# Robust Algorithm Software for NACA 4-Digit Airfoil Shape Optimization Using the Adjoint Method
## 2 Bibliographic Review  
The introduction of the gradient-based CFD method by [PERSON] [4] in 1988 marked the first application of adjoint equations to transient flow [4]. They are also among the most prominent developers of this method and have utilized adjoint equations in various computational fluid dynamics test cases, including the optimization of aerodynamic shapes for Eulerian flows that govern the Navier-Stokes equations. [PERSON] was the first person to use optimization to solve differential equations [5], while adjoint equations were originally developed for shape optimization in fluid dynamics to solve a defined control problem [5; 6]. The coupling of perturbation equations with this method was achieved by integrating the Reynolds averaged Navier-Stokes (RANS) equations with the Spalart-Allmaras single equation turbulence model [7; 8]. Several researchers, including [PERSON] et al. [9], utilized the adjoint optimization method in fluid dynamics and Eulerian equations and conducted significant research in this field [9; 10]. Later, [PERSON] and [PERSON] [11] used Green's functions for the adjoint equations [11]. To design an airfoil, they used the adjoint method under incompressible non-viscous conditions in the far-field region [12], while [PERSON] [13] used the adjoint equations by considering the angle of attack based on the boundary conditions of the distant region [13]. In the multi-objective optimization field, various types of research studies have been conducted. [PERSON] and [PERSON] [14] presented a method for shape optimization utilizing B-splines, which are piecewise polynomial curves that can represent intricate shapes, making them valuable in engineering designs [14]. [PERSON] et al. [15] investigated the effect of initial population sampling on the convergence of multi-objective genetic algorithms (MOGAs) [15]. The authors conducted experiments using different techniques for initial population sampling and evaluated their impact on the performance of MOGA. Additionally, they introduced a new evolutionary algorithm for multi-objective optimization, MOGA-II, and tested its performance on single-objective optimization problems, comparing it to two successful algorithms: differential evolution and a standard evolutionary algorithm. [PERSON] et al. [16] conducted experiments using particle image velocimetry (PIV) and discovered that incorporating contoured cavities on the upper surface of the diffuser helps to reduce the extent of the flow separation region, thus enhancing the diffuser's efficiency. They also presented a study that aimed to improve diffuser efficiency using multiple local recirculation regions (MLRR) [17]. The Aerodynamic Shape Optimization Design Group is currently researching optimization methods and developing new techniques and algorithms. In recent studies, they have employed an adjoint-based solver to compute sensitivity derivatives. This approach is demonstrated in two articles: [PERSON] et al. [18] used the B-Spline method to adjust volume parameters, and calculated sensitivity derivatives using an adjoint equation solver and a jet stream solver. This approach was used to obtain an optimized shape for the NACA 0012 airfoil [18]. In a study by [PERSON] et al. [19], three methods for the shape optimization of airfoils were compared [19]. The first two methods used free-form deformation, and the third method used the radial basis function (RBF) coupled with the adjoint method. Sensitivity derivatives were obtained using an adjoint-based solver with an unstable flow solver and a finite difference method. The drag coefficients of the NACA 0012 and RAE 2822 airfoils were reduced under free deformation conditions. In an article by [PERSON] et al. (2010), the Bezier curve technique was used to modify the shapes of the NACA 0012 and RAE 2822 airfoils for shape optimization (2010). The sensitivity derivatives were calculated using a flow solver and an adjoint solver. The conjugate gradient method was used as an optimization algorithm in this study. [PERSON] et al.