function is numerically determined in terms of complex fractional moments. Since inverting the numerical version of characteristic function to be the corresponding probability distribution is not an easy task, two analytical expressions of characteristic function are proposed according to its different features, where the undetermined parameters are obtained via fitting an analytical expression based on the data estimated by complex fractional moments. Then, the probability distribution of the limit state function is obtained by using the inverse Fourier transform of characteristic function, where a one-to-one mapping relationship is involved. Finally, the failure probability can be readily obtained. Several analytical distributions and numerical examples are extensively investigated to validate the proposed method.