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Mathematical Problems in Engineering
Volume 2013, Article ID 325025, 13 pages
Research Article

Generalized Neumann Expansion and Its Application in Stochastic Finite Element Methods

1Department of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing 100084, China
2High Performance Computing Center, School of Aerospace, Tsinghua University, Beijing 100084, China
3Key Laboratory of Applied Mechanics, School of Aerospace, Tsinghua University, Beijing 100084, China
4College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK

Received 18 July 2013; Accepted 13 August 2013

Academic Editor: Zhiqiang Hu

Copyright © 2013 Xiangyu Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


An acceleration technique, termed generalized Neumann expansion (GNE), is presented for evaluating the responses of uncertain systems. The GNE method, which solves stochastic linear algebraic equations arising in stochastic finite element analysis, is easy to implement and is of high efficiency. The convergence condition of the new method is studied, and a rigorous error estimator is proposed to evaluate the upper bound of the relative error of a given GNE solution. It is found that the third-order GNE solution is sufficient to achieve a good accuracy even when the variation of the source stochastic field is relatively high. The relationship between the GNE method, the perturbation method, and the standard Neumann expansion method is also discussed. Based on the links between these three methods, quantitative error estimations for the perturbation method and the standard Neumann method are obtained for the first time in the probability context.