# An Automated Framework for Streamlined CFD-Based Design and Optimization of Fixed-Wing UAV Wings
## 2 Design Space
### Geometric Design Space and Parametrization Techniques
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Planform** & **Description** & **Units** & **Limits** \\
**Parameter** & & & \\ \hline \(c_{1}\) & Centerline chord length & m & \(c_{1}\in[0.2,\ 10]\) \\ \(c_{2}\) & Root chord length & m & \(c_{2}\in[0.3<c_{1},\ c_{1}]\) \\ \(c_{3}\) & Tip chord length & m & \(c_{3}\in[0.3<c_{2},\ c_{2}]\) \\ \hline \(b_{1}\) & \(c_{2}\) spanwise location & m & \(b_{1}\in[0.2\times b_{2},\ 0.7\times b_{2}]\) \\ \(b_{2}\) & \(c_{3}\) spanwise location (semispan) & m & \(b_{2}\in[1,\ 10]\) \\ \hline \(\Lambda_{1}\) & Sweep angle of section 1@\(c/4\) & deg & \(A_{1}\in[0,\ 60]\) \\ \(\Lambda_{2}\) & Sweep angle of section 2@\(c/4\) & deg & \(A_{2}\in[0,\ 60]\) \\ \hline \(i_{1}\) & Twist angle of \(c_{2}\)@\(c/4\) & deg & \(i_{1}\in[-5,\ 5]\) \\ \(i_{2}\) & Twist angle of \(c_{3}\)@\(c/4\) & deg & \(i_{2}\in[-7,\ 7]\) \\ \hline \(\gamma_{1}\) & Dihedral angle of section 1 & deg & \(\gamma_{1}\in[-10,\ 10]\) \\ \(\gamma_{2}\) & Dihedral angle of section 2 & deg & \(\gamma_{2}\in[-10,\ 10]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design variables used to parameterize the wing planform and their respectable range. Figure 1: The two distinct sections used to generate a conventional and a BWB wing. A total of 11 low-level parameters are required to fully define the planform of a two-section wing, without including any details about the airfoil shape. It should be noted that all parameters are referenced from the coordinate system with an origin on the leading edge of the centerline chord. The x-axis spans along the chordwise direction and the y-axis spans along the spanwise direction of the wing. Design variable constraints have been defined to exclude unrealistic wing geometries such as having taper ratio values larger than 1, wings with very thin sections that would be impossible to manufacture, or wings with excessive aft and forward sweep angles (Figure 3). These parameters, combined with constraints, ensure that the design space remains realistic and aligned with established aerodynamic standards. The wing's cross-section is defined by three distinct airfoils: one at the root, one at the kink (midsection break), and one at the tip. Airfoil profiles are assigned to these spanwise locations, with interpolation between them. The airfoil shape greatly affects  
Figure 3: Unrealistic wing designs produced when the design space is not being constrained. The planform surfaces are displayed with the leading edge being on the left. Figure 2: Design variables used to parameterize the wing planform. the aerodynamic characteristics of wing geometry. The precise shape of an airfoil can vary greatly depending on its intended use, flight conditions, and performance requirements. One of the most common airfoil databases is the one from UIUC [27], which contains more than 1500 airfoil shapes from real cases. Selecting airfoils based on their unique categorical identifiers transforms the design space into a high-dimensional, discrete domain. This discrete nature lacks the continuity required for efficient aerodynamic shape optimization, where smooth transitions between design variables are essential. To address this, parameterization techniques that represent airfoil shapes through continuous variables are employed. Common parametrization techniques include the Bezier-PARSEC method, Class Function/Shape Function Transformations (CST), B-Splines combined with dimensionality reduction techniques such as Principal Component Analysis (PCA), and others [28]. However, the main drawbacks of those methods are the generation of non-smooth trailing edges with the frequent crossing of the lower pressure surface and the upper suction one and the demand of a large amount of design to accurately represent the airfoil in terms of geometric similarity and aerodynamic performance. Given these challenges, a novel approach to airfoil parameterization was required--one that could achieve high shape diversity while maintaining a low dimensionality for efficient optimization.