A cubature collocation based sparse polynomial chaos expansion for efficient structural reliability analysis

Polynomial chaos expansion (PCE) is widely used to build a surrogate meta-model of the performance function for structural reliability analysis. The number of terms to be determined in PCE grows exponentially with the number of input random variables, which makes the computational effort intractable in practices. Although several sparse PCEs have been developed, a large number of deterministic model evaluations may be still required to achieve a satisfactory accuracy since equal-weighted collocation samples are used. To address such problems, this paper proposes a cubature collocation based sparse PCE for efficient structural reliability analysis. An iterative scheme is actually involved in the proposed method, which automatically selects the significant terms in PCE contributing to the variance of the performance function. The cubature formula not only generates unequal-weighted collocation samples, which has much faster convergent rate, but also provides the target variance of the performance function to terminate the iterative process. In this regard, a weighted regression method is employed in each step to determine the coefficients of PCE. As a consequence, a rather small number of terms in PCE are retained. Since the number of cubature collocation points is relatively small, the construction of a sparse PCE is quite efficient. Several numerical examples are investigated to validate the proposed method for structural reliability analysis. The results show the effectiveness of the proposed method for different reliability problems.