Uncertainty analysis by dimension reduction integration and Saddlepoint Approximations

Uncertainty analysis, which assesses the impact of the uncertainty of input variables on responses, is an indispensable component in engineering design under uncertainty such as reliability based design and robust design. However, uncertainty analysis is not an affordable computational burden in many engineering problems. In this paper, a new uncertainty analysis method is proposed with the purpose of accurately and efficiently estimating the cumulative distribution function (CDF), probability density function (PDF) and statistical moments of a response given the distributions of input variables. The bivariate dimension-reduction method and numerical integration are used to calculate the moments of the response; then Saddlepoint Approximations are employed to estimate the CDF and PDF of the response. The proposed method requires neither the derivatives of the response nor the search of the Most Probable Point (MPP), which is needed in the commonly used First - or Second - Order Reliability Methods (FORM or SORM). The efficiency and accuracy of the proposed method is illustrated with three example problems. The method is more accurate and efficient for estimating the full range of the distribution of a response than FORM and SORM. This method provides results as accurate as Monte Carlo simulation, with a significantly reduced computational effort.