An adaptive high-order stochastic perturbation collocation method for uncertainty quantification and propagation

In this paper, an adaptive high-order stochastic perturbation collocation method is proposed with the aim of effectively quantifying and accurately propagating uncertainty in engineering problems. By using the stochastic perturbation theory, this proposed method first performs a seventh-order perturbation expansion, and then combines the collocation method to cleverly construct a non-intrusive calculation format of the seventh-order perturbation expansion. This proposed format avoids the derivation of the stiffness matrix and exhibits the characteristic of a simple and unified form for different problems. In addition, an adaptive point selection method is strategically introduced, which can identify the perturbation terms that have a greater effect on the statistical characteristics of the responses and disregard the perturbation terms that have a lesser effect on them, thus achieving more efficient uncertainty analysis. In combination with the proposed method and the maximum entropy method, the numerical calculation format for the probability density function of the stochastic response is established, so as to fully and intuitively describe the stochastic characteristics of the responses. Through four numerical examples, the proposed method is compared with various uncertainty methods. The results demonstrate that the adaptive seventh-order stochastic perturbation collocation method has the characteristics of higher accuracy and greater efficiency. Especially for the examples with large coefficient of variation of the response, such as those with a coefficient of variation greater than 0.3, the proposed method can also provide the results with high accuracy.