# The Objective Space and the Formulation of Design Requirement in Natural Laminar Flow Optimization
## 2 CFD Solver and Its Validation
### Information Extracted from Objective Space  
If there is some information that represents some features to be deployed in optimization process, then it should produce result that is irrelevant of sampling strategy. With the approach of enumeration, a grid strategy is used for sampling of different scales. An index \(n\) indicates the evenly distribution of sampling seeds in every CST parameter, or sampling scale, as is shown in Table 3. The total number of samples easily exceeds reasonable calculation limitation if the value of \(n\) is large enough because of iteration through all the parameters. This traditional sampling strategy is for revealing the intrinsic nature hidden in objective space. The huge amount of calculation corresponding to different values of \(n\) is not necessary in the real practice: if the relevant data is enough, then the eigenvectors can be estimated within approximation. For airfoil samples corresponding to given \(n\) value, their aerodynamic performance of \(n\) airfoils obtained through CFD constitute a database represented by array (7), where the two columns are skin friction drag and pressure drag force, respectively. \begin{table}
\begin{tabular}{l c} \hline \hline \(n\) **Value** & **Normalized Location of Sample Seeds for Each Parameter** \\ \hline
2 & 0.250, 0.750 \\
3 & 0.166, 0.500, 0.833 \\
4 & 0.125, 0.375, 0.625, 0.875 \\
5 & 0.100, 0.300, 0.500, 0.700, 0.900 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Illustration of grid sampling determined by \(n\) for each CST parameter. \[\mathbf{A}=\left[ \begin{array}{cc}a_{1,1}&a_{1,2}\\ \vdots&\vdots\\ a_{n,1}&a_{n,2}\end{array} \right] \tag{7}\]  
The data from objective space systematically forms the covariance matrix \(\mathbf{P}\) in Equation (8). The spectral decomposition is shown in Equation (9) where \(\mathbf{\Lambda}\) represents diagonal matrix of eigenvalues and \(\mathbf{U}\) represents agglomeration of eigenvectors. \[\mathbf{P}_{n\times n}=\frac{1}{n}\mathbf{A}\mathbf{A}^{T} \tag{8}\]  
\[\mathbf{P}_{n\times n}=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{T} \tag{9}\]  
The objective space and eigenvectors are obtained for each \(n\) in Table 3, the result of which is shown in Figure 7 where green, solid lines represent first principal component and the red, dashed lines represent second principal component. Since the design requirement in this case involves two aerodynamic performances, only two components are required. The boundary of the objective space can be clearly seen, which is caused by the various geometrical constraint posed by the problem. Similar to the 'convergence' of mesh independence check, the directions of eigenvectors are also 'converged' to a stable direction with the increasing of \(n\) (i.e., the directions of eigenvectors for objective space for \(n\) equals 4 and 5 are close), which shows the intrinsic information in the objective space determined by the problem description can be captured. The paper thinks that the understanding of the objective space and eigenvectors is directly associated with a more reasonable and problem-oriented formulation of design requirement. With the sampling, a comprehensive pattern that the data is arranged in objective space can be perceived, where the eigenvectors can be a quantitative description of it. Therefore, if the formulation of design requirement adopts such information, the optimization can be better guided by leading every sample to move in the direction that is parallel to the best direction decided by the nature of the problem. Also, the eigenvectors has engineering meaning in that a better  
Figure 7: Global view of objective space for each \(n\) and their eigenvectors. optimization criterion can be proposed in industrial aerodynamic design process as a balanced solution to the ultimate effect of total drag reduction and the perspective of emphasizing part of drag reduction via methodological guidance.