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Research Papers

Autonomous Uncertainty Quantification for Discontinuous Models Using Multivariate Padé Approximations

[+] Author and Article Information
Richard Ahlfeld

Uncertainty Quantification Lab,
Department of Aeronautics Imperial
College London,
London SW7 2AZ, UK
e-mail: r.ahlfeld14@imperial.ac.uk

Francesco Montomoli

Uncertainty Quantification Lab,
Department of Aeronautics Imperial
College London,
London SW7 2AZ, UK

Mauro Carnevale

Osney Thermo-Fluids Laboratory,
Department of Engineering Science,
University of Oxford,
Oxford OX2 0ES, UK

Simone Salvadori

Department of Industrial Engineering,
University of Florence,
Florence 50121, Italy

1Corresponding author.

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received May 12, 2017; final manuscript received October 14, 2017; published online January 23, 2018. Editor: Kenneth Hall.

J. Turbomach 140(4), 041004 (Jan 23, 2018) (10 pages) Paper No: TURBO-17-1062; doi: 10.1115/1.4038826 History: Received May 12, 2017; Revised October 14, 2017

Problems in turbomachinery computational fluid dynamics (CFD) are often characterized by nonlinear and discontinuous responses. Ensuring the reliability of uncertainty quantification (UQ) codes in such conditions, in an autonomous way, is challenging. In this work, we suggest a new approach that combines three state-of-the-art methods: multivariate Padé approximations, optimal quadrature subsampling (OQS), and statistical learning. Its main component is the generalized least-squares multivariate Padé–Legendre (PL) approximation. PL approximations are globally fitted rational functions that can accurately describe discontinuous nonlinear behavior. They need fewer model evaluations than local or adaptive methods and do not cause the Gibbs phenomenon like continuous polynomial chaos methods. A series of modifications of the Padé algorithm allows us to apply it to arbitrary input points instead of optimal quadrature locations. This property is particularly useful for industrial applications, where a database of CFD runs is already available, but not in optimal parameter locations. One drawback of the PL approximation is that it is nontrivial to ensure reliability. To improve stability, we suggest to couple it with OQS. Our reasoning is that least-squares errors, caused by an ill-conditioned design matrix, are the main source of error. Finally, we use statistical learning methods to check smoothness and convergence. The resulting method is shown to efficiently and correctly fit thousands of partly discontinuous response surfaces for an industrial film cooling and shock interaction problem using only nine CFD simulations.

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Figures

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Fig. 1

Response surfaces of hyperbolic tangent using a 16 points full tensor grid (a) and an 18 points optimally subsampled grid (b). The analytical function is shown in (c): (a) full tensor grid, (b) optimal quadrature subsampled, and (c) true function

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Fig. 2

Demonstration of automatic parameter selection

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Fig. 3

(a) Control volume and (b) flow features

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Fig. 4

Grid details in (a) shock interaction region and (b) coolant region

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Fig. 5

(a) Comparison between CFD (upper) and experimental data (lower) for the deterministic reference configuration case. (b) ηaw distribution over the centerline

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Fig. 6

(a) Flow streamlines and interaction with the shock and (b) details of the tornado vortex after the shock

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Fig. 7

Temperature maps and streamlines between hole and shock

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Fig. 8

The graphs illustrates how the shock moves through the stochastic response surface of the thermal efficiency

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Fig. 9

Example of choice of the automatic response surface error and smoothness control

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Fig. 10

Three examples of automatically fitted Padé response surfaces before (1), at (2) and after (3) the shock location. The bottom row shows three PDFs to the x-locations before, at and after the shock.

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Fig. 11

(a) Mean and standard deviation of the film cooling effectiveness and (b) sensitivity analysis of input parameters using sobol indices

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