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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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Figures

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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).

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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

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Fig. 5

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

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Fig. 6

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

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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.

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Fig. 7

Conception diagram of the SSBJ test case

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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

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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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