# Design and Optimization of Modular Solid Rocket Grain Matching Multi-Thrust Performance Curve
## 3 Results and Discussions
### Dual-Thrust Case  
The use of a dual-thrust grain in missiles is primarily aimed at extending their flight range. This configuration consists of a boost stage and an endurance stage. While there are various methods to design a dual-thrust grain, ensuring the stability of pressure platforms is often a challenging task. In this regard, the utilization of modular grain offers a better solution. The design objective for this particular case is to achieve a pressure of 10 MPa for a duration of 5 s, followed by a pressure of 5 MPa for an additional 5 s. The throat area of the nozzle remains constant at 0.007143 m\({}^{2}\), and the mass flow rate during each phase can be calculated using Equation (23). The optimization problem is subject to certain constraints, as outlined in Table 2, where the number of segments \(N\) is 2. To be simple, we assume that different grain segments share the same geometry parameters, for example, \(\epsilon_{1}=\epsilon_{2}\). The optimization process is conducted by the Nelder-Mead algorithm. Interestingly, it is observed that the majority of the geometry parameters remain the same as those presented in Table 3. The propellant loading fraction is 81.37%, the throat-to-port ratio is 0.306 and the burning-to-port ratio is 90.3 (<100), suggesting that erosive burning is unlikely to occur for typical double-base propellants [22]. However, there are differences in the burning rate and grain length, as indicated in Table 5.  
One possible explanation is that when a fixed grain radius and a specific throat-to-port ratio are considered, the optimal solutions remain the same. It means that Table 3 provides the optimal configuration for the modular grain, which proves to be suitable not only for single-thrust scenarios but also for dual-thrust and even triple-thrust applications. Using the PEF method, the burning surface regression and the pressure-time curve can be obtained and shown in Figures 12 and 13. During every phase, the relative maximum deviation between designed pressure and target pressure is 3.0%. The relative mean squared error (RMSE) between two curves is 0.001588. The performance matching degree (\(r^{2}\)), calculated by Equation (34), is 0.9897 (>0.95), indicating a very accurate match to the target performance curve. Due to the engineering considerations mentioned earlier, a modification has been made to the slot grain design. The resulting burning surface regression and pressure-time curve can be observed in Figures 14 and 15, respectively. These graphical representations provide insights into the effects of the modified slot on the combustion process. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Items** & **Type** & **Unit** \\ \hline \multirow{4}{*}{Star grains} & \(r_{\text{Ref, st,1}}\) & 0.0188853 & m/s \\ \cline{2-4}  & \(L_{\text{st,1}}\) & 0.450059 & m \\ \cline{2-4}  & \(r_{\text{Ref, st,2}}\) & 0.0104209 & m/s \\ \cline{2-4}  & \(L_{\text{st,2}}\) & 1.306028 & m \\ \hline \multirow{4}{*}{Slot grains} & \(r_{\text{Ref, sl,1}}\) & 0.0188972 & m/s \\ \cline{2-4}  & \(L_{\text{sl,1}}\) & 0.361296 & m \\ \cline{2-4}  & \(r_{\text{Ref, sl,2}}\) & 0.010427 & m/s \\ \cline{2-4}  & \(L_{\text{sl,2}}\) & 0.40305 & m \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Optimal result of dual-thrust grain. Figure 14: Burning surface regression of the dual-thrust grain (modified slot grain). Figure 12: Burning surface regression of the dual-thrust grain. Figure 13: Internal ballistic of the dual-thrust grain. Figure 15: Internal ballistic of the dual-thrust grain (modified slot grain). Based on the obtained results, it is evident that the maximum deviation between the design outcome and the target is 5.9%. The relative mean squared error (RMSE) between the two curves is 0.001179. The performance matching degree (\(r^{2}\)), calculated by Equation (34), is 0.9913 (>0.95). The modification made to the slot-grain design not only ensures its manufacturability but also enhances the initial pressure platform, resulting in a better match with the target profile.