## 3 Sensitivity Analysis of Aerodynamic Characteristics  
In this section, the Sobol sensitivity method based on an ARBF surrogate model [21; 26; 27] is adopted to analyze the influence of design parameters on the aerodynamic characteristics. The radial basis function (RBF) [26; 27] uses a forward neural network model with local approximation capability to express the relationship between design variables (the radius, length, and centroid position of the cylindrical projectile, the inlet flow field velocity, and the initial angle of attack) \(\mathbf{d}=\left[d_{1},d_{2},\ldots,d_{N_{D}}\right]^{\mathrm{T}}\) and output responses (the lift coefficient, drag coefficient, dynamic derivative, etc.) \(r(\mathbf{d})\), that is  
\[r(\mathbf{d})=\widetilde{r}(\mathbf{d})+\varepsilon=\sum_{i=1}^{N_{S}}\beta_{ i}\phi_{i}(\left\|\mathbf{d}-\mathbf{d}_{S}^{(i)}\right\|)+\varepsilon \tag{14}\]  
where \(\widetilde{r}(\mathbf{d})\) is the approximate value of the response \(r(\mathbf{d})\); \(\phi_{i}(\left\|\mathbf{d}-\mathbf{d}_{S}^{(i)}\right\|)\) is the basis function of the design variable \(\mathbf{d}\); \(\beta_{i}(i=1,\cdots,N_{S})\) is the undetermined coefficient; \(\varepsilon\) is the error term; \(N_{D}\) and \(N_{S}\) are the numbers of design variables and basis functions, respectively; and \(\left\|\cdot\right\|\) is the Euclidean norm. Commonly used basis functions include thin-plate spline functions, Gaussian functions, multi-quadric functions, etc., which determine the properties of RBF model. Taking the Gaussian function as an example  
\[\phi_{i}(\left\|\mathbf{d}-\mathbf{d}_{S}^{(i)}\right\|)=\exp\left(-\frac{ \left(\mathbf{d}-\mathbf{d}_{S}^{(i)}\right)^{\mathrm{T}}\left(\mathbf{d}- \mathbf{d}_{S}^{(i)}\right)}{2\delta_{l}^{2}}\right) \tag{15}\]  
\(\delta_{l}\) is the normalization parameter of the Gaussian function of the \(i\)th hidden layers. Considering that RBF model has high fitting accuracy in nonlinear problems but poor performance in linear problems, this paper adopts an ARBF model [26; 27], which adds linear polynomials to the original model, that is  
\[\widetilde{r}(\mathbf{d})=\sum_{i=1}^{N_{S}}\beta_{i}\phi_{i}(\left\|\mathbf{ d}-\mathbf{d}_{S}^{(i)}\right\|)+\sum_{j=1}^{N_{D}}b_{j}f_{j}(\mathbf{d}) \tag{16}\]  
where \(f_{j}(\mathbf{d})\) is the polynomial function of the design variable  
By introducing the orthogonality condition [26; 27]  
\[\sum_{i=1}^{N_{i}}\beta_{i}f_{j}(\mathbf{d}_{S}^{(i)})=0\ (j=1,2,\cdots,N_{P}) \tag{19}\]  
The following matrix form can be obtained  
\[ \begin{bmatrix}\mathbf{\Phi}_{S}&\mathbf{F}_{S}\\ \mathbf{F}_{S}^{\mathrm{T}}&\mathbf{O}\end{bmatrix}\begin{bmatrix}\mathbf{\hat {p}}\\ \mathbf{b}\end{bmatrix} = \begin{bmatrix}\mathbf{r}_{S}\\ \mathbf{0}\end{bmatrix}  \tag{20}\]  
By solving the above equation, one can obtain the unknown coefficients of Equation (17). It must be pointed out that the sampling strategy of the surrogate model determines the spatial distribution of sampling points, and a reasonable number of sampling points and their spatial distribution will have a crucial impact on the accuracy of the surrogate model. In this study, the LH method [21; 27] is used for obtaining the sampling point \(\mathbf{d}_{S}^{(i)}=\Big{[}d_{1}^{(i)},d_{2}^{(i)},\cdots,d_{N_{D}}^{(i)} \Big{]}^{\mathrm{T}}(i=1,2,\cdots,N_{S})\), and the corresponding output response \(r(\mathbf{d}_{S}^{(i)})\) of each \(\mathbf{d}_{S}^{(i)}\) is simulated by the method in Section 2. Furthermore, based on the cross-validation method, the average relative error (MRE) and the complex correlation coefficient (\(R\) Square, \(R^{2}\)) are used to evaluate the accuracy of surrogate model [21; 27]. Based on the surrogate model of aerodynamic characteristics, the Sobol global sensitivity method [21; 27] is further applied to obtain the sensitivity of design parameters to aerodynamic characteristics. The basic principle of the Sobol method can be found in reference [21; 27], which will not be elaborated here.