## 2 Methodology  
Three representative stochastic algorithms are integrated into the optimization process, all included in the Inspyred library of Python [11]: Genetic Algorithm (GA), Evolution Strategies (ES) and Particle Swarm Optimizer (PSO). All of them are members of the Evolutionary Algorithms (EA) [12; 13] and share a great number of common features. They are population-based performing with best-to-survive criteria. EA algorithms are considered suitable for design optimization of airfoils, therefore they have been selected in the present work instead of other methods, such as deterministic gradient-based algorithms, which may present convergence difficulties during the optimization process. GA [14] is the most traditional category, especially in the fields of Aerodynamics and Turbomachinery. It resembles the members of population with chromosomes and applies the typical operators of crossover and mutation to evolve to the next generation. ES algorithm [15] applies the mutation and recombination operators not only on the design variables, but also on endogenous strategic parameters such as the mutation strength. It usually employs encoding of design variables in the field of real and not binary numbers as GA does. PSO algorithm [16] has been in application for the last two decades. It represents each solution as a particle and the population of solutions is called a swarm of particles. The best positions of each particle and of the swarm are continuously updated and the velocities are adjusted based on the experiences of the particles. The procedure stops when the position of the swarm satisfies the criterion for the objective function. In the following the optimization problem is outlined. This requires definition of the objective function, the design variables of the problem and the necessary geometric design constraints, which ensure that physically acceptable solutions are obtained, while at the same time prevent unnecessary computations for geometrically inconsistent candidate solutions. In the present analysis, as objective function the maximum lift coefficient (\(\text{C}_{\text{L,max}}\)) of the morphing airfoil shape is chosen, under specific conditions with respect to the freestream, i.e., specific values of Reynolds (Re) and Mach (Ma) numbers. The baseline airfoil (undeformed shape of wing cross section) is chosen as the starting design of the optimization process and its coordinates are normalized with respect to the chord. In order to maximize \(\text{C}_{\text{L,max}}\), the shape of the baseline airfoil is modified at the LE and TE regions, defined by the intervals (0, 0.25) and (0.8, 1) of the x-coordinate, respectively. The design variables are the y-coordinates of the control points of the Bezier curves used in approximating the airfoil morphing shapes. The control points are only allowed to move vertically. In the present work, the Bezier curves approximation method has been chosen due to its flexibility in representing smooth geometries, such as airfoil deformations, by altering the positions of the control points. Curvature and slope constraints are imposed to avoid irregularities and discontinuities of the derived geometries and to ensure a smooth connection between deformed and undeformed parts. These constraints are related to:  
1. The slope variations of the airfoil geometry. Large changes in the geometry gradients should not be permitted and this is checked by the product of the gradients (first derivatives) of consecutive segments of the discretized geometry. In the case that negative values of this product less than \(-10^{-3}\), are obtained, the current airfoil geometry is rejected. 2. The maximum curvature of the LE geometry, which undergoes the highest deformations. An upper limit is set for the curvature of the upper and lower parts of the deformable LE geometry. The curvature is calculated with respect to the second and the first derivative.