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Research Papers

Random Field Characterization Considering Statistical Dependence for Probability Analysis and Design

[+] Author and Article Information
Zhimin Xi

Department of Mechanical Engineering, University of Maryland, College Park, MD 20742zxi@umd.edu

Byeng D. Youn1

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

Aleatory uncertainty is defined as objective and irreducible uncertainty with sufficient information on the random variable.

Epistemic uncertainty can be classified as subjective and reducible uncertainty due to the lack of knowledge on the random variable.

1

Corresponding author.

J. Mech. Des 132(10), 101008 (Oct 04, 2010) (12 pages) doi:10.1115/1.4002293 History: Received July 13, 2009; Revised July 29, 2010; Published October 04, 2010; Online October 04, 2010

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.

Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Random field over samples

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Figure 2

Flowchart for determining the number of the important signatures

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Figure 3

Statistical dependence of v1 and v2

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Figure 4

Nonlinearity of YT,k with the transformation sequence [v1,v2]

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Figure 5

Nonlinearity of YT,k with the transformation sequence [v2,v1]

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Figure 6

EDR results with the transformation sequence [v1,v2]

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Figure 7

EDR results with the transformation sequence [v2,v1]

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Figure 8

Simulation model of a cantilever beam with the random field

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Figure 9

The first two normalized signatures

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Figure 10

Comparison of the exact and approximate random fields (first random field realization)

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Figure 11

Statistical properties of V1 and V2

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Figure 12

Histograms of the maximum beam deflection using RFA and RPA

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Figure 13

Bistable mechanism

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Figure 14

Force displacement curve

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Figure 15

Creation of the random field for one beam

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Figure 16

The first two normalized signatures

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Figure 17

Comparison of exact and approximate random fields (first random field realization)

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Figure 18

Statistical properties of V1 and V2

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Figure 19

Comparison of RFA and RPA

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Figure 20

Comparison of the proposed RFA and MCS

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Figure 25

Reliability error by ignoring statistical dependence of random field variables

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Figure 24

Comparison of RFA and RPA

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Figure 23

Statistical dependence of random field variables

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Figure 22

Approximation of the random field with different numbers of signatures

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Figure 21

Three random field snapshots

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