## 3 Rotor Blade Design Approximation with RBFNN  
In the rotor blade design, it is important to minimize the distance between aerodynamic and shear centers in order to reduce both the flutter instability and the restrictions on the flight envelope. Also, the weight of rotor blade should be improved while the structural stiffness properties are almost maintained at the level of the baseline design. The design optimization problem of rotor blade can be formulated as follows:  
\[Mininize\ f=|\alpha-\gamma| \tag{9}\]  
\[subject\ to\ \frac{m}{m_{base}}\leq 1\]  
\[0.95\ \leq\ \frac{\text{GJ}}{\text{Gly}_{base}}\leq 1.05\]  
\[0.95\ \leq\ \frac{\text{El}_{22}}{\left(\text{El}_{22}\right)_{base}}\leq 1.05\]  
\[0.95\ \leq\ \frac{\text{El}_{33}}{\left(\text{El}_{33}\right)_{base}}\leq 1.05\]  
where \(\alpha\) is the aerodynamic center, and \(\gamma\) is the shear center. The properties GJ, EI\({}_{22}\) and EI\({}_{33}\) are torsional, flap-wise bending, and chord-wise bending stiffnesses, respectively. The notation \(m\) is mass per unit length. The subscript _base_ denotes the value representing the baseline model. An application of optimization methods to the structural design problems of rotor blade often requires a large amount of numerical costs. An alternative for reducing the numerical cost in the process of design optimization is to employ the meta-model by establishing the accurate levels of approximate objective and constraint functions over the exploratory design space. Using rationally constructed meta-models, near-optimal approximate solutions can be obtained in addition to the estimation of design behaviors, that is, the sensitivity-like information. The practical design problem of rotor blade was chosen to investigate the characteristics of the RBFNN approximation with respect to the learning parameters such as the range of spline smoothing parameter, the number of training data, and the number of hidden layers. The RBFNN based meta-models were generated for the rotor blade design problem. The architecture of RBFNN consisted of one to three layers in the hidden layers to evaluate the effect of the number of hidden layers on the performance of RBFNN. The number of neurons in that layer was defined as the mean of the neurons in the input and output layers referring to the related research book [28]. Each neural network in the RBFNN was trained by back-propagation algorithm which is one of the most reliable algorithms having such characteristics as the simplicity of implementation and the ability to quickly generate networks that have the capability to generalize [28]. Error analyses were carried out to evaluate the variation of performance according to the variation learning parameters. In order to complete the RBFNN meta-modeling for the rotor blade design problem, the calculation time was 8.3 h using a server equipped with six core Intel Xeon CPUs and 128 GB RAM. In order to investigate the characteristics of the RBFNN approximation with respect to the learning parameters for the practical design problem with a number of performance functions, the design problem of composite rotor blade was explored [32]. The composite rotor blade was designed to use for the tilt rotor aircraft combines the advantages of vertical takeoff and landing capabilities. The design performance functions of rotor blade were calculated a rotor blade cross-sectional modeling and analysis program, which is Variational Asymptotic Beam Section (VABS) [33] developing for analyzing the composite blade structure using an equivalent beam model. In the VABS program, the cross-sectional stiffness matrix is calculated from a two-dimensional finite element discretization. In this study, the VABS was used for obtaining both training and testing data for use in RBFNN based meta-modeling.