## 3 Rotor Blade Design Approximation with RBFNN
Regarding to the cross-section of rotor blade with specificgeometry information, a NURBS curve was generated for a given number of points, and the non-uniform rational B-spline (NURBS) was transformed into the cross-section of rotor blades with specific geometry data. Such information was adopted to calculate the design performance functions of rotor blade by VABS. The structure design configuration and the design variables of the rotor blade were shown in Figure 6.  
As shown in Figure 6, the design variables were thicknesses of D-spar (h\({}_{\text{D}}\)), skin (h\({}_{\text{skin}}\)), and web (h\({}_{\text{web}}\)), as well as the web location (d\({}_{\text{loc}}\)) and the web connection orientation (d\({}_{\text{ang}}\)). Both initial values and design range for the design variables were represented in Table 3.  
In the calculation of design performances, the chord length was considered 20.2 in., and the aerodynamic center was at the quarter chord. In the structure design problem of composite rotor blade, the distance between aerodynamic and shear centers should be minimized to avoid flutter instability of the rotor blade, and to avert the restrictions on the flight envelope. As design performance functions, the structural mass of rotor blade should be minimized while some important internal structural stiffness properties such as torsional (GJ), flap-wise bending (EI\({}_{22}\)) and chord-wise bending stiffnesses (EI\({}_{33}\)) were almost maintained at the level of the initial design, i.e., within \(\pm 5\%\) variations. In the present study, the RBFNN-based approximation was carried out for the design performance functions. With 500 training data and 90 testing data, the RBFNN based meta-models were fitted for each design performance function. The fitted RBFNN meta-models of rotor blade design are shown in Figures 7-11. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Variable** & **Lower Bound** & **Initial Design** & **Upper Bound** \\ \hline h\({}_{\text{D}}\) (in.) & 0.3 & 0.35 & 0.4 \\ h\({}_{\text{skin}}\) (in.) & 0.03 & 0.04 & 0.05 \\ h\({}_{\text{web}}\) (in.) & 0.01 & 0.04 & 0.05 \\ d\({}_{\text{loc}}\) (in.) & \(-9.0\) & \(-8.4\) & \(-7.0\) \\ d\({}_{\text{ang}}\) (deg.) & 60 & 90 & 120 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Initial value and design range in design variable. Figure 6: Structure design configuration and design variables of rotor blade. Figure 8: Meta-models for design variable—h\({}_{\text{skin}}\). Figure 7: Meta-models for design variable—h\({}_{\text{D}}\). Figure 10: Meta-models for design variable—d\({}_{\text{loc}}\). Figure 9: Meta-models for design variable—h\({}_{\text{web}}\). From the meta-modeling results, remarkable multi-modal variations were found in the relations between two design variables (\(\rm{h_{web}}\) and \(\rm{d_{ang}}\)) and the design performance functions. In other design variables such as the thicknesses of D-spar (\(\rm{h_{D}}\)), the skin (\(\rm{h_{skin}}\)) and the web location, their meta-modeling results represented almost linear characteristics. Statistical analyses were also performed to evaluate the main effects of design variables on the responses of design performance functions. The results of statistical analyses were shown in Figure 12.  
The main effects of \(\rm{h_{D}}\) and \(\rm{d_{loc}}\) among the design performance functions were higher than other design variables. The most sensitive design variable on the mass and the EI\({}_{22}\) was the \(\rm{h_{D}}\), and the main effect values on the mass and the EI\({}_{22}\) were 47% and 44%, respectively. The most sensitive design variable on the GJ and the EI\({}_{33}\) was the \(\rm{d_{loc}}\), and the main effect values on the GJ and the EI\({}_{33}\) were \(-52\%\) and \(-48\%\), respectively. The statistical results also showed that \(\rm{d_{loc}}\) and \(\rm{d_{ang}}\) had negative effects on all the design performance functions while other design variables had positive effects.