# Interactive Airfoil Optimization Using Parsec Parametrization and Adjoint Method
## 1 Introduction
Despite the computational efficiency of the adjoint method, it is good practice to parametrize the optimized geometry with an appropriate number of design variables and to avoid the use of redundant ones, thereby preventing (or at least minimizing) problems related to the geometrical or physical infeasibility of the solution obtained. This is particularly applicable when the original problem has to be simplified for optimization. To obtain the design variables required by the optimization method, there are, at least in the case of an airfoil, literally dozens of parametrization methods [10,11]. From those, parsec parametrization is only one of the many well-proven parametrizations available. Those parametrizations involve 4- and 5-digit NACA series and their modifications [12], class shape transformation (CST) [13], the analytic equation, as presented in [14], and so on. From those parametrizations, the 4- and 5-digit NACA series and analytic equation from [14] have less than or equal to six parameters. CST can have an arbitrary number of parameters, and parsec has 11 or 12, depending on its variant [13,15]. There are even some modified parsec methods [15]. The parsec method has been extensively used for airfoil optimization, such as in [13,16, 17, 18]. It has strong control over the airfoil's leading edge, crest locations, and curvatures at the crest locations [15]. On top of that, its parameters have a clear and straightforward geometrical meaning, giving better insight into the airfoil shape. Parsec is a very powerful parametrization for covering general airfoil shapes with a reasonable amount of design variables [10,19]. Its performance in geometrical accuracy is roughly comparable to that of the Bezier, RBF, and Hicks-Henne parametrizations, with a similar number of design variables [11]. In contrast, its performance in the accuracy of aerodynamical properties is comparable to all the other parametrization methods covered in [11], with a similar number of design variables. However, parsec lacks precise control of the trailing edge shape [10], but this can be beneficial in this case, as the trailing edge shape has a very influential position in the panel method that may not be related to real physical phenomena. For this article, the panel method proposed by [PERSON] and [PERSON] in 1966 [20] was used as a flow solver, allowing a very low computational cost. The simplicity of the panel method means that great care must be taken when selecting the parametrization. The ability to prescribe a nearly arbitrary trailing edge geometry (which is possible with many parametrizations) may hinder the practical usability of the optimization, either through manufacturing difficulties or through the flow phenomena the panel method does not account for. On top of that, too many design variables or excessively strong control over the trailing edge shape can lead to geometrically impossible (distorted) shapes arising during optimization. The parsec parametrization was used because a proven parametrization method was needed, and regarding the flow solver used, the assurance of the geometrical and physical feasibility of the resulting shape was required. The next thing to consider is the uncertainty of the input data, together with the finite manufacturing precision. The dependence of the optimal shape on the input data uncertainties, together with the influence that the finite manufacturing precision has on the performance of both the original and optimized airfoils would be a very interesting research topic. Many methods for assessing the influence of uncertain operating conditions on the solution exist. Those methods and their possible usage can be found for example in [21; 22]. This topic, however, is so broad that a full-length article would be needed to cover it in appropriate detail; thus, it will not be discussed here.