The proper orthogonal decomposition method has been employed to extract the important field signatures of random field observed in an engineering product or process. Our preliminary study found that the coefficients of the signatures are statistically uncorrelated but may be dependent. To this point, the statistical dependence of the coefficients has been ignored in the random field characterization for probability analysis and design. This paper thus proposes an effective random field characterization method that can account for the statistical dependence among the coefficients for probability analysis and design. The proposed approach has two technical contributions. The first contribution is the development of a natural approximation scheme of random field while preserving prescribed approximation accuracy. The coefficients of the signatures can be modeled as random field variables, and their statistical properties are identified using the chi-square goodness-of-fit test. Then, as the paper’s second technical contribution, the Rosenblatt transformation is employed to transform the statistically dependent random field variables into statistically independent random field variables. The number of the transformation sequences exponentially increases as the number of random field variables becomes large. It was found that improper selection of a transformation sequence among many may introduce high nonlinearity into system responses, which may result in inaccuracy in probability analysis and design. Hence, this paper proposes a novel procedure of determining an optimal sequence of the Rosenblatt transformation that introduces the least degree of nonlinearity into the system response. The proposed random field characterization can be integrated with any advanced probability analysis method, such as the eigenvector dimension reduction method or polynomial chaos expansion method. Three structural examples, including a microelectromechanical system bistable mechanism, are used to demonstrate the effectiveness of the proposed approach. The results show that the statistical dependence in the random field characterization cannot be neglected during probability analysis and design. Moreover, it is shown that the proposed random field approach is very accurate and efficient.

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# Random Field Characterization Considering Statistical Dependence for Probability Analysis and Design

Zhimin Xi

,
Zhimin Xi

Department of Mechanical Engineering,

zxi@umd.edu
University of Maryland

, College Park, MD 20742
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Byeng D. Youn

,
Byeng D. Youn

Assistant Professor

School of Mechanical and Aerospace Engineering,

bdyoun@snu.ac.kr
Seoul National University

, Seoul, Korea
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Chao Hu

Chao Hu

Department of Mechanical Engineering,

huchaost@umd.edu
University of Maryland

, College Park, MD 20742
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Zhimin Xi

Byeng D. Youn
Assistant Professor

School of Mechanical and Aerospace Engineering,

Seoul National University

, Seoul, Koreabdyoun@snu.ac.kr

Chao Hu

Department of Mechanical Engineering,

University of Maryland

, College Park, MD 20742huchaost@umd.edu

*J. Mech. Des*. Oct 2010, 132(10): 101008 (12 pages)

**Published Online:**October 4, 2010

Article history

Received:

July 13, 2009

Revised:

July 29, 2010

Online:

October 4, 2010

Published:

October 4, 2010

Citation

Xi, Z., Youn, B. D., and Hu, C. (October 4, 2010). "Random Field Characterization Considering Statistical Dependence for Probability Analysis and Design." ASME. *J. Mech. Des*. October 2010; 132(10): 101008. https://doi.org/10.1115/1.4002293

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