Error estimate of point selection in uncertainty quantification of nonlinear structures involving multiple nonuniformly distributed parameters

Summary The error analysis for the selection of representative point sets (RPSs) in multidimensional random‐variate space assigned with arbitrary nonuniform distribution with compact support for uncertainty quantification is developed by extending the Koksma‐Hlawka inequalities, which bound the worst error with some kind of discrepancy of the RPS and the variation of the integrand. The novel concepts of the EF‐discrepancy and the GF‐discrepancy are introduced, and the connection and equivalence between them are inquired into. Based on such theoretical basis, the error estimate of selecting RPSs in the standardized space instead of that in the original physical space is studied, showing that the standardization of the input random variables does not increase, but usually reduce, the error bound by GF‐discrepancy. The extended Koksma‐Hlawka inequality also establishes the theoretical basis for the error estimate of the probability density evolution method. Besides, the closed‐form expressions for the EF‐discrepancy are given in the Appendix. A numerical example involving a complex nonlinear engineering structure modeled by the finite element method is studied, showing the accuracy of the proposed approach.