A GF-discrepancy for point selection in stochastic seismic response analysis of structures with uncertain parameters

• The extended Koksma–Hlawka inequality for generic non-uniform distributions is proved.
• The relationship between EF- and GF-discrepancy is suggested.
• A point set rearrangement strategy for stochastic dynamics analysis is proposed.
• Numerical examples involving up to 50 basic random variables by PDEM is illustrated.

In the stochastic dynamic analysis of nonlinear structures, the strategy of point selection plays a critical role in achieving the tradeoffs between the accuracy and efficiency. To this end, cooperated with the concept of the extended F-discrepancy (EF-discrepancy), the Koksma–Hlawka inequality, which bounds the worst error of cubature formulae, is extended to the cases involving non-uniform distributions in the present paper. Further, in order to avoid the computational complexity of EF-discrepancy, the Generalized F-discrepancy (GF-discrepancy) is introduced. In light of the quantitative equivalence between the EF-discrepancy and GF-discrepancy, the extended Koksma–Hlawka inequality could be modified by replacing the EF-discrepancy with the GF-discrepancy. Thereby the rationality of adopting GF-discrepancy as the objective function of point selection is theoretically supported. Thus, by reducing the GF-discrepancy, a new strategy of representative point set determination via rearrangement is proposed. The proposed approach is then applied to stochastic dynamical response analysis of strong nonlinear structures by incorporated into the probability density evolution method, showing its effectiveness for practical applications. Problems to be further studied are outlined.