Research Papers: Design Automation

Propagation of Modeling Uncertainty by Polynomial Chaos Expansion in Multidisciplinary Analysis

[+] Author and Article Information
S. Dubreuil

Onera—The French Aerospace Lab,
Toulouse F-31055, France
e-mail: sylvain.dubreuil@onera.fr

N. Bartoli, T. Lefebvre

Onera—The French Aerospace Lab,
Toulouse F-31055, France

C. Gogu

Université de Toulouse,
UPS, INSA, Mines Albi, ISAE,
Institut Clément Ader (ICA),
3 rue Caroline Aigle,
Toulouse F-31400, France

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 7, 2016; final manuscript received June 20, 2016; published online September 12, 2016. Assoc. Editor: Samy Missoum.

J. Mech. Des 138(11), 111411 (Sep 12, 2016) (11 pages) Paper No: MD-16-1194; doi: 10.1115/1.4034110 History: Received March 07, 2016; Revised June 20, 2016

Multidisciplinary analysis (MDA) is nowadays a powerful tool for analysis and optimization of complex systems. The present study is interested in the case where MDA involves feedback loops between disciplines (i.e., the output of a discipline is the input of another and vice versa). When the models for each discipline involve non-negligible modeling uncertainties, it is important to be able to efficiently propagate these uncertainties to the outputs of the MDA. The present study introduces a polynomial chaos expansion (PCE)-based approach to propagate modeling uncertainties in MDA. It is assumed that the response of each disciplinary solver is affected by an uncertainty modeled by a random field over the design and coupling variables space. A semi-intrusive PCE formulation of the problem is proposed to solve the corresponding nonlinear stochastic system. Application of the proposed method emphasizes an important particular case in which each disciplinary solver is replaced by a surrogate model (e.g., kriging). Three application problems are treated, which show that the proposed approach can approximate arbitrary (non-Gaussian) distributions very well at significantly reduced computational cost.

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Chittick, I. R. , and Martins, J. R. R. A. , 2008, “ An Asymmetric Suboptimization Approach to Aerostructural Optimization,” Optim. Eng., 10(1), pp. 133–152. [CrossRef]
Martins, J. R. R. A. , and Lambe, A. B. , 2013, “ Multidisciplinary Design Optimization: A Survey of Architectures,” AIAA J., 51(9), pp. 2049–2075. [CrossRef]
Liang, C. , Mahadevan, S. , and Sankararaman, S. , 2015, “ Stochastic Multidisciplinary Analysis Under Epistemic Uncertainty,” ASME J. Mech. Des., 137(2), p. 021404. [CrossRef]
Zang, T. , Hemsch, M. , Hilburger, M. , Kenny, S. , Luckring, J. , Maghami, P. , Padula, S. , and Stroud, W. J. , 2002, “ Needs and Opportunities for Uncertainty Based Multidisciplinary Design Methods for Aerospace Vehicles,” NASA Langley Research Center, Technical Report No. TM-2002-211462. http://ntrs.nasa.gov/search.jsp?R=20020063596
Jones, D. R. , Schonlau, M. , and Welch, W. J. , 1998, “ Efficient Global Optimization of Expensive Black-Box Functions,” J. Global Optim., 13(4), pp. 455–492. [CrossRef]
Brevault, L. , Balesdent, M. , Bérend, N. , and Le Riche, R. , 2015, “ Decoupled MDO Formulation for Interdisciplinary Coupling Satisfaction Under Uncertainty,” AIAA J., 54(1), pp. 186–205. [CrossRef]
Sankararaman, S. , and Mahadevan, S. , 2012, “ Likelihood-Based Approach to Multidisciplinary Analysis Under Uncertainty,” ASME J. Mech. Des., 134(3), p. 031008. [CrossRef]
Leotardi, C. , Diez, M. , Serani, A. , Iemma, U. , and Campana, E. F. , 2014, “ A Framework for Efficient Simulation-Based Multidisciplinary Robust Design Optimization With Application to a Keel Fin of a Racing Sailboat,” International Conference on Engineering and Applied Sciences Optimization (OPTI2014), M. Papadrakakis , M. G. Karlaftis, and N. D. Lagaros , eds., Kos Island, Greece, June 4–6, pp. 1177–1193. http://s3.amazonaws.com/academia.edu.documents/39293386/0046353965db274864111411.pdf?AWSAccessKeyId=AKIAJ56TQJRTWSMTNPEA&Expires=1468946783&Signature=18cJ7qYifM5aTSgwwAdbNSWUdPc%3D&response-content-disposition=inline%3B%20filename%3DA_framework_for_efficient_simulation-bas.pdf
Koch, P. , Wujek, B. , Golovidov, O. , and Simpson, T. , 2002, “ Facilitating Probabilistic Multidisciplinary Design Optimization Using Kriging Approximation Models,” AIAA Paper No. 2002-5415.
Oakley, D. R. , Sues, R. H. , and Rhodes, G. S. , 1998, “ Performance Optimization of Multidisciplinary Mechanical Systems Subject to Uncertainties,” Probab. Eng. Mech., 13(1), pp. 15–26. [CrossRef]
Forrester, A. , Sobester, A. , and Keane, A. , 2008, Engineering Design Via Surrogate Modelling: A Practical Guide, Wiley, Hoboken, NJ.
Jaulin, L. , 2001, Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics, Vol. 1, Springer-Verlag, London.
Zadeh, L. A. , 1965, “ Fuzzy Sets,” Inf. Control, 8(3), pp. 338–353. [CrossRef]
Shafer, G. , 1976, A Mathematical Theory of Evidence, Vol. 1, Princeton University Press, Princeton, NJ.
Dubois, D. , and Prade, H. , 2012, Possibility Theory: An Approach to Computerized Processing of Uncertainty, Springer, New York.
Ferson, S. , Joslyn, C. A. , Helton, J. C. , Oberkampf, W. L. , and Sentz, K. , 2004, “ Summary From the Epistemic Uncertainty Workshop: Consensus Amid Diversity,” Reliab. Eng. Syst. Saf., 85(1), pp. 355–369. [CrossRef]
Roy, C. J. , and Oberkampf, W. L. , 2011, “ A Comprehensive Framework for Verification, Validation, and Uncertainty Quantification in Scientific Computing,” Comput. Methods Appl. Mech. Eng., 200(25–28), pp. 2131–2144. [CrossRef]
Sudret, B. , 2008, “ Global Sensitivity Analysis Using Polynomial Chaos Expansions,” Reliab. Eng. Syst. Saf., 93(7), pp. 964–979. [CrossRef]
Ghanem, R. G. , and Spanos, P. D. , 1991, Stochastic Finite Elements: A Spectral Approach, Springer, New York.
Soize, C. , and Ghanem, R. , 2004, “ Physical Systems With Random Uncertainties: Chaos Representations With Arbitrary Probability Measure,” SIAM J. Sci. Comput., 26(2), pp. 395–410. [CrossRef]
Xiu, D. , and Karniadakis, G. E. , 2002, “ The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations,” SIAM J. Sci. Comput., 24(2), pp. 619–644. [CrossRef]
Poëtte, G. , and Lucor, D. , 2012, “ Non Intrusive Iterative Stochastic Spectral Representation With Application to Compressible Gas Dynamics,” J. Comput. Phys., 231(9), pp. 3587–3609. [CrossRef]
Wan, X. , and Karniadakis, G. E. , 2005, “ An Adaptive Multi-Element Generalized Polynomial Chaos Method for Stochastic Differential Equations,” J. Comput. Phys., 209(2), pp. 617–642. [CrossRef]
Lebrun, R. , and Dutfoy, A. , 2009, “ A Generalization of the Nataf Transformation to Distributions With Elliptical Copula,” Probab. Eng. Mech., 24(2), pp. 172–178. [CrossRef]
Lebrun, R. , and Dutfoy, A. , 2009, “ Do Rosenblatt and Nataf Isoprobabilistic Transformations Really Differ?,” Probab. Eng. Mech., 24(4), pp. 577–584. [CrossRef]
Krige, D. G. , 1951, “ A Statistical Approach to Some Mine Evaluations and Allied Problems at the Witwatersrand,” Master's thesis, University of Witwatersrand, Johannesburg, South Africa.
Rasmussen, C. E. , and Williams, C. K. I. , 2006, Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning), MIT Press, Cambridge, MA.
Pedregosa, F. , Varoquaux, G. , Gramfort, A. , Michel, V. , Thirion, B. , Grisel, O. , Blondel, M. , Prettenhofer, P. , Weiss, R. , Dubourg, V. , Vanderplas, J. , Passos, A. , Cournapeau, D. , Brucher, M. , Perrot, M. , and Duchesnay, E. , 2011, “ Scikit-Learn: Machine Learning in Python,” J. Mach. Learn. Res., 12, pp. 2825–2830. http://www.jmlr.org/papers/v12/pedregosa11a.html
Sellar, R. S. , Batill, S. M. , and Renaud, J. E. , 1996, “ Response Surface Based, Concurrent Subspace Optimization for Multidisciplinary System Design,” AIAA Paper No. 96-0714.
Shizgal, B. D. , and Jung, J.-H. , 2003, “ Towards the Resolution of the Gibbs Phenomena,” J. Comput. Appl. Math., 161(1), pp. 41–65. [CrossRef]
Jaeger, L. , Gogu, C. , Segonds, S. , and Bes, C. , 2013, “ Aircraft Multidisciplinary Design Optimization Under Both Model and Design Variables Uncertainty,” J. Aircr., 50(2), pp. 528–538. [CrossRef]
Ruijgrok, G. , 1990, Elements of Airplane Performance, Delft University Press, Delft, The Netherlands.
Lefebvre, T. , Schmollgruber, P. , Blondeau, C. , and Carrier, G. , 2012, “ Aircraft Conceptual Design in a Multi-Level, Multi-Fidelity, Multi-Disciplinary Optimization Process,” 28th ICAS Congress, Brisbane, Australia, ICAS2012-1.6.4. https://www.researchgate.net/profile/Gerald_Carrier/publication/256842085_Aircraft_Conceptual_Design_In_A_Multi-Level_Multi-Fidelity_Multi-Disciplinary_Optimization_Process/links/0c960523ee68f917c1111411.pdf
Sobieszczanski, J. , Agte, J. , and Sandusky, R. , 1998, “ Bi-Level Integrated System Synthesis (BLISS),” Technical Report No. NASA/TM-1998-20871.
Sobol, I. M. , 2001, “ Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates,” Math. Comput. Simul., 55(1), pp. 271–280. [CrossRef]


Grahic Jump Location
Fig. 1

(a) Illustration of the sellar test case with uniform model uncertainties. Square markers are used for the deterministic case; cross markers are used for a random realizations; solid lines are used for the boundaries of the uniform random fields and (b) PDF of (Y1, Y2), MCS results (100,000 simulations).

Grahic Jump Location
Fig. 2

PDF of Ŷ1,Ŷ2 obtained by PCE for degree d = 2 to d = 5. The same scale as Fig.1(b) is used.

Grahic Jump Location
Fig. 3

(a) Illustration of the sellar test case with kriging metamodels, (b) PDF of (Y1, Y2), MCS results (100,000 simulations), (c) PDF of Ŷ1,Ŷ2 obtained by PCE kriging case, and (d) simplified kriging case

Grahic Jump Location
Fig. 4

(a) Kriging case with a supplementary DOE point for GPY1, PDF of (Y1, Y2), MCS results (100,000 simulations), (b) PDF of Ŷ1,Ŷ2 obtained by PCE, kriging case, and (c) simplified kriging case

Grahic Jump Location
Fig. 5

Illustration of the regression approach based on databases. Linear regressions and 99% CI constructed by the method proposed in Ref. [31].

Grahic Jump Location
Fig. 6

(a) PDF of (MTOW, ZFW) obtained by MCS (100,000 simulations) and (b) PDF of MTOŴ,ZFŴ obtained by the proposed approach

Grahic Jump Location
Fig. 7

Conception diagram of the SSBJ test case

Grahic Jump Location
Fig. 8

Comparison between the MCS and the proposed method results. The lower triangular part presents the MCS PDF whereas the upper triangular part presents the PDF obtained with the proposed method. On the diagonal, the histograms of the marginal obtained by both methods are superposed.



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