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research-article

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
r.ahlfeld14@imperial.ac.uk

Francesco Montomoli

Uncertainty Quantification Lab, Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
f.montomoli@imperial.ac.uk

Mauro Carnevale

Osney Thermo-Fluids Laboratory, Department of Engineering Science, University of Oxford, Oxford OX2 0ES, UK
m.carnevale@imperial.ac.uk

Simone Salvadori

Department of Industrial Engineering, University of Florence, Florence 50121, Italy
simone.salvadori@unifi.it

1Corresponding author.

ASME doi:10.1115/1.4038826 History: Received May 12, 2017; Revised October 14, 2017

Abstract

Problems in turbomachinery Computational Fluid Dynamics (CFD) are often characterised by non-linear and discontinuous responses. Ensuring the reliability of Uncertainty Quantification (UQ) codes in such conditions, in an autonomous way, is a challenging problem. In this work, we suggest a new approach that combines three state-of-the-art methods: multivariate Padé approximations, Optimal Quadrature Subsampling and Statistical Learning. Its main component is the generalised least squares multivariate Padé -Legendre (PL) approximation. PL approximations are globally fitted rational functions that can accurately describe discontinuous non-linear behaviour. 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 allow 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 non-trivial to ensure reliability. To improve stability we suggest to couple it with Optimal Quadrature Subsampling. 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 9 CFD simulations.

Copyright (c) 2017 by ASME
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