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• Developing an efficient bi-fidelity method for uncertainty quantification of high dimensional CFD problems. • Implementing the method in uncertainty quantification of two real engineering problems. • Reducing the computational cost in both test cases in comparison to the combination of POD and full PCE methods. In the current paper, an efficient surrogate model based on combination of Proper Orthogonal Decomposition (POD) and compressed sensing is developed for affordable representation of high dimensional stochastic fields. In the developed method, instead of the full (or classical) Polynomial Chaos Expansion (PCE), the ℓ 1 -minimization approach is utilized to reduce the computational work-load of the low-fidelity calculations. To assess the model capability in the real engineering problems, two challenging high-dimensional CFD test cases namely; i) turbulent transonic flow around RAE2822 airfoil with 18 geometrical uncertainties and ii) turbulent transonic flow around NASA Rotor 37 with 3 operational and 21 geometrical uncertainties are considered. Results of Uncertainty Quantification (UQ) analysis in both test cases showed that the proposed multi-fidelity approach is able to reproduce the statistics of quantities of interest with much lower computational cost than the classical regression-based PCE method. It is shown that the combination of the POD with the compressed sensing in RAE2822 and Rotor 37 test cases gives respectively computational gains between 1.26–7.72 and 1.79–9.05 times greater than the combination of the POD with the full PCE. [ABSTRACT FROM AUTHOR]

The presence of uncertainties is inevitable in engineering design and analysis, where failure in understanding their effects might lead to the structural or functional failure of the systems. The role of global sensitivity analysis in this aspect is to quantify and rank the effects of input random variables and their combinations to the variance of the random output. In problems where the use of expensive computer simulations is required, metamodels are widely used to speed up the process of global sensitivity analysis. In this paper, a multi-fidelity framework for global sensitivity analysis using polynomial chaos expansion (PCE) is presented. The goal is to accelerate the computation of Sobol sensitivity indices when the deterministic simulation is expensive and simulations with multiple levels of fidelity are available. This is especially useful in cases where a partial differential equation solver computer code is utilized to solve engineering problems. The multi-fidelity PCE is constructed by combining the low-fidelity and correction PCE. Following this step, the Sobol indices are computed using this combined PCE. The PCE coefficients for both low-fidelity and correction PCE are computed with spectral projection technique and sparse grid integration. In order to demonstrate the capability of the proposed method for sensitivity analysis, several simulations are conducted. On the aerodynamic example, the multi-fidelity approach is able to obtain an accurate value of Sobol indices with 36.66% computational cost compared to the standard single-fidelity PCE for a nearly similar accuracy. [ABSTRACT FROM AUTHOR]

Surrogate models are indispensable tools in uncertainty quantification and global sensitivity analysis. Polynomial chaos expansion (PCE) is one of the most widely used surrogate models, thanks to its faster convergence rate compared to Monte Carlo simulation. In some cases, especially for complex problems, analyzing the complexity of the random input–output relationship (e.g., nonlinearity and interactions between input variables) may reveal additional information and useful insight. To that end, this paper introduces the use of Shapley additive explanations (SHAP) to help the explanation of a PCE model. Originating from game theory and machine learning, SHAP computes the contribution of the input variables to the single prediction level. SHAP enables visual inspection of the nonlinearity and interaction between variables from a PCE model. In addition, as an alternative to Sobol indices, SHAP also quantifies the relative importance of the inputs to the output. This paper introduces a procedure to calculate SHAP values from a PCE model without explicitly building multiple PCE models. A fast and exact algorithm that enables the calculation of SHAP for high-dimensional problems is presented. The usefulness of SHAP with PCE is demonstrated on several algebraic and non-algebraic problems. • An explainable AI technique is implemented within polynomial chaos expansion (PCE). • The Shapley Additive Explanations (SHAP) enhances the explainability of PCE. • A method to analytically calculate SHAP for PCE is presented. • SHAP allows visualization of nonlinearity and interactions from the PCE model. • The method is demonstrated on three nonlinear uncertainty quantification problems. [ABSTRACT FROM AUTHOR]