Importance sampling algorithm for generalized failure probability function

Under probabilistic inputs and fuzzy states, the generalized failure probability function (GFPF) gives the GFPF value as a function of the random-input distribution parameter (DP) within the design domain. By quantifying how these DPs affects structural safety, the GFPF decouples generalized reliability-based design optimization from generalized failure probability constraints. To efficiently estimate GFPF, a single-loop method utilizing cross-entropy-based importance sampling (SL-CEIS) is developed. Initially, an optimal importance sampling density (ISD) is designed, which is independent of DPs yet encompasses the entire design domain. Based on this unified density, a single-loop framework is proposed whereby different GFPF values corresponding to various DP realizations are capable of being estimated using a single importance sample pool drawn from the unified ISD. Subsequently, an importance sampling density based on cross-entropy is formulated to approximate the unified ISD with implicit and irregular expressions, enabling efficient exploration of the fuzzy failure domain and extraction of important samples. Finally, a Kriging model is adaptively trained to forecast the performance function values of the importance samples. Numerical tests show SL-CEIS delivers accurate GFPF estimates at a cost that scales neither with the dimension of DPs nor with the number of their realizations.