Multi‐Sigmoid‐Enhanced Global Multimodal Density Modeling Framework for Failure Probability Assessment

ABSTRACT

Probabilistic modeling is fundamental to engineering reliability analysis. Multimodality may arise from mixed operating conditions, variations between material batches, and drift in manufacturing and assembly parameters. Such multimodality can obscure valleys and tail features that often dominate failure probability. To address this issue, this paper proposes a multimodal density modeling and reliability assessment framework based on fractional moments and the maximum entropy principle. The proposed framework uses a multi‐sigmoid transformation in the density modeling process. A polynomial‐corrected maximum entropy model is then constructed to capture global trends as well as local valleys and tail structures while satisfying non‐negativity and normalization constraints. An adaptive numerical optimization module is used to determine the number and orders of fractional moments, Lagrange multipliers, and lower‐bound parameters. A Jacobian‐corrected inverse mapping ensures consistent assessment in the original physical space. Using paired input‐output samples, the resulting model can estimate failure probability and local input‐variable sensitivity without requiring an explicit analytical performance function. As a post‐processing diagnostic, a nonparametric grouped sensitivity module is introduced to examine how local input regions affect the response distribution and failure probability. Analytical derivations, numerical benchmarks, and engineering case studies demonstrate that the proposed method achieves accurate density fitting and robust failure probability estimation.