Laurence W. Cook

Robust Airfoil Optimization and the Importance of Appropriately Representing Uncertainty

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 n...

Using stochastic dominance in multi-objective optimizers for aerospace design under uncertainty

© 2018, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved. In optimization under uncertainty for aerospace design, statistical moments of the quantity of interest are often treated as separate objectives and are traded off in a multi-objective optimization formulation. However, in many design problems the trade-off between statistical moments can be large and the Pareto front representing this trade-off can include designs with undesirable behavior, such as being robust but being guaranteed to give a worse performance than another design. When a simulation of a system is computationally expensive, obtaining the full Pareto front is unfeasible and so spending optimization time obtaining such undesirable designs wastes time that could be spent obtaining more desirable alternatives. As a remedy, we propose an optimization formulation that can use multiple dominance criteria to avoid generating potentially inferior designs. We consider various orders of stochastic dominance as criteria to use alongside statistical moment based Pareto dominance, and illustrate how this gives rise to improved designs using a limited computational budget in an acoustic horn design problem and a transonic airfoil design problem.

Generalized information reuse for optimization under uncertainty with non-sample average estimators: Generalized information reuse for optimization under uncertainty with nonsample average estimators

© 2018 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons, Ltd. In optimization under uncertainty for engineering design, the behavior of the system outputs due to uncertain inputs needs to be quantified at each optimization iteration, but this can be computationally expensive. Multifidelity techniques can significantly reduce the computational cost of Monte Carlo sampling methods for quantifying the effect of uncertain inputs, but existing multifidelity techniques in this context apply only to Monte Carlo estimators that can be expressed as a sample average, such as estimators of statistical moments. Information reuse is a particular multifidelity method that treats previous optimization iterations as lower fidelity models. This work generalizes information reuse to be applicable to quantities whose estimators are not sample averages. The extension makes use of bootstrapping to estimate the error of estimators and the covariance between estimators at different fidelities. Specifically, the horsetail matching metric and quantile function are considered as quantities whose estimators are not sample averages. In an optimization under uncertainty for an acoustic horn design problem, generalized information reuse demonstrated computational savings of over 60% compared with regular Monte Carlo sampling.

Horsetail matching for optimization under probabilistic, interval and mixed uncertainties

© 2017, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved. The importance of including uncertainties in the design process of aerospace systems is becoming increasingly recognized, leading to the recent development of many techniques for optimization under uncertainty. Most existing methods represent uncertainties in the problem probabilistically; however, in many real life design applications it is often difficult to assign probability distributions to uncertainties without making strong assumptions. Existing approaches for optimization under different types of uncertainty mostly rely on treating combinations of statistical moments as separate objectives, but this can give rise to stochastically dominated designs. Horsetail matching is a flexible approach to optimization under any mix of probabilistic and interval uncertainties that overcomes some of the limitations of existing approaches. The formulation delivers a single, differentiable metric as the objective function for optimization. It is demonstrated on algebraic test problems and the design of a flying wing using a coupled aero-structural analysis code.

Research Data Supporting "Robust Airfoil Optimization and the Importance of Appropriately Representing Uncertainty"

This data comprises python source code, along with scripts that illustrate how to use this source code to recreate results in the publication. Further details are given in the README.txt file.

On the Importance of Appropriately Representing Uncertainty in Robust Airfoil Optimization

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. Nomenclature x Design variables nx Number of design variables x∗ Design variables for optimal solutions Y System model output / quantity of interest u Uncertainty vector xl i Lower bound of i th design variable xu i Upper bound of i th design variable g j jth constraint function ∗PhD Candidate, Department of Engineering, University of Cambridge. Student Member AIAA. Email: lwc24@cam.ac.uk †University Lecturer, Department of Engineering, University of Cambridge. Senior Member AIAA. Email: jpj1001@cam.ac.uk ng Number of constraints φk Polynomial basis function ak Expansion coefficient M Truncation of expansion p Polynomial chaos order in each dimension μ Mean σ Standard deviation fp Measure of performance fr Measure of robustness wp Performance measure weighting wr Robustness measure weighting ∆ Magnitude of uncertainty v Parameter subject to uncertainty u Underlying uncertain variable CL Lift coefficient CD Drag coefficient L/D Lift to drag ratio