Stochastic Collocation with CFD robustness concepts for multi-dimensional stochastic space: Application to a transonic airfoil

TL;DR: It has been proved that SC-ENO is able to choose automatically either a piecewise or a global polynomial approximation based on the smoothness of the target solutions in each dimension of the stochastic space while the efficiency of Stochastic Collocation method is maintained.

Abstract: Computational simulations have developed to a phase where the inherent physical variability prevalent in computational models exerts a larger effect on the predictive results than the deterministic numerical errors. To aim for more accurate and realistic simulations of the physical systems, it is imperative to include the input uncertainties into the computational models and investigate their effects on the outputs of interest. In the field of Computational Fluid Dynamics (CFD), which features high non-linearity and complexity, the non-intrusive Stochastic Collocation method (SC) gains great popularity by the virtue of easy implementation and high convergence rate of its spectral basis. The main idea of SC is to constructs a surrogate response surface in the stochastic space by globally interpolating the sampling values obtained from the deterministic simulations. Therefore, like any other spectral method, it shows limitations in capturing local parametric steep gradient or discontinuity in the stochastic space. Besides, the convergence rate is deteriorated due to the Gibbs oscillations. These spurious oscillations, which amount to unphysical realizations, could result in falsely enlarged full confidence interval, e.g., the pressure distribution along the upper surface of a transonic airfoil. To provide more robust stochastic analysis, the Gibbs oscillations in the stochastic space need to be eliminated. To this end, the robustness concepts from the CFD community, Local Extremum Conserving (LEC), Monotonicity Preserving (MP) and Essentially Non-Oscillatory (ENO), are reformulated for the multi-dimensional stochastic space and incorporated into the Stochastic Collocation method. The proposed method is termed Stochastic Collocation with Essentially Non-Oscillatory (SC-ENO) robustness. Different from the traditional Stochastic Collocation method, the SC-ENO method first resolves the locations of the discontinuity in the stochastic space by enforcing the robust limiter to the surrogate response surface. Then the whole stochastic space is partitioned into disjoint smooth sub-domains bounded by the discontinuities. All the deterministic sampling points are classified into each smooth sub-domain. Finally, the model surrogate in each smooth element is constructed by interpolating the sampling points of the same class. The proposed discontinuity detection method is implemented for the $1$-dimensional space and extended to the multi-dimensional space by the so-called dimension-by-dimension approach. Hence, the deterministic multi-dimensional sampling points are structured and formed by the tensor product of $1$-dimensional nodes. As for the surrogate construction, the performance multivariate interpolation methods, \textit{i.e.}, Sauer-Xu Lagrange interpolation and the least interpolation, are compared in terms of the robustness and the accuracy. The least interpolation method is better than Sauer-Xu algorithm but it lacks robustness for certain distribution of the sampling points. To remedy this, a element-wise interpolation method matching the proposed discontinuity detection method is developed. One remarkable feature of the proposed SC-ENO method is being completely non-parametric. It is essentially a post-processing of the input sample realizations. When the deterministic sampling points are structured, it has been proved that SC-ENO is able to choose automatically either a piecewise or a global polynomial approximation based on the smoothness of the target solutions in each dimension of the stochastic space. Therefore, the robustness of the surrogate model is ensured while the efficiency of Stochastic Collocation method is maintained. The accuracy and convergence property of the proposed SC-ENO method is investigated for some numerical test functions with jump discontinuities. Although there is no obvious improvement of the convergence rate, especially for the statistical quantities, the issue of the Gibbs oscillations is solved. To see how SC-ENO performs for real problems with jump discontinuities or steep gradient in the stochastic space, it is applied to the shock tube problem with uncertainties in initial conditions and the transonic viscous flow over the RAE 2822 airfoil with uncertainties in inflow conditions. In both of these test cases, the accuracy, efficiency and robustness of the SC-ENO method are illustrated. It is shown that there are no unphysical overshoots in the full confidence intervals of the interested quantities with sufficient deterministic sampling points. Meanwhile, the performance of another approach dealing with the stochastic discontinuities, Subcell Resolution (SR), is investigated for these two test cases. It is concluded that the SC-ENO method is more suitable for the viscous flow problems whereas the SR approach performs better for the inviscid case. Using the same set of the deterministic sampling values, the results of some interested quantities of smooth nature are presented, where the spectral convergence property is obtained. Since the SC-ENO method works on the tensor product sampling points, it is prone to the curse of dimensionality for high-dimensional stochastic spaces. To reduce the computational resources, the proposed approach is combined with the sparse grid approach of both isotropic type and dimension-adaptive type. However, the numerical experiments shows that the robustness property of the SC-ENO is lost for the sparse grid case. A more efficient and robust scheme suitable for high-dimensional space is a future research topic.