Comparing optimization under uncertainty and multi-point optimization for airfoil shape design

Jason S Bank, Purdue University

Abstract

The 2-D cross-section shape, or the airfoil, of an aircraft wing significantly impacts the lift and drag of the aircraft. While the aircraft may have a dominant design condition (e.g. cruise at a specified altitude and velocity), it will operate at various altitudes and speeds during a mission. To allow good performance across the flight envelope, the traditional airfoil shape optimization approach formulates a multi-point problem that seeks to minimize a weighted sum of the drag coefficient at multiple flight conditions, subject to appropriate constraints. The multi-point approach can be effective, but it treats all design variables and parameters as though they were deterministic while many of these may follow probability distributions. This thesis investigates a design under uncertainty formulation for airfoil optimization as an alternative to the traditional deterministic, multi-point problem formulations. The implementation and investigations of these different methods will occur within a low- speed airfoil design problem. Due to the low-speed approach, the calculation of drag results and other aerodynamic properties is completed using XFOIL. Using velocity as the uncertain flight condition, it is shown that a beta probabilistic distribution results in lower drag coefficient values at specified flight speeds than an uniform distribution at the same specified speeds. The sampling of the distributions is completed using a basic Monte Carlo Sampling. Thus, due to this probabilistic sampling, a genetic algorithm provides the optimization. While the non-deterministic solution does not gain any improvement on the deterministic results at on-design conditions, the off- design performance results in at least a 32% reduction in drag at other conditions. There appears to be some distinct advantages to using design under uncertainty in lieu of multi-point optimization for airfoil shape optimization.

Degree

M.S.E.

Advisors

Crossley, Purdue University.

Subject Area

Engineering|Aerospace engineering

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