# The Objective Space and the Formulation of Design Requirement in Natural Laminar Flow Optimization
## 2 CFD Solver and Its Validation
### Design Variables in Geometry Description  
The parametric geometric representation approach Class-Shape Transformation (CST) conquered many aerodynamic designs with its smoothness and convenience in shape description [31; 41; 42; 43]. Conventional CST (Equation (5)) imposes shape functions (Equation (3)) on specific class function (Equation (4)) with accumulated Bernstein polynomials assuring shape smoothness [44; 45; 46], where \(n\) is the scale of the method; \(u\) is the non-dimensional position, \(u_{TE}\) is the trailing edge thickness correction (equal to zero in a sharp edge case); \(A_{i}\) are the parameters. For ordinary airfoil surfaces, two parameters: \(N_{1}\) and \(N_{2}\) take the value of 1 and 0.5, respectively. \[Shape(u)=CF(u)\cdot(\sum_{i=0}^{n}A_{i}\cdot BP_{i}^{n}(u))+u_{TE}u \tag{3}\]  
\[Class(u,N_{1},N_{2})=u^{N_{1}}(1-u)^{N_{2}} \tag{4}\]  
\[Class\_Shape(u)=(u^{N_{1}}(1-u)^{N2})\cdot(\sum_{i=0}^{n}A_{i}\frac{n!}{i!(n-i )!}u^{i}(1-u)^{n-i})+u_{TE}u \tag{5}\]  
The difference in form of \(L_{2}\)-norm of vertical coordinates between \(Class\_Shape(u)\) function and original data can be minimized until the parameters of \(A_{i}\) are obtained via least square method:  
\[\sqrt{\frac{n}{i}(C(u_{i})_{original}-C(u_{i})_{fitted})^{2}} \tag{6}\]  
A CST method of 11 scale is applied to get airfoil RAE2822's shape variation, the Bernstein polynomial decomposition of which is shown in Figure 4. The x-y plot represents coordinates of different Bernstein polynomials. For given x position, the summation of local values of Bernstein polynomials is the actual y coordinate of airfoil. The difference between the original RAE2822 airfoil surface and fitted one via CST is shown in Figure 5, which meets the demand of accuracy of model tolerance of \(1\times 10^{-7}\) m recommended by [35]. The design space is created based on the variation of the first 4 CST parameters representing top, front region of the airfoil RAE2822. Each valid airfoil sample needs to meet detailed geometric constraints: not only the parameter is variated within bounding limits, the generated shapes have to be verified later in terms of curvature distribution, area check, and monotonicity check (cf. Figure 6):  
* The curvature (\(\kappa\equiv\frac{y^{{}^{\prime\prime}}}{(1+y^{{}^{\prime}2})^{3/2}}\)) distribution has to be within \(\pm 0.1\) based on that of the original airfoil surface. * The monotonicity must be consistent with that of the original airfoil surface to avoid twisted shape which is definitely harmful to the approaching towards the target via Differential Evolution algorithm. * And constraint of maximum area of 125% of the initial area is enforced to avoid shape with too high drag before CFD calculation. These constraints affect the data domain and its distribution in design space as well as objective space, which has impact on the final formulation of design requirement. Principal Component Analysis as Data Manipulator  
PCA is introduced to reduce the number of variables or design space dimensions on the premise of maintaining/capturing intrinsic information (e.g., the dominant physics of a system) in the original empirical data, when the problem requires expensive high-fidelity simulations [47; 48; 49]. PCA is also referred to as Karhunen-Loeve transformation, Proper Orthogonal Decomposition, singular systematic  
Figure 4: The decomposition in terms of Bernstein polynomials of airfoil RAE2822’s shape. Figure 5: The difference between the original RAE2822 airfoil surface and fitted one via CST. Figure 6: Illustration of valid airfoil samples in form of x-y profiles as well as their corresponding curvature distribution along curve. analysis, singular value decomposition under different circumstances, which has other variations in methodologies, e.g., kernel PCA, probabilistic nonlinear PCA/Gaussian process latent variable models, etc. [50].