Probabilistic approach to free-form airfoil shape optimization under uncertainty

Free-form shape optimization of airfoils to minimize drag directly poses unexpected difficulties. Practical experience has indicated that a deterministic optimization for discrete operating conditions can result in dramatically inferior performance when the actual operating conditions are different from the (somewhat arbitrarily selected) design conditions used during the optimization. Extensions from single-point to multipoint optimization have proven unable to remedy this problem of localized optimization adequately near the sampled operating conditions. An intrinsically statistical approach is presented, and how the shortcomings of traditional optimization methods can be overcome is demonstrated. We discuss two algorithms. The first one is based on a numerical evaluation of the expectation integral during each optimization step. The second one is based on a closed-form second-order analytic approximation of the integration. The airfoil geometry optimization example (inviscid Euler flow) also reveals how the relative likelihood of each of the operating conditions is automatically taken into consideration during the optimization process. This is a key advantage over the use of multipoint methods.