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Research Papers: Design Automation

Stochastic Multidisciplinary Analysis Under Epistemic Uncertainty

[+] Author and Article Information
Chen Liang

Department of Civil and Environmental
Engineering,
Vanderbilt University,
Nashville, TN 37235

Sankaran Mahadevan

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

Shankar Sankararaman

SGT, Inc.,
Moffett Field, CA 94035

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 8, 2014; final manuscript received November 12, 2014; published online December 11, 2014. Assoc. Editor: Christopher Mattson.

J. Mech. Des 137(2), 021404 (Feb 01, 2015) (12 pages) Paper No: MD-14-1155; doi: 10.1115/1.4029221 History: Received March 08, 2014; Revised November 12, 2014; Online December 11, 2014

This paper presents a probabilistic framework to include the effects of both aleatory and epistemic uncertainty sources in coupled multidisciplinary analysis (MDA). A likelihood-based decoupling approach has been previously developed for probabilistic analysis of multidisciplinary systems, but only with aleatory uncertainty in the inputs. This paper extends this approach to incorporate the effects of epistemic uncertainty arising from data uncertainty and model errors. Data uncertainty regarding input variables (due to sparse and interval data) is included through parametric or nonparametric distributions using the principle of likelihood. Model error is included in MDA through an auxiliary variable approach based on the probability integral transform. In the presence of natural variability, data uncertainty, and model uncertainty, the proposed methodology is employed to estimate the probability density functions (PDFs) of coupling variables as well as the subsystem and system level outputs that satisfy interdisciplinary compatibility. Global sensitivity analysis (GSA), which has previously considered only aleatory inputs and feedforward or monolithic problems, is extended in this paper to quantify the contribution of model uncertainty in feedback-coupled MDA by exploiting the auxiliary variable approach. The proposed methodology is demonstrated using a mathematical MDA problem and an electronic packaging application example featuring coupled thermal and electrical subsystem analyses. The results indicate that the proposed methodology can effectively quantify the uncertainty in MDA while maintaining computational efficiency.

Copyright © 2015 by ASME
Topics: Errors , Uncertainty
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References

Cramer, E. J., Dennis, J. E., Frank, P. D., and Shubin, G. R., 1993, “Problem Formulation for Multidisciplinary Optimization Problem Formulation for Multidisciplinary Optimization,” SIAM J. Optim., 4(4), pp. 754–776. [CrossRef]
Ilan, K., Steve, A., Robert, B., Peter, G., and Ian, S., 1994, “Multidisciplinary Optimization Methods for Aircraft Preliminary Design,” AIAA Paper No. 4325.
Belytschko, T., 1980, “Fluid–Structure Interaction,” Comput. Struct., 12(4), pp. 459–469. [CrossRef]
Thornton, E. A., 1992, “Thermal Structures: Four Decades of Progress,” J. Aircr., 29(3), pp. 485–498. [CrossRef]
Wieting, A. R., Dechaumphai, P., Bey, K. S., Thornton, E. A., and Morgan, K., 1991, “Application of Integrated Fluid-Thermal-Structural Analysis Methods,” Thin Walled Struct., 11(1–2), pp. 1–23. [CrossRef]
Sobieszczanski-Sobieski, J., and Haftka, R., 1997, “Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments,” Struct. Optim., 14(1) pp. 1–23. [CrossRef]
Felippa, C. A., Park, K. C., and Farhat, C., 2001, “Partitioned Analysis of Coupled Mechanical Systems,” Comput. Methods Appl. Mech. Eng., 190(24–25), pp. 3247–3270. [CrossRef]
Michler, C., and Hulshoff, S., 2004, “A Monolithic Approach to Fluid–Structure Interaction,” Comput. Fluids, 33(5–6), pp. 839–848. [CrossRef]
Haldar, A., and Mahadevan, S., 2000, Probability, Reliability, and Statistical Methods in Engineering Design, Wiley, New York.
Sankararaman, S., and Mahadevan, S., 2011, “Likelihood-Based Representation of Epistemic Uncertainty due to Sparse Point Data and/or Interval Data,” Reliab. Eng. Syst. Saf., 96(7), pp. 814–824. [CrossRef]
Agarwal, H., Renaud, J. E., Preston, E. L., and Padmanabhan, D., 2004, “Uncertainty Quantification Using Evidence Theory in Multidisciplinary Design Optimization,” Reliab. Eng. Syst. Saf., 85(1–3), pp. 281–294. [CrossRef]
Helton, J. C., Johnson, J. D., Oberkampf, W. L., and Storlie, C. B., 2007, “A Sampling-Based Computational Strategy for the Representation of Epistemic Uncertainty in Model Predictions With Evidence Theory,” Comput. Methods Appl. Mech. Eng., 196(37–40), pp. 3980–3998. [CrossRef]
Du, L., Choi, K. K., Youn, B. D., and Gorsich, D., 2006, “Possibility-Based Design Optimization Method for Design Problems With Both Statistical and Fuzzy Input Data,” ASME J. Mech. Des., 128(4), pp. 928–935. [CrossRef]
Zhang, X., and Huang, H.-Z., 2009, “Sequential Optimization and Reliability Assessment for Multidisciplinary Design Optimization Under Aleatory and Epistemic Uncertainties,” Struct. Multidiscip. Optim., 40(1–6), pp. 165–175. [CrossRef]
Zadeh, L. A., 2002, “Toward a Perception-Based Theory of Probabilistic Reasoning With Imprecise Probabilities,” J. Sta. Plan. Inference, 105(1), 233–264.
Ferson, S., Kreinovich, V., Ginzburg, L., Myers, D. S., and Sentz, K., 2003, “Constructing Probability Boxes and Dempster-Shafer Structures,” Technical Report No. SAND2002-4015.
Matsumura, T., and Haftka, R. T., 2013, “Reliability Based Design Optimization Modeling Future Redesign With Different Epistemic Uncertainty Treatments,” ASME J. Mech. Des., 135(9), p. 091006. [CrossRef]
Zaman, K., Rangavajhala, S., McDonald, M. P., and Mahadevan, S., 2011, “A Probabilistic Approach for Representation of Interval Uncertainty,” Reliab. Eng. Syst. Saf., 96(1), pp. 117–130. [CrossRef]
Gu, X., Renaud, J. E. E., Batill, S. M. M., Brach, R. M. M., and Budhiraja, A. S., 2000, “Worst Case Propagated Uncertainty of Multidisciplinary Systems in Robust Optimization,” Struct. Optim., 20(3), pp. 190–213. [CrossRef]
Kokkolaras, M., Mourelatos, Z. P., and Papalambros, P. Y., 2006, “Design Optimization of Hierarchically Decomposed Multilevel Systems Under Uncertainty,” ASME J. Mech. Des., 128(2), pp. 503–508. [CrossRef]
Liu, H., Chen, W., Kokkolaras, M., Papalambros, P. Y., and Kim, H. M., 2006, “Probabilistic Analytical Target Cascading: A Moment Matching Formulation for Multilevel Optimization Under Uncertainty,” ASME J. Mech. Des., 128(4), pp. 991–1000. [CrossRef]
Du, X., and Chen, W., 2005, “Collaborative Reliability Analysis Under the Framework of Multidisciplinary Systems Design,” Optim. Eng., 6(1), pp. 63–84. [CrossRef]
Mahadevan, S., and Smith, N., 2006, “Efficient First-Order Reliability Analysis of Multidisciplinary Systems,” Int. J. Reliab. Saf., 1(1), pp. 137–154. [CrossRef]
Sankararaman, S., and Mahadevan, S., 2012, “Likelihood-Based Approach to Multidisciplinary Analysis Under Uncertainty,” ASME J. Mech. Des., 134(3), p. 031008. [CrossRef]
Rebba, R., Mahadevan, S., and Huang, S., 2006, “Validation and Error Estimation of Computational Models,” Reliab. Eng. Syst. Saf., 91(10–11), pp. 1390–1397. [CrossRef]
Mahadevan, S., and Liang, B., 2011 “Error and Uncertainty Quantification and Sensitivity Analysis in Mechanics Computational Models,” Int. J. Uncertainty Quantif., 1(2), pp. 147–161. [CrossRef]
Kennedy, M. C., and O’Hagan, A., 2001, “Bayesian Calibration of Computer Models,” J. R. Stat. Soc., 63(3), pp. 425–464. [CrossRef]
Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., and Tarantola, S., 2008, Global Sensitivity Analysis: The Primer, Wiley-Interscience, Hoboken, NJ. [CrossRef]
McKeeman, W. M., 1962, “Algorithm 145: Adaptive Numerical Integration by Simpson’s Rule,” Commun. ACM, 5(12), p. 604. [CrossRef]
Mahadevan, S., and Rebba, R., 2006, “Inclusion of Model Errors in Reliability-Based Optimization,” ASME J. Mech. Des., 128(4), pp. 936–944. [CrossRef]
Chen, W., Baghdasaryan, L., Buranathiti, T., and Cao, J., 2004, “Model Validation via Uncertainty Propagation and Data Transformations,” AIAA J., 42(7), pp. 1403–1415. [CrossRef]
Sankararaman, S., Ling, Y., Shantz, C., and Mahadevan, S., 2011, “Inference of Equivalent Initial Flaw Size Under Multiple Sources of Uncertainty,” Int. J. Fatigue, 33(2), pp. 75–89. [CrossRef]
Huang, S., Mahadevan, S., and Rebba, R., 2007, “Collocation-Based Stochastic Finite Element Analysis for Random Field Problems,” Probab. Eng. Mech., 22(2), pp. 194–205. [CrossRef]
Clarke, S. M., Griebsch, J. H., and Simpson, T. W., 2005, “Analysis of Support Vector Regression for Approximation of Complex Engineering Analyses,” ASME J. Mech. Des., 127(6), pp. 1077–1087. [CrossRef]
Wang, G. G., and Shan, S., 2007, “Review of Metamodeling Techniques in Support of Engineering Design Optimization,” ASME J. Mech. Des., 129(4), pp. 370–380. [CrossRef]
Hombal, V., and Mahadevan, S., 2011, “Bias Minimization in Gaussian Process Surrogate Modeling for Uncertainty Quantification,” Int. J. Uncertainty Quantif., 1(4), pp. 321–349. [CrossRef]
Zhu, P., Zhang, S., and Chen, W., 2014, “Multi-Point Objective-Oriented Sequential Sampling Strategy for Constrained Robust Design,” Eng. Optim., pp. 1–21. [CrossRef]
Pearson, E., 1938, “The Probability Integral Transformation for Testing Goodness of Fit and Combining Independent Tests of Significance,” Biometrika, 30(1–2), pp. 134–148. [CrossRef]
Sankararaman, S., and Mahadevan, S., 2012, “Separating the Contributions of Variability and Parameter Uncertainty in Probability Distributions,” Reliab. Eng. Syst. Saf., 112(4), pp. 187–199. [CrossRef]
Padula, S., Alexandrov, N., and Green, L., 1996, “MDO Test Suite at NASA Langley Research Center,” AIAA Paper, (96-4028).
Rangavajhala, S., Sura, V., Hombal, V., and Mahadevan, S., 2011 “Discretization Error Estimation in Multidisciplinary Simulations,” AIAA J., 49(12), pp. 2673–2683. [CrossRef]
Liu, Y., Chen, W., Arendt, P., and Huang, H.-Z., 2011, “Toward a Better Understanding of Model Validation Metrics,” ASME J. Mech. Des., 133(7), p. 071005. [CrossRef]
Ling, Y., and Mahadevan, S., 2013, “Quantitative Model Validation Techniques: New Insights,” Reliab. Eng. Syst. Saf., 111, pp. 217–231. [CrossRef]
Rangavajhala, S., Liang, C., and Mahadevan, S., 2012, “Design Optimization Under Aleatory and Epistemic Uncertainties,” Proceedings of 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Indianapolis, IN, Sept.17–19. [CrossRef]

Figures

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Fig. 1

Multidisciplinary system

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Fig. 2

One iteration of coupled analysis

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Fig. 3

Multidisciplinary system: partially decoupled

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Fig. 4

Family of distributions

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Fig. 5

Optimization framework for maximum likelihood method

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Fig. 6

FORM with auxiliary variable

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Fig. 7

Output uncertainty due to model errors

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Fig. 10

PDFs of coupling variables U12 (left) and U21 (right)

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Fig. 9

Nonparametric PDF of x5

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Fig. 8

Functional relations of the mathematical MDA model

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Fig. 12

Mean and 95% bound of GP prediction, accounting for discretization error (thermal analysis)

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Fig. 11

Electronic packaging problem: feedback-coupled MDA. (a) Geometry of a regular heatsink. (b) Disciplinary analyses and coupling variables.

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Fig. 13

PDF of coupling variables: component heat y4. (left) and temperature y5 (right)

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Fig. 14

PDF of system output: power density

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Fig. 15

Comparison of results from LAMDA and SOFPI for (a) temperature, (b) component heat, and (c) power density

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