# A General Integrated Method for Design Analysis and Optimization of Missile Structure
## 2 Theory
### Augmented Fourier Series-Based Polynomial Surrogate Model  
Generally speaking, the optimization process needs to call the parametric modeling module and analysis module several times, which directly obstructs the implementation of the optimization strategy due to the vast amount of computation. In recent years, the surrogate model has been proven to be a useful technique according to its excellent computational efficiency. The system response, approximated as a simple explicit function, can be easily derived without executing tedious computation. In this section, a novel Augmented Fourier series-based polynomial (AFP) surrogate model will be constructed to replace the original finite element calculation [19]. Figure 1: The CST parametrization of an NACA0006 airfoil and the corresponding fitting residual with BPO = 5: **(a)** The airfoil geometric shape and **(b)** the residual differences. Figure 2: The CST parametrization of an NACA4412 airfoil and the corresponding fitting residual with BPO = 5: **(a)** The airfoil geometric shape and **(b)** the residual differences. \[T_{i_{1},i_{2},\cdots,i_{s}}(x_{1},x_{2},\cdots,x_{s}) =T_{i_{1},i_{2},\cdots,i_{s}}(\theta_{1},\theta_{2},\cdots,\theta_{s})\] \[=T_{i_{1}}(x_{1})T_{i_{2}}(x_{2})\cdots T_{i_{s}}(x_{s})\] \[=T_{i_{1}}(\theta_{1})T_{i_{2}}(\theta_{2})\cdots T_{i_{s}}( \theta_{s})\]  
where \(m_{i}(i=1,2,\cdots,s)\) is the number of interpolation points for the corresponding variable. Based on the theory of Fourier series polynomials approximation in the previous subsection, we can construct a Fourier series-based polynomials surrogate model:  
\[g(x_{1},\cdots,x_{s})\approx\sum_{i_{1}=0}^{2n_{1}}\cdots\sum_{i_{s}=0}^{2n_{s }}\left(\frac{1}{2}\right)^{s}g_{i_{1},\cdots,i_{s}}T_{i_{1},\cdots,i_{s}}(x_{ 1},\cdots,x_{s})=\beta^{T}\alpha \tag{18}\]  
where  
\[\beta=[\beta_{1},\cdots,\beta_{m}]^{T}=(1/2)^{s}[g_{0,\cdots,0},\cdots,g_{i_{ 1},\cdots,i_{s}},\cdots,g_{2n_{1},\cdots,2n_{s}}]^{T}\]  
\[\alpha=[\alpha_{1},\cdots,\alpha_{m}]^{T}=[T_{0,\cdots,0}(x),\cdots,T_{i_{1}, \cdots,i_{s}}(x),\cdots,T_{2n_{1},\cdots,2n_{s}}(x)]^{T}\]  
then  
\[\left[ \begin{array}{ccc}\alpha_{1}(x_{1})&\cdots&\alpha_{m}(x_{1})\\ \vdots&\ddots&\vdots\\ \alpha_{1}(x_{p})&\cdots&\alpha_{m}(x_{p})\end{array} \right]\left[ \begin{array} []{c}\beta_{1}\\ \vdots\\ \beta_{m}\end{array} \right]=\left[ \begin{array}{c}g(x_{1})\\ \vdots\\ g(x_{p})\end{array} \right] \tag{19}\]  
where \(x_{1},\cdots,x_{p}\) denotes the sample vector. From Equation (19), the corresponding coefficient vector can be calculated numerically using the Gauss elimination method. Though the approximation accuracy of the higher order polynomial-based Fourier polynomial model is more suitable for strong non-linear problems than the traditional regression polynomial models (TRP) [14; 21], especially for periodic problems, it is inappropriate for linear responses. Therefore, in order to make the Fourier series-based polynomial model suitable for linear response problems, the model should be augmented with a linear polynomial, given as [22]:  
\[g_{a}(x)\approx\sum_{i_{1}=0}^{2n_{1}}\cdots\sum_{i_{s}=0}^{2n_{s}}\left( \frac{1}{2}\right)^{s}g_{i_{1},\cdots,i_{s}}T_{i_{1},\cdots,i_{s}}(x)+\sum_{j =1}^{q}c_{j}f_{j}(x), \tag{20}\]  
where \(f(x)\) is a linear polynomial function, \(q\) is the number of terms in the polynomial, and \(c_{j}\) represent the unknown coefficients of the linear polynomial. It should be noted that the default order of the linear polynomial was set as 1 in this study. Correspondingly, we can derive a group of linear equations with respect to the unknown expansion coefficients:  
\[ \begin{bmatrix}\alpha_{1}(x_{1})&\cdots&\alpha_{m}(x_{1})&f_{1}(x_{1})&\cdots &f_{q}(x_{1})\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ \alpha_{1}\Big{(}x_{p})&\cdots&\alpha_{m}\Big{(}x_{p})&f_{1}\Big{(}x_{p})& \cdots&f_{q}\Big{(}x_{p}\Big{)}\end{bmatrix} \left( \begin{array}{c}\beta\\ c\end{array} \right)=\left[ \begin{array}{c}g(x_{1})\\ \vdots\\ g(x_{p})\end{array}