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Research Papers

Probability Bounds Analysis Applied to the Sandia Verification and Validation Challenge Problem

[+] Author and Article Information
Aniruddha Choudhary

Mem. ASME
Aerospace and Ocean Engineering Department,
Virginia Tech,
460 Old Turner Street,
Blacksburg, VA 24061
e-mail: aniruddhac@gmail.com

Ian T. Voyles

Mem. ASME
Aerospace and Ocean Engineering Department,
Virginia Tech,
460 Old Turner Street,
Blacksburg, VA 24061
e-mail: itvoyles@vt.edu

Christopher J. Roy

Mem. ASME
Aerospace and Ocean Engineering Department,
Virginia Tech,
460 Old Turner Street,
Blacksburg, VA 24061
e-mail: cjroy@vt.edu

William L. Oberkampf

Mem. ASME
W. L. Oberkampf Consulting,
5112 Hidden Springs Trail,
Georgetown, TX 78633
e-mail: wloconsulting@gmail.com

Mayuresh Patil

Aerospace and Ocean Engineering Department,
Virginia Tech,
460 Old Turner Street,
Blacksburg, VA 24061
e-mail: mpatil@vt.edu

1Corresponding author.

Manuscript received February 8, 2015; final manuscript received July 31, 2015; published online February 19, 2016. Guest Editor: Kenneth Hu.

J. Verif. Valid. Uncert 1(1), 011003 (Feb 19, 2016) (13 pages) Paper No: VVUQ-15-1010; doi: 10.1115/1.4031285 History: Received February 08, 2015; Revised July 31, 2015; Accepted August 07, 2015

Our approach to the Sandia Verification and Validation Challenge Problem is to use probability bounds analysis (PBA) based on probabilistic representation for aleatory uncertainties and interval representation for (most) epistemic uncertainties. The nondeterministic model predictions thus take the form of p-boxes, or bounding cumulative distribution functions (CDFs) that contain all possible families of CDFs that could exist within the uncertainty bounds. The scarcity of experimental data provides little support for treatment of all uncertain inputs as purely aleatory uncertainties and also precludes significant calibration of the models. We instead seek to estimate the model form uncertainty at conditions where the experimental data are available, then extrapolate this uncertainty to conditions where no data exist. The modified area validation metric (MAVM) is employed to estimate the model form uncertainty which is important because the model involves significant simplifications (both geometric and physical nature) of the true system. The results of verification and validation processes are treated as additional interval-based uncertainties applied to the nondeterministic model predictions based on which the failure prediction is made. Based on the method employed, we estimate the probability of failure to be as large as 0.0034, concluding that the tanks are unsafe.

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References

Figures

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Fig. 1

Side view (left) and axial view (right) of the tank

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Fig. 2

An example of probability box (p-box) for a parameter (x) that is a mixture of both aleatory (random) and epistemic (lack of knowledge) uncertainty

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Fig. 4

Effect of perturbing various input variables on maximum stress: (a) tank radius, (b) tank surface thickness, and (c) Young's modulus. Note that input distribution for radius is truncated at R=30 in.

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Fig. 5

Effect of perturbing various input variables on maximum normal displacement: (a) tank radius, (b) tank surface thickness, and (c) Young's modulus. Note that input distribution for radius is truncated at R=30 in.

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Fig. 6

Uncertainty propagation at nominal conditions. M=50 epistemic samples and N=200 aleatory samples were used (for a total of 10,000 simulations) on medium grid (m=2): (a) all CDFs and (b) p-box.

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Fig. 7

Locations on the tank surface where the displacement data are measured during field tests. Twenty locations are marked with circles. Filled circles are locations where we convert the displacement data to stress data. Location#16 is where all the experimental data are pooled for MAVM calculation.

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Fig. 8

The u-pooling process explained for pooling data to one location at H=51 in. liquid height: (a)–(e) probing for probability levels (u values) based upon the surrogate stress data and (f) collecting stress data for the experimental CDF

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Fig. 9

MAVM calculation using simulation p-box and experimental (u-pooled) discrete CDF at H=51 in. operating condition

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Fig. 10

Total prediction uncertainty

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Fig. 11

Prediction p-box for maximum stress (solid curves) and p-box for yield stress (dotted curves): (left) CDFs for maximum stress and CCDFs for yield stress, and (right) enlarged view. The relevant point of intersection is shown with a solid circle which results in maximum probability of failure as: 1−0.9966=0.0034.

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