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Research Papers

Simple Effective Conservative Treatment of Uncertainty From Sparse Samples of Random Variables and Functions

[+] Author and Article Information
Vicente J. Romero

Mem. ASME
Sandia National Laboratories,
P.O. Box 5800,
Albuquerque, NM 87185-0828
e-mail: vjromer@sandia.gov

Benjamin B. Schroeder

Sandia National Laboratories,
P.O. Box 5800,
Albuquerque, NM 87185-0828
e-mail: bbschro@sandia.gov

James F. Dempsey

Sandia National Laboratories,
P.O. Box 5800,
Albuquerque, NM 87185-0840
e-mail: jfdemps@sandia.gov

Nicole L. Breivik

Sandia National Laboratories,
P.O. Box 5800,
Albuquerque, NM 87185-0386
e-mail: nlbreiv@sandia.gov

George E. Orient

Mem. ASME
Sandia National Laboratories,
P.O. Box 5800,
Albuquerque, NM 87185-0386
e-mail: georien@sandia.gov

Bonnie R. Antoun

Mem. ASME
Sandia National Laboratories,
P.O. Box 969,
Livermore, CA 94551-9042
e-mail: brantou@sandia.gov

John R. Lewis

Sandia National Laboratories,
P.O. Box 5800,
Albuquerque, NM 87185-0830
e-mail: jrlewi@sandia.gov

Justin G. Winokur

Sandia National Laboratories,
P.O. Box 5800,
Albuquerque, NM 87185-0828
e-mail: jgwinok@sandia.gov

1Corresponding author.

Manuscript received June 16, 2017; final manuscript received February 24, 2018; published online April 30, 2018. Assoc. Editor: Siu-Kui Au.This paper has been authored by Sandia Corporation under Contract No. DE-NA0003525 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the paper for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this paper, or allow others to do so, for United States Government purposes.

ASME J. Risk Uncertainty Part B 4(4), 041006 (Apr 30, 2018) (17 pages) Paper No: RISK-17-1070; doi: 10.1115/1.4039558 History: Received June 16, 2017; Revised February 24, 2018

This paper examines the variability of predicted responses when multiple stress–strain curves (reflecting variability from replicate material tests) are propagated through a finite element model of a ductile steel can being slowly crushed. Over 140 response quantities of interest (QOIs) (including displacements, stresses, strains, and calculated measures of material damage) are tracked in the simulations. Each response quantity's behavior varies according to the particular stress–strain curves used for the materials in the model. We desire to estimate or bound response variation when only a few stress–strain curve samples are available from material testing. Propagation of just a few samples will usually result in significantly underestimated response uncertainty relative to propagation of a much larger population that adequately samples the presiding random-function source. A simple classical statistical method, tolerance intervals (TIs), is tested for effectively treating sparse stress–strain curve data. The method is found to perform well on the highly nonlinear input-to-output response mappings and non-normal response distributions in the can crush problem. The results and discussion in this paper support a proposition that the method will apply similarly well for other sparsely sampled random variable or function data, whether from experiments or models. The simple TI method is also demonstrated to be very economical.

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References

Romero, V. , Schroeder, B. , Dempsey, J. F. , Lewis, J. , Breivik, N. , Orient, G. , Antoun, B. , Winokur, J. , Glickman, M. , and Red-Horse, J. , 2017, “Evaluation of a Simple UQ Approach to Compensate for Sparse Stress-Strain Curve Data in Solid Mechanics Applications,” AIAA Paper No. 2017-0818.
Romero, V. , Dempsey, J. F. , Wellman, G. , and Antoun, B. , 2012, “A Method for Projecting Uncertainty From Sparse Samples of Discrete Random Functions—Example of Multiple Stress-Strain Curves,” AIAA Paper No. 2012-1365.
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Figures

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Fig. 1

Half-symmetry finite element model of can crush configuration showing (a) geometry and components, (b) finite element mesh, and (c) example simulation results of can deformation

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Fig. 2

Engineering stress versus engineering strain tensile-test results with several repeats for 304 L stainless steel at room temperature through 800C, strain rate = 0.001/s

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Fig. 3

Two substantially different sets of synthetic MLEP s-s curves (1000 curves per set) used to characterize performance of TI sparse-data UQ method

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Fig. 4

Multiplier f on calculated standard deviation used to form 0.95/0.90 TI ranges versus number of random samples (ignore confidence interval curve in the plot)

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Fig. 5

Computational experiments and assessment strategy for TI method performance characterization

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Fig. 6

1000 random realizations of transient response for each of 18 output quantities (using the 1000 input stress–strain curves in 200C set)

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Fig. 7

Tolerance interval method success rates over 10,000 trials, and some response histograms, for the 70 QOIs and N = 4 stress–strain curves (200C) drawn at random per trial

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Fig. 8

Variation of TI method success rates versus the number N of 200C stress–strain curves drawn at random per trial (e.g., N = 4 in Fig. 5), and for conservative and nonconservative 2.5–97.5 percentile ranges per item 1 in this section, and for noncentral 95% ranges also counted as successes per item 2. Dots are averages over the 70 QOI cases; uncertainty bars are ±1 standard deviation of the 70 individual success rates.

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Fig. 9

Some response QOIs with the 200C and 400C sets of 1000 synthetic input stress–strain curves

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Fig. 10

Tolerance interval method success rates over 10,000 trials for the 70 QOIs and N = 4 stress–strain curves (400C) drawn at random per trial

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Fig. 11

Variation of TI success rates versus the number of 400C stress–strain curves drawn at random per trial, for conservative 2.5–97.5 percentile ranges. Dots are averages over the 70 QOI cases; uncertainty bars are ±1 standard deviation of the 70 individual success rates.

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Fig. 12

Definition of mismatch errors between TI and reference percentile range

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Fig. 13

Average normalized absolute error magnitudes from 700,000 trial TIs in the labeled error categories (10,000 TI trials for each of 70 QOIs)

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Fig. 14

Normalized performance metric results versus number of samples for said error preferences/weights and penalized and nonpenalized undershoot errors. Mean results for each no. of samples are averaged over 680,000 TI trials for the 200C s-s curve data.

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