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Research Papers

Propagation of Input Uncertainty in Presence of Model-Form Uncertainty: A Multifidelity Approach for Computational Fluid Dynamics Applications

[+] Author and Article Information
Jian-Xun Wang, Christopher J. Roy

Department of Aerospace and
Ocean Engineering,
Virginia Tech,
Blacksburg, VA 24060

Heng Xiao

Department of Aerospace and
Ocean Engineering,
Virginia Tech,
Blacksburg, VA 24060
e-mail: hengxiao@vt.edu (Heng Xiao)

1Corresponding author.

Manuscript received August 10, 2016; final manuscript received December 27, 2016; published online September 7, 2017. Assoc. Editor: Yan Wang.

ASME J. Risk Uncertainty Part B 4(1), 011002 (Sep 07, 2017) (8 pages) Paper No: RISK-16-1107; doi: 10.1115/1.4037452 History: Received August 10, 2016; Revised December 27, 2016

Proper quantification and propagation of uncertainties in computational simulations are of critical importance. This issue is especially challenging for computational fluid dynamics (CFD) applications. A particular obstacle for uncertainty quantifications in CFD problems is the large model discrepancies associated with the CFD models used for uncertainty propagation. Neglecting or improperly representing the model discrepancies leads to inaccurate and distorted uncertainty distribution for the quantities of interest (QoI). High-fidelity models, being accurate yet expensive, can accommodate only a small ensemble of simulations and thus lead to large interpolation errors and/or sampling errors; low-fidelity models can propagate a large ensemble, but can introduce large modeling errors. In this work, we propose a multimodel strategy to account for the influences of model discrepancies in uncertainty propagation and to reduce their impact on the predictions. Specifically, we take advantage of CFD models of multiple fidelities to estimate the model discrepancies associated with the lower-fidelity model in the parameter space. A Gaussian process (GP) is adopted to construct the model discrepancy function, and a Bayesian approach is used to infer the discrepancies and corresponding uncertainties in the regions of the parameter space where the high-fidelity simulations are not performed. Several examples of relevance to CFD applications are performed to demonstrate the merits of the proposed strategy. Simulation results suggest that, by combining low- and high-fidelity models, the proposed approach produces better results than what either model can achieve individually.

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Figures

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Fig. 1

The dependence of the lift coefficient CL of a NACA-0012 airfoil section on AoA and Reynolds number Re. The three different flow regimes are demarcated with dash-dotted lines.

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Fig. 2

Plots of the synthetic response surface f(x1, x2) in Eq. (6). The three-dimensional elevated surface with contour is shown in panel (a), and the cross sections at several values x1 are shown in panel (b).

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Fig. 3

(a) The prior distribution of the model discrepancy function δ is represented as a GP with a zero-mean function and a stationary kernel. (b) The posterior GP of δ given observation data at five input locations. In both the prior and the posterior GPs, the mean functions, 95% confidence intervals, and several random realizations from the GPs are indicated.

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Fig. 4

Corrected propagated uncertainty (bold solid line) based on the multimodel method compared with original results (light solid line) from the low-fidelity model

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Fig. 5

Response surfaces of (a) the mapping CL(α, M) constructed by RANS solutions and (b) model discrepancy of low-fidelity model (i.e., thin airfoil theory with the Prandtl–Glauert compressibility correction)

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Fig. 6

Corrected propagated uncertainty (bold solid line) based on the multimodel method compared with original results (light solid line) from the low-fidelity model. Ten high fidelity simulations are used.

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Fig. 7

Individual CDFs of QoI corrected by using 30 realizations of the GP (before aggregation) with (a) ten high-fidelity simulations and (b) 40 high-fidelity simulations

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Fig. 8

Propagated uncertainty distribution obtained by using the proposed multimodel method compared with those obtained by using the single-model approaches, i.e., with either high-fidelity model or the low-fidelity model alone

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