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Research Papers

Probabilistic Framework for Uncertainty Propagation With Both Probabilistic and Interval Variables

[+] Author and Article Information
Kais Zaman

Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235kaiszaman@yahoo.com

Mark McDonald

Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235mark.p.mcdonald@vanderbilt.edu

Sankaran Mahadevan1

Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235sankaran.mahadevan@vanderbilt.edu

1

Corresponding author.

J. Mech. Des 133(2), 021010 (Feb 08, 2011) (14 pages) doi:10.1115/1.4002720 History: Received September 04, 2009; Revised May 12, 2010; Published February 08, 2011; Online February 08, 2011

This paper develops and illustrates a probabilistic approach for uncertainty representation and propagation in system analysis, when the information on the uncertain input variables and/or their distribution parameters may be available as either probability distributions or simply intervals (single or multiple). A unique aggregation technique is used to combine multiple interval data and to compute rigorous bounds on the system response cumulative distribution function. The uncertainty described by interval data is represented through a flexible family of probability distributions. Conversion of interval data to a probabilistic format enables the use of computationally efficient methods for probabilistic uncertainty propagation. Two methods are explored for the implementation of the proposed approach, based on (1) sampling and (2) optimization. The sampling-based strategy is more expensive and tends to underestimate the output bounds. The optimization-based methodology improves both aspects. The proposed methods are used to develop new solutions to challenge problems posed by the Sandia epistemic uncertainty workshop (Oberkampf, 2004, “Challenge Problems: Uncertainty in System Response Given Uncertain Parameters,” Reliab. Eng. Syst. Saf., 85, pp. 11–19). Results for the challenge problems are compared with earlier solutions.

Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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PBO and EBO bounds

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Figure 3

Optimization methods for output uncertainty quantification (case 1)

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Optimization methods for output uncertainty quantification (case 2)

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Figure 5

Mass-spring-damper system acted on by an excitation function (21)

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Family of Johnson distributions for input variable a for problem A-1

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Bounds on CDF of system response for problem A-1

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Family of Johnson distributions for input variable b for problem A-2

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Figure 9

Bounds on CDF of system response for problem A-2

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Bounds on CDF of system response for problem A-3

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Figure 11

Family of log-normal distributions for input variable b for problem A-4

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Figure 12

Bounds on CDF of system response for problem A-4

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Figure 13

Family of log-normal distributions for input variable b for problem A-5

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Figure 14

Bounds on CDF of system response for problem A-5

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Bounds on CDF of system response for problem A-6

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Figure 16

Family of triangular distributions for input variable k for problem B

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Family of triangular distributions for input variable ω for problem B

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Figure 18

Bounds on CDF of system response for problem B

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Figure 19

Rigorous versus optimal bounds

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Figure 1

Johnson distribution family identification

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