Improved Neumann Expansion Method for Stochastic Finite Element Analysis

Abstract

This paper presents an improved Neumann expansion method for calculating statistical moments of either a structural response or system inverse matrix under stochastic uncertainties in system properties and loads. The essentials of the proposed method are the successive matrix inversion and partial bivariate dimension reduction method that enable efficient multidimensional probability integration. The successive matrix inversion computes an exact realization of a stochastic system response, or an inverse matrix, by using the concept of inverse Neumann expansion. In characterizing the stochastic response, multidimensional probability integration is solved for statistical moments and covariance matrices. The existing univariate decomposition method is highly efficient to approximate the multidimensional integration. However, due to the missing interaction effects, the univariate decomposition method could lead to a huge error. In this study, a partial bivariate decomposition method is proposed to capture bivariate interaction effects approximately with a marginal increase in computational cost. Several numerical examples, including representative structural problems, are discussed to demonstrate the accuracy and efficiency of the proposed method compared with other existing methods. It is found that the proposed method provides better accuracy with significant computational cost savings when random fluctuations make locally limited changes in a system.