The main goal of this paper is to document a comparative study of different computational-fluid-dynamics-based optimization techniques applied to the solution of a three-dimensional wing drag minimization problem. To achieve this objective, three optimization tools were used: SYN107 (Intelligent Aerodynamics International), MDOPT (The Boeing Company), and OPTIMAS (Israel Aerospace Industries). The first tool employs gradient-based search techniques using the continuous adjoint equation, the second one is a response-surface method, and the last one uses a floating-point genetic algorithm as its search engine. As the starting geometry, the public domain DPW-W1 wing (a test case for the Third AIAA Drag Prediction Workshop) was used. The comparisons included herein are provided in three stages: cross analysis of the initial geometry by the computational fluid dynamics tools employed in the optimizations, optimization of the initial geometry to minimum drag, and cross analysis of optimal shapes achieved by the optimization tools using all computational fluid dynamics tools employed. The cross analysis also includes results from an independent computational fluid dynamics method that was not used in any of the optimization efforts. These results help quantify the level of variation that is inherent in, and can be expected from, application of the current state-of-the-art aerodynamic optimization methods. The present work may be regarded as a move toward the construction of reliable test cases for an aerodynamic shape optimization problem. Another goal of this collaborative investigation is to collect lessons learned from this pilot project to help develop a model for an aerodynamic optimization workshop.

A computational-uid-dynamics tool capable to simulate accurately viscous ows around complex aerodynamic con gurations is described. The method combines a multiblock technique with a low-dissipation numerical method incorporated into a multigrid framework. Structured subdomains (blocks) are united in multigrid/multiblock structures, and the blocks are treated independently at each stage of the numerical procedure, maintaining a regular information exchange between the neighboring blocks. In the numerical procedure the convection part of the equations is approximated by a low-order upwind-biased scheme employed for multigrid relaxation in combination with a higher-order essentially nonoscillatory scheme used to supply a defect to the right-hand side of the discrete equations on the nest multigrid level in a way ensuring the overall high accuracy of the scheme. Computational examples demonstrate the ability of the resulting method to perform accurate large-scale computations of complex three-dimensional turbulent ows around realistic aerodynamic con gurations.

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The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. In this paper we consider a variant of the problem and explore its connections to data-driven online selection. Specifically, we are given n cards with arbitrary non-negative numbers written on both sides. The cards are randomly placed on n consecutive positions on a table, and for each card, the visible side is also selected at random. The player sees the visible side of all cards and wants to select the card with the maximum hidden value. To this end, the player flips the first card, sees its hidden value and decides whether to pick it or drop it and continue with the next card.We study algorithms for two natural objectives. In the first one, similar to the secretary problem, the player wants to maximize the probability of selecting the maximum hidden value. We show that this can be done with probability at least 0.45292. In the second objective, similar to the prophet inequality, the player wants to maximize the expectation of the selected hidden value. Here we show a guarantee of at least 0.63518 with respect to the expected maximum hidden value.Our algorithms result from combining three basic strategies. One is to stop whenever we see a value larger than the initial n visible numbers. The second one is to stop the first time the last flipped card's value is the largest of the currently n visible numbers in the table. And the third one is similar to the latter but to stop it additionally requires that the last flipped value is larger than the value on the other side of its card.We apply our results to the prophet secretary problem with unknown distributions, but with access to a single sample from each distribution. In particular, our guarantee improves upon 1 − 1/e for this problem, which is the currently best known guarantee and only works for the i.i.d. prophet inequality with samples.

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SUMMARYA new approach to the robust handling of non-linear constraints for GAs (genetic algorithms) optimization is proposed. A speciÿc feature of the approach consists of the change in the conventional search strategy by employing search paths which pass through both feasible and infeasible points (contrary to the traditional approach where only feasible points may be included in a path). The method (driven by full Navier-Stokes computations) was applied to the problem of multiobjective optimization of aerodynamic shapes subject to various geometrical and aerodynamic constraints. The results demonstrated that the method retains high robustness of conventional GAs while keeping CFD computational volume to an acceptable level, which allowed the algorithm to be used in a demanding engineering environment.