Least squares approximation-based polynomial chaos expansion for uncertainty quantification and robust optimization in aeronautics

For many engineering problems, reliability and robustness are far more important than the nominal performance when it comes to the choice of a design over another. Availability of state-of-the-art simulation codes and trusted modeling approaches help in reducing the epistemic uncertainties, but what is needed for decision-making is ultimately the knowledge of the aleatory uncertainty for the system of interest, and methods capable of minimizing such uncertainty while limiting the impact on the expected performance. A reliable measure of performance is only achieved if all uncertainties affecting the system’s geometry, physical properties and operational conditions can be propagated through the model equations to the quantities of interest. Optimization in a deterministic sense, providing the best possible performance for nominal values of the design variables, is then replaced by Robust Optimization, which aims at maximizing the expected performance while minimizing the probability of its degradation. Today, multi-physics problems and demanding high-fidelity simulations are common denominators in Aeronautics as well as in many other industrial fields. It is then necessary to employ techniques that are able to perform uncertainty quantification and propagation at reasonable costs, without requiring the development of dedicated solvers: Non-Intrusive Polynomial Chaos is a prime candidate, but not all of its variants are equally accurate and cost-effective. Efficiency and flexibility of Least Squares surrogates and gradient-enhanced Polynomial Chaos expansions for Uncertainty Quantification and Robust Optimization are discussed in this paper employing an in-house framework. Advantages of the proposed methods over Collocation approaches are demonstrated by comparing the results for a selection of algebraic test cases and for the shape optimization of the NACA 0012 airfoil under transonic conditions.