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This paper presents the results of applying direct and surrogate-based optimization (SBO) algorithms to two-dimensional aerodynamic benchmark problems, both involving transonic flow, one invisvid and the other viscous. The direct optimization methods used in this study are the adjoint-based FUN3D and Stanford University Unstructured solvers. The SBO algorithms include the SurroOpt framework, which exploits approximation-based models, the multi-level optimization (MLO) algorithm, which relies on physics-based models, as well as the adjoint-enhanced MLO algorithm. The results demonstrate that direct optimization and the approximation-based methods are able to yield designs that are comparable to those obtained with high-dimensional shape parameterization methods. Physics-based SBO shows a rapid design improvement at a low computational cost compared to the direct and the approximation-based SBO techniques, which indicates that—for certain problems—derivative-free methods may be competitive to adjoint-based algorithms when embedded in surrogate-assisted frameworks. On the other hand, global search approaches, while more expensive, exhibit the potential to produce the best quality results.

The objective of this paper is to present a robust optimization algorithm for computationally efficient airfoil design under mixed (inherent and epistemic) uncertainty using a multi-fidelity approach. This algorithm exploits stochastic expansions derived from the Non-Intrusive Polynomial Chaos (NIPC) technique to create surrogate models utilized in the optimization process. A combined NIPC expansion approach is used, where both the design and the mixed uncertain parameters are the independent variables of the surrogate model. To reduce the computational cost, the high-fidelity Computational Fluid Dynamics (CFD) model is replaced by a suitably corrected low-fidelity one, the latter being evaluated using the same CFD solver but with a coarser mesh. The model correction is implemented to the low-fidelity CFD solutions utilized for the construction of stochastic surrogate by using multi-point Output Space Mapping (OSM) technique. The proposed algorithm is applied to the design of NACA 4-digit airfoils with four deterministic design variables (the airfoil shape parameters and the angle of attack), one aleatory uncertain variable (the Mach number) and one epistemic variable (β, a geometry parameter) to demonstrate robust optimization under mixed uncertainties. In terms of computational cost, the proposed technique outperforms the conventional approach that exclusively uses the high-fidelity model to create the surrogates. The design cost reduces to only 34 equivalent high-fidelity model evaluations versus 168 obtained with the conventional method.

The trade-offs between the aerodynamic performance and aerodynamic noise signature of two-dimensional airfoil shapes in low-speed, high-lift flow are investigated. The figures of interest are calculated using Reynolds-Averaged Navier-Stokes-based computational fluid dynamics (CFD) simulations. A computationally efficient procedure for obtaining the Pareto front of the figures of interest is presented. The proposed approach utilizes a multi-objective evolutionary algorithm (MOEA) that works with a fast surrogate model of the aerodynamic surface under design, obtained with kriging interpolation of low-fidelity CFD simulations. The surrogate is enhanced by means of space mapping response correction based on a limited number of high-fidelity CFD simulation training points allocated in the design space. The Pareto set generated by the multi-objective optimization of the surrogate using MOEA is iteratively refined by local enhancements of the surrogate model. The proposed method allows us to obtain—at a low computational cost—a set of airfoil geometries representing the trade-offs between the figures of interest. We illustrate the approach using an example of an airfoil at a Mach number of 0.208, lift coefficient of 1.5, and a Reynolds number of 0.665 million.

In this paper, we present a robust optimization algorithm for low computational cost airfoil design under aleatory uncertainty. Our approach exploits stochastic expansions derived from the Non-Intrusive Polynomial Chaos (NIPC) technique to create response surface approximation (RSA) models utilized in the optimization process. In this work, we employ a combined NIPC expansion approach, where both the design and the uncertain parameters are the input arguments of the RSA model. In order to reduce the computational complexity of the design process, the high-delity computational uid dynamic (CFD) model is replaced by a suitably corrected low-delity one, the latter being evaluated using the same CFD solver but with a coarser mesh and relaxed convergence criteria. The model correction is realized at the response level using multi-point output space mapping (OSM). The OSM correction can be obtained without costly parameter extraction procedure and ensures that the low-delity model represents the high-delity one with sucient accuracy. The proposed robust optimization algorithm is applied to the design of transonic airfoils with four deterministic design variables (the airfoil shape parameters and the angle of attack) and one aleatory uncertain variable (the Mach number). In terms of computational cost, the proposed surrogate-based technique outperforms the conventional approach that exlusively uses the high-delity model to create the RSA models: the design cost corresponds to only 12 equivalent high-delity model evaluations versus 42 for the conventional method.

A low-speed wind tunnel is developed for fluid dynamics research at Reykjavik University. The tunnel is intended for conducting research on the flow past micro air vehicles, as well as fundamental research on turbulence. High flow quality is elemental for both research projects. The tunnel is of open suction type and is composed of a square inlet with a honeycomb and turbulence screens, settling chamber, contraction, experimental section housing, diffuser, and axial fan. In this paper, we describe the details of the design optimization procedure of the contraction, which is a key to getting a high quality flow in the experimental section. A high-fidelity computational fluid dynamic (CFD) flow solver is used to capture the nonlinear flow physics. Due to the high computational expense of the CFD simulations, surrogate-based optimization (SBO) is used to accelerate the design process. The SBO approach replaces direct optimization of the high-fidelity (accurate but computationally expensive) model by iterative optimization of a properly corrected lowfidelity model. Here, we exploit variable-resolution CFD simulations, as well as a simple multiplicative response correction technique to construct the surrogate model of the wind tunnel contraction, allowing us to optimize its shape at a low computational cost. To our knowledge, it is the first application of SBO to wind tunnel contraction design. The optimum nozzle design is verified using a high-fidelity CFD simulation.

An adaptive response correction (ARC) technique for computationally efficient surrogate-based optimization (SBO) of transonic airfoils is developed. SBO replaces a direct optimization of the high-fidelity (accurate but computationally expensive) airfoil model by an iterative optimization of a properly corrected low-fidelity one. Here, both models are based on the same governing fluid flow equations, however, the low-fidelity one uses a coarser discretization and relaxed convergence criteria. The ARC technique exploits the knowledge contained in the low-fidelity model to construct a fast and yet an accurate prediction tool (surrogate) that is used to find the high-fidelity model optimum. The optimization algorithm is embedded in the trust-region framework to ensure convergence. Examples demonstrate that the optimized airfoil design can be obtained at a substantially lower computational cost when compared to the direct high-fidelity model optimization.

In the paper, a low-cost design technique for transonic airfoils is discussed. We exploit the surrogate-based optimization principle where the direct optimization of the high-fidelity (accurate but computationally expensive) airfoil model is replaced by iterative optimization of a properly corrected low-fidelity model. Both models are based on the same governing fluid flow equations (the Euler equations), however, the latter uses coarser discretization and relaxed convergence criteria. The shape-preserving response prediction (SPRP) technique is utilized to predict the high-fidelity model response, here, the airfoil pressure distribution. In this prediction process, SPRP employs the actual changes of the low-fidelity model response due to the design variable adjustments. The optimization algorithm is embedded in the trustregion framework to ensure good convergence properties. The airfoils are parameterized using Bezier curves with a total of eleven design variables. Examples demonstrate that the optimized airfoil design can be obtained at a low computational cost.

A computationally efficient and robust methodology for transonic airfoil design is presented. The approach replaces the direct optimization of an accurate, but computationally expensive, high-fidelity airfoil model by an iterative re-optimization of a corrected low-fidelity model. The low-fidelity model is based on the same governing fluid flow equations as the high-fidelity one, but uses coarser discretization and relaxed convergence criteria. The shape-preserving response prediction technique is utilized to align the pressure distribution of the low-fidelity model with that of the high-fidelity model. Our method is applied to constrained airfoil lift maximization and drag minimization in twodimensional inviscid transonic flow. The optimized designs are obtained at substantially lower computational cost when compared to the direct high-fidelity model optimization. The savings are 85 to over 90 percent depending on the test case.