# Design and Optimization of Modular Solid Rocket Grain Matching Multi-Thrust Performance Curve
## 2 Design Methods
### Optimization Approach  
For a simple design problem, the above approach is enough. However, in order to obtain better results for a given objective and constraints, an optimization loop should be introduced. We suggest optimizing \(\varepsilon_{j}\), \(\theta_{j}\), \(n_{\text{st},j}\), \(h_{i}\), \(n_{\text{st},j}\) and \(R\) to maximize the total propellant loading fraction and avoid erosive burning by giving an upper limit of the throat-port ratio \(J_{\text{max}}\), that is,  
\[ \begin{array}{l}\max_{\varepsilon_{j}\theta_{j},n_{\text{st},j},n_{\text{st}, j},R}\eta_{\text{total}}\\ \text{st.}\quad J\leq J_{\text{max}},\,R_{\text{min}}\leq R\,\leq R_{\text{max}} \\ \quad\varepsilon_{j,\text{min}}\leq\varepsilon_{j}\leq\varepsilon_{j,\text{ max}},\,\theta_{j,\text{min}}\leq\theta_{j}\leq\theta_{j,\text{max}},\,n_{\text{st},j,\text{min}}\leq n _{\text{st},j}\leq n_{\text{st},j\text{max}},n_{\text{st},j}\in\mathbb{Z}\\ \quad h_{j,\text{min}}\leq h_{j}\leq h_{j,\text{max}},\,n_{\text{st},j,\text{ min}}\leq n_{\text{st},j}\leq n_{\text{st},j,\text{max}},n_{\text{st},j}\in\mathbb{Z} \end{array}  \tag{26}\]  
The port area of a single star grain is calculated as follows:  
\[A_{\text{p,st}}=2n_{\text{st}}\Bigg{(} \begin{array}{l}\frac{1}{2}l^{2}(1- \varepsilon)\beta_{\text{st}}+\frac{1}{2}\Big{(}\cos(\varepsilon\beta_{\text{ st}})-\sin(\varepsilon\beta_{\text{st}})\cot\Big{(}\frac{\theta}{2}\Big{)} \Big{)}l^{2}\sin(\varepsilon\beta_{\text{st}})\\ +lr\Big{(}\frac{\sin(\varepsilon\beta)}{\sin\Big{(}\frac{\varepsilon}{2}\Big{)} }+(1-\varepsilon)\beta_{\text{st}}\Big{)}+\frac{1}{2}r^{2}\Big{(}\frac{\pi}{ 2}+\beta_{\text{st}}-\frac{\theta}{2}-\cot\Big{[}\frac{\theta}{2}\Big{]} \Big{)}\end{array} \Bigg{)} \tag{27}\]  
and the port area of a slot grain can be integrated as follows:  
\[A_{\text{p,sl}}=n_{\text{sl}}\bigg{(}R^{2}\text{arctan}\bigg{(}\frac{h}{\sqrt {R^{2}-h^{2}}}\bigg{)}-h^{2}\cot\beta_{\text{sl}}+h\sqrt{R^{2}-h^{2}}\bigg{)} \tag{28}\]  
Therefore, the total propellant loading fraction can be calculated by  
\[\eta_{\text{total}}=1-\frac{\sum\limits_{j=1}^{N}\Big{(}A_{\text{p,st},j}L_{ \text{st},j}+A_{\text{p,sl},j}L_{\text{sl},j}\Big{)}}{\pi R^{2}\sum\limits_{j= 1}^{N}\big{(}L_{\text{st},j}+L_{\text{sl},j}\big{)}} \tag{29}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Items** & **Type \({}^{*}\)** \\ \hline \multirow{8}{*}{Star grains} & \(r_{\text{Ref, st},j}\) & Needs solving \\ \cline{2-3}  & \(L_{\text{st},j}\) & Needs solving \\ \cline{2-3}  & \(l_{j}\) & Needs solving \\ \cline{2-3}  & \(\varepsilon_{j}\) & Given by designer \\ \cline{2-3}  & \(\theta_{j}\) & Given by designer \\ \cline{2-3}  & \(r_{j}\) & Given by designer \\ \cline{2-3}  & \(n_{\text{st},j}\) & Given by designer \\ \cline{2-3}  & \(n_{\text{p, st},j}\) & Given by designer \\ \hline \multirow{8}{*}{Slot grains} & \(r_{\text{Ref, sl},j}\) & Needs solving \\ \cline{2-3}  & \(L_{\text{sl},j}\) & Needs solving \\ \cline{2-3}  & \(h_{i}\) & Given by designer \\ \cline{2-3}  & \(n_{\text{sl},j}\) & Given by designer \\ \cline{2-3}  & \(n_{\text{p, sl},j}\) & Given by designer \\ \hline Common & \(R\) & Given by designer \\ \hline \hline \end{tabular} \({}^{*}\)Needs solving\({}^{*}\) means the variable should be solved by the governing equations. \({}^{*}\)Given by designer\({}^{*}\) means the variable should be provided by the designer or an optimizer.  
\end{table} 
Table 1: Unknowns in the governing equations. The throat-to-port ratio can be calculated by  
\[J=\frac{A_{\rm t}}{\min_{j=1,2\cdots N}\left(A_{\rm p,st,j},A_{\rm p,st,j}\right)} \tag{30}\]  
Since \(J\leq J_{\rm max}\) may be a nonlinear constraint, the best practice is to add a penalty to the objective function, and Equation (26) becomes  
\[