It is important to design engineering systems to be robust with respect to uncertainties in the design process. Often, this is done by considering statistical moments, but over-reliance on statistical moments when formulating a robust optimization can produce designs that are stochastically dominated by other feasible designs. This article instead proposes a formulation for optimization under uncertainty that minimizes the difference between a design's cumulative distribution function and a target. A standard target is proposed that produces stochastically non-dominated designs, but the formulation also offers enough flexibility to recover existing approaches for robust optimization. A numerical implementation is developed that employs kernels to give a differentiable objective function. The method is applied to algebraic test problems and a robust transonic airfoil design problem where it is compared to multi-objective, weighted-sum and density matching approaches to robust optimization; several advantages over these existing methods are demonstrated. ARTICLE HISTORY

The computational economy of Reynolds-Averaged Navier-Stokes solvers encourages their widespread use in the optimization of aerospace designs. Unfortunately, the real-world performance of the resulting optimized designs may have shortcomings. A common contributor to this shortfall is a lack of adequately accounting for the uncertainty introduced by the structure of the turbulence model. We investigate whether including measures of turbulence-based uncertainty, as predicted by the eigenspace perturbation method, in an optimization under uncertainty framework can result in designs that are more robust with respect to turbulence model-form uncertainty. In an asymmetric diffuser design problem and a transonic airfoil design problem, our optimization formulation taking account of turbulence-based uncertainty obtained designs that were more robust to turbulence model uncertainty than optimal designs obtained via deterministic approaches.

Nomenclature C D Drag coefficient C L Lift coefficient F Cumulative distribution function F −1 Quantile function (inverse cumulative distribution function) g Constrained quantity q Quantity of interest Q Set of feasible values of the quantity of interest Q Superquantile function S Set of non-dominated designs u Vector of values of uncertain parameters U Random vector of uncertain parameters x Vector of design variables Y Set of possible designs ω Underlying random event µ Mean value of a random variable σ Standard deviation of a random variable Ω Sample space of underlying random event x For a given design

A challenge in engineering design is to choose suitable objectives and constraints from many quantities of interest, while ensuring an optimization is both meaningful and computationally tractable. We propose an optimization formulation that can take account of more quantities of interest than existing formulations, without reducing the tractability of the problem. This formulation searches for designs that are optimal with respect to a binary relation within the set of designs that are optimal with respect to another binary relation. We then propose a method of finding such designs in a single optimization by defining an overall ranking function to use in optimizers, reducing the cost required to solve this formulation. In a design under uncertainty problem, our method obtains the most robust design that is not stochastically dominated faster than a multiobjective optimization. In a car suspension design problem, our method obtains superior designs according to a k-optimality condition than previously suggested multiobjective approaches to this problem. In an airfoil design problem, our method obtains designs closer to the true lift/drag Pareto front using the same computational budget as a multiobjective optimization. K E Y W O R D Saerodynamics, engineering design, optimization, probabilistic methods, shape design, suspension design, transonic *A binary relation is a comparison of two designs that determines whether one is better than the other, and for an optimal design no better design exists with respect to the relation. 3 Int J Numer Methods Eng. 2020;121:2481-2502.wileyonlinelibrary.com/journal/nme

The importance of designing airfoils to be robust with respect to uncertainties in operating conditions is well recognized. However, often the probability distribution of such uncertainties does not exist or is unknown, and a designer looking to perform a robust optimization is tasked with deciding how to represent these uncertainties within the optimization framework. This paper asks "how important is the choice of how to represent input uncertainties mathematically in robust airfoil optimization?", specifically comparing probabilistically based aleatory uncertainties and interval based epistemic uncertainties. This is first investigated by considering optimizations on several algebraic test problems, which illustrate the mechanisms by which the representation of uncertainty becomes significant in a robust optimization. This insight is then used to predict and subsequently demonstrate that for two airfoil design problems the advantage of doing a robust optimization over a deterministic optimization is similar regardless of how the input uncertainties are represented mathematically. The benefit of this is potentially eliminating the time required to establish an accurate representation of the uncertainties from the preliminary stage of design, where time is a valuable resource.