Research Papers

A General Methodology for Uncertainty Quantification in Engineering Analyses Using a Credible Probability Box

[+] Author and Article Information
Mark E. Ewing, Brian C. Liechty, David L. Black

Northrop Grumman Innovation Systems,
Promontory, UT 84302

Manuscript received April 12, 2018; final manuscript received September 10, 2018; published online October 8, 2018. Assoc. Editor: Christopher J. Roy.

J. Verif. Valid. Uncert 3(2), 021003 (Oct 08, 2018) (12 pages) Paper No: VVUQ-18-1013; doi: 10.1115/1.4041490 History: Received April 12, 2018; Revised September 10, 2018

Uncertainty quantification (UQ) is gaining in maturity and importance in engineering analysis. While historical engineering analysis and design methods have relied heavily on safety factors (SF) with built-in conservatism, modern approaches require detailed assessment of reliability to provide optimized and balanced designs. This paper presents methodologies that support the transition toward this type of approach. Fundamental concepts are described for UQ in general engineering analysis. These include consideration of the sources of uncertainty and their categorization. Of particular importance are the categorization of aleatory and epistemic uncertainties and their separate propagation through an UQ analysis. This familiar concept is referred to here as a “two-dimensional” approach, and it provides for the assessment of both the probability of a predicted occurrence and the credibility in that prediction. Unique to the approach presented here is the adaptation of the concept of a bounding probability box to that of a credible probability box. This requires estimates for probability distributions related to all uncertainties both aleatory and epistemic. The propagation of these distributions through the uncertainty analysis provides for the assessment of probability related to the system response, along with a quantification of credibility in that prediction. Details of a generalized methodology for UQ in this framework are presented, and approaches for interpreting results are described. Illustrative examples are presented.

Copyright © 2018 by American Society of Mechanical Engineers
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Fig. 1

Illustration of credible intervals: (a) intervals defined on the log-normal PDF and (b) intervals defined on the log-normal CDF

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Fig. 2

One-dimensional Monte Carlo approach

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Fig. 3

One-dimensional Monte Carlo results for the model of Eq. (1)

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Fig. 4

Two-dimensional Monte Carlo approach

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Fig. 7

Two-dimensional Monte Carlo approach using an inner loop for aleatory and an outer loop for epistemic uncertainty: (a) ensemble of CDFs and credible P-box for the SRQ of Eq. (1), (b) SRQ ranges for 10% and 90% probabilities interpreted from the P-box, and (c) probability ranges for an SRQ value interpreted from the P-box.

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Fig. 11

Length distribution

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Fig. 9

Weight distribution

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Fig. 10

Strength distribution

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Fig. 5

Two-dimensional Monte Carlo results for the model of Eq. (1)

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Fig. 6

One-dimensional approach combining epistemic and aleatory uncertainties in a single Monte Carlo iteration scheme

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Fig. 12

Credible P-box for the original beam design

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Fig. 13

Credible P-box for the updated design

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Fig. 14

Solid rocket motor

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Fig. 15

Nozzle contour with design stations at various area ratios (A) and supporting analysis stations (S)

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Fig. 16

The CDF ensemble and the 90% credible P-box for the 200 °F isotherm depth (Station 3)

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Fig. 17

Results for the 200 °F isotherm depths at the nozzle analysis stations



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