Abstract

This paper presents the eigenvector dimension reduction (EDR) method for probability analysis that makes a significant improvement based on univariate dimension reduction (DR) method. It has been acknowledged that the DR method is accurate and efficient for assessing statistical moments of mildly nonlinear system responses in engineering applications. However, the recent investigation on the DR method has found difficulties of instability and inaccuracy for highly nonlinear system responses while maintaining reasonable efficiency. The EDR method integrates the DR method with three new technical components: (1) eigenvector sampling, (2) one-dimensional response approximation, and (3) a stabilized Pearson system. First, 2N+1 and 4N+1 eigenvector sampling schemes are proposed to resolve correlated and asymmetric random input variables. The eigenvector samples are chosen along the eigenvectors of the covariance matrix of random parameters. Second, the stepwise moving least squares (SMLS) method is proposed to accurately construct approximate system responses along the eigenvectors with the response values at the eigenvector samples. Then, statistical moments of the responses are estimated through recursive numerical integrations. Third, the stabilized Pearson system is proposed to predict probability density functions (PDFs) of the responses while eliminating singular behavior of the original Pearson system. Results for some numerical and engineering examples indicate that the EDR method is a very accurate, efficient, and stable probability analysis method in estimating PDFs, component reliabilities, and qualities of system responses