## 3 Application
### Optimization  
In the experiment design, a full factorial design was used. When this method is applied, equally spaced design points were distributed in the design space and the distance between the neighbor design points was set to 0.25 in both the \(\omega_{1}\) and \(\omega_{2}\) directions. This allowed us to distribute a total of 19 design points in the experiment design step (see Figure 8). At each design point, the morphed BWB aircraft was created with the flattening-based morphing method, and the CFD simulation was performed to compute its lift-to-drag ratio. After all calculations were finished, a surrogate model computing the relation between the lift-to-drag ratio and weights was constructed using an ANN. When constructing the ANN, all 19 simulations performed in the experiment design step were used as training data. To evaluate the accuracy of the ANN with data that the ANN have never seen before, four additional simulations were performed at arbitrarily chosen design parameter' sets (which are \((\omega_{1}\), \(\omega_{2})=(0.08,0.15)\), \((0.12,0.84)\), \((0.32,0.41)\), and \((0.66,0.29)\)), and these simulation results were used as the test data. The ANN was trained to minimize the cost function \(J\), which is defined as  
\[J=\sum_{i=1}^{n}\left[\left(\
icefrac{{L}}{{D}}\right)_{i,\,ANN}-\left( \
icefrac{{L}}{{D}}\right)_{i,\,CFD}\right]^{2}+\lambda||\mathbf{W}||_{2}, \tag{10}\]  
where \(\left(\
icefrac{{L}}{{D}}\right)_{i,\,ANN}\) and \(\left(\
icefrac{{L}}{{D}}\right)_{i,\,CFD}\) are the \(i\)-th lift-to-drag ratios computed from ANN and CFD, respectively, \(\lambda\) is the regularization constant, and \(\mathbf{W}\) is the vector of the neural network connectivity weights. The multi-layer perceptron approach is used to construct the neural network. The Relu function is used as the activation function, and the Adam optimizer is applied as the optimization algorithm. The regularization constant was set to \(10^{-3}\), and the  
Figure 7: Several BWB aircraft created with the flattening-based morphing method. The weights of each morphed BWB aircraft are presented in the caption. The weights of the aircraft plotted in (**a–d**) exist in the current study’s design space, but the ones plotted in (**e,f**) are located outside of the design space. ANN with two hidden layers, in which each hidden layer consists of 32 neurons, was used in the optimization process. The way these ANN-related parameters were determined is described in Section 4. After the surrogate model was constructed, an optimal set of weights that maximized the lift-to-drag ratio was searched for. As an optimization algorithm, a genetic algorithm that stochastically finds an optimal solution was used [40; 53; 54]. When solving the optimization problem, two geometrical constraints can be considered because these are needed to secure proper loading capacity of a new BWB aircraft. To ensure proper loading capacity, a certain amount of centerbody's volume should be maintained for the morphed BWB aircraft. In the current study, proper volume of the centerbody was assumed to be secured by defining the lower limits of the surface area of the centerbody and cross-sectional area at the symmetry plane. With these two constraints, the optimization problem handled in the current study can be stated as  
\[ \begin{split}\underset{\omega_{1},\,\omega_{2}}{\text{maximize}}& L/D\\ \text{subject to}& A_{c}\geqq\gamma A_{c,ref}\\ & A_{s}\geqq\gamma A_{s,ref}\end{split}  \tag{11}\]  
where \(A_{c}\) and \(A_{c,ref}\) are the surface area of the centerbody of the morphed and reference aircraft, respectively, \(A_{s}\) and \(A_{s,ref}\) are the cross-sectional areas of the morphed and reference aircraft on the symmetry plane, respectively, and \(\gamma\) is the constant that determines the relative loading capacity of the morphed aircraft compared with the reference aircraft.