# Interactive Airfoil Optimization Using Parsec Parametrization and Adjoint Method
## 1 Introduction  
The process used in the aerodynamic optimization of an airfoil relies on two main features. The first is a computationally efficient optimization method that can effectively minimize (or maximize) the cost function. The second feature, which is perhaps even more important than the first one, is the ability to represent an airfoil in terms of design variables (to parametrize the airfoil). Currently, there are many parametrizations for airfoils and many optimization methods, yielding dozens of possible combinations. However, the ability to perform CFD-based optimization reliably and reasonably quickly relies heavily on the discovery of the adjoint method in the 1970s and 1980s and its development in the 1990s and early 2000s. Currently, the objective of aerodynamic optimization research and development is shifting towards fast interactive design optimization [1], where the results of the optimization have to be known almost instantly. The CFD-based adjoint optimization method, which is now an extremely well thought-out and proven concept that is implemented in both commercial (like Ansys Fluent) and open source (such as OpenFOAM) big CFD codes, is not well suited to this task. This is due to the vast computational complexity ofmodern-day CFD simulations, which makes the optimization too slow to be considered interactive, even with the use of the most efficient optimization methods [1]. It follows that the computational complexity of both the flow solution and the optimization algorithm must be substantially reduced to allow interactive design optimization [1]. This can be conducted in at least two ways. The first is presented in [1] and its references and involves the use of modern machine learning (ML) methods to estimate the flow variables (and sometimes even adjoint variables when coupled with the adjoint method) and force coefficients. The other way, which is presented in this article, is to stick to the adjoint method for optimization while making some assumptions about the flow, which allows the use of much simpler (and thus faster) methods for solving the flow variables, such as the panel method. This approach (when cautiously formulated) almost completely avoids three main hindrances of the ML approach: extrapolatory predictions, the curse of dimensionality, and the need for vast sets of good-quality training data [1]. However, in the adjoint-based assumption approach, great care has to be taken when placing the design point and simplifying the flow, as some important flow phenomena may be omitted in the simplification. This means that for some problems, especially where compressible and/or viscous flow must be assumed, it may be more feasible to use ML methods to deal with the challenges of interactive optimization. The adjoint method is a computationally efficient optimization method, and thanks to its efficiency and accuracy, it is well suited for optimization in fluid mechanics, as stated in [2,3]. However, its complexity in both the mathematical apparatus and the programming implementation made initial progress after its proposal rather slow-paced [4]. Since its proposal, it has been successfully used in a variety of optimization problems in the field of fluid mechanics (both external and internal), such as those in [2, 3, 4, 5, 6, 7]. The adjoint method comes from the theory of optimal control, as stated in [3,8], and was first proposed by [PERSON] (*1945) and [PERSON] (*1934) [7]. [PERSON] developed and successfully used the first optimization code based on the adjoint approach [9]. The adjoint method, unlike the finite difference method, evaluates the gradient of the cost function indirectly using the adjoint variables [2,3,8]. Consequently, the computational effort tied to the computation of the gradient of the cost function is not coupled to the number of design variables and is always roughly equal to the computation of two flow solutions [2,3,8].