Gappy Proper Orthogonal Decomposition-Based Two-Step Optimization for Airfoil Design

A ERODYNAMIC shape optimization (ASO) is one of the key techniques of aircraft design. In recent years, with the rapid development of computational fluid dynamics techniques and much research on the optimization algorithm, researchers have made considerable progress in ASO. The design period is especially shortened by high-performance computers. However, there are still two problems inASO.Thefirst problem is thatwe have not yet gotten any efficient global optimal methods, and the second is that we have to find a high-fidelity model that can obtain the desirable solution as soon as possible. Optimization methods can generally be divided into global optimization methods and local optimization methods. Theoretically, global optimization methods are more likely to obtain the global optimal solution but with greater cost. Two-dimensional ASO using global optimization methods may be affordable with current available computer resources [1,2]. With three-dimensional problems, there is almost noway towork out. Local optimizationmethods are applied to ASO in a wide range of uses because of their rapid rate of convergence and reliable optimal results [3,4]. But these methods are available only for functions with one optimum; when they are used for functions with many local optima, it is almost impossible to find the global optimal solution. In practice, the problems of ASO are too complicated to obtain the global optimum by local optimization methods. In recent years, global optimization methods and local optimization methods put together to collaborate is a new orientation [5] that can obtain better results than that of any single optimal method. The description of collaborative multimethod optimization in [6] gives a comprehensive summary about collaborative strategy. Though we can obtain better results by collaborative multimethod optimization, the great expense of global optimization methods is still the limitation. To approximately calculate the cost function by the surrogate model [7–9] is an effective way to reduce the computational cost. Proper orthogonal decomposition (POD) [10] can be used to construct the surrogate model by providing a set of bases and describing the system by linear combination. Basic POD needs to solve an eigenvalue problem of the autorelation matrix whose order is as high as the original problem, which is very hard or even impossible to solve. Sirovich [11] introduced the method of snapshots as a way to efficiently determine the POD basis for large problems. Legresley and Alonso [12,13] obtained a reduced-order model by projecting Euler equations to a set of POD bases, which is used successfully in both direct and inverse airfoil designs. Gappy POD [14] is a modification of the snapshot POD method that can reconstruct incomplete, or gappy, data independent of governing equations of described problems. Bui-Thanh et al. [15] used it to reconstruct the data of the flowfield and give a simple method on airfoil inverse design. Cai [16] compared the method in [12,15] for inverse design and believed that the latter has higher accuracy. Zhang et al. [17] obtained an optimum of multidisciplinary design about an S-shape intake by using gappy POD. In this paper, the method of snapshots generation and Hicks– Henne bump functions [18] are improved, and the flow analysis of airfoils based on gappy POD is presented. For the purpose of obtaining the optimal solutions within the given time, two-step optimization is introduced. Finally, the results of the optimization design are shown.