# Design and Optimization of Modular Solid Rocket Grain Matching Multi-Thrust Performance Curve
## 3 Results and Discussions
### Triple-Thrust Case  
Designing a triple-thrust grain configuration poses a significant challenge. While the three-segmented modular grain theoretically has the potential to achieve the desired outcome, there is an alternative approach to consider. As shown in Figure 16, in this case, we propose replacing the third segment with an end-burning grain. This substitution may offer a viable solution in achieving the desired triple-thrust configuration. The design objective is to generate a pressure of 10 MPa for a duration of 5 s, followed by a pressure of 6 MPa for 5 s, and then a pressure of 3 MPa for 10 s. In order to reduce the overall length of the grain, the allowable range for the grain radius has been adjusted from [0.05, 0.2] to [0.05, 0.25]. Since the third segment is replaced by an end-burning grain, Equation (23) needs to consider the burning of the third segment, that is,  
\[ \begin{split} Q_{1,\text{target}}&=\frac{A_{1}p_{ \text{c,1}}}{c*}-\rho_{\text{ed,3}}\Big{(}\frac{p_{\text{c,1}}}{P_{\text{Ref}} }\Big{)}^{n_{\text{pred,3}}}r_{\text{Ref, ed,3}}\tau R^{2},\\ Q_{2,\text{target}}&=\frac{A_{1}p_{\text{c,2}}}{c* }-\rho_{\text{ed,3}}\Big{(}\frac{p_{\text{c,2}}}{P_{\text{Ref}}}\Big{)}^{n_{ \text{pred,3}}}r_{\text{Ref, ed,3}}\tau R^{2}\end{split}  \tag{35}\]  
where \"ed\" means the end-burning grain, and the reference burning rate \(r_{\text{Ref, ed,3}}\) can be decided by the chamber pressure of the third phase, as follows:  
\[r_{\text{Ref, ed,3}}=\frac{A_{1}p_{\text{c,3}}}{c*\pi R^{2}\rho_{\text{ed,3}} r_{\text{Ref, ed,3}}\Big{(}\frac{p_{\text{c,1}}}{P_{\text{Ref}}}\Big{)}^{n_{\text{pred,3}}}} \tag{36}\]  
The length of the third segment is calculated by burning time, as follows:  
\[L_{\text{ed,3}}=\sum_{i=1}^{3}\biggl{(}\biggl{(}\frac{p_{\text{c,i}}}{p_{ \text{Ref}}}\biggr{)}^{n_{\text{pred,3}}}r_{\text{Ref, ed,3}}t_{\text{b,i}}\biggr{)} \tag{37}\]  
The other solution process is similar to the previous sections and will not be repeated here. The burning surface regression and the pressure-time curve are shown in Figures 17 and 18. Figure 16: Geometry of triple-thrust grain. The designed grain successfully forms the triple-thrust profile. During every phase, the relative maximum deviation between designed pressure and target pressure is 2.8%. The relative mean squared error (RMSE) between the two curves is 0.000758. The performance matching degree (\(r^{2}\)), calculated by Equation (34), is 0.9940 (>0.95). The propellant loading fraction is 87.88%, the throat-to-port ratio is 0.301, and the burning-to-port ratio is 43.8 (<100), suggesting that erosive burning is unlikely to occur for typical double-base propellants [22]. It is worth noting that the transient period between phases is shorter than 1 s and there is no tail-off period at the end of burning. In this case, three types of grain templates are used, including star, modified slot, and end-burning grain, demonstrating the flexibility of modular grain. Till now, we have examined the four features or advantages of modular grain that are highlighted in the Introduction section. 1. Simplicity of Basic Grain Shape Template: The basic template for grain shape is designed to be simple, thereby minimizing the overall complexity of the grain shape. The segmented grain technology is relatively mature, and the manufacturability of 2D grain configurations can be effectively addressed with existing techniques. In contrast, the design results of other existing grain optimization methods may be complicated and not suitable for real-world manufacturing. 2. Mathematical Assurance of Steadiness: The steadiness of the pressure platform is guaranteed by the governing equations. In contrast, other existing grain optimization methods cannot mathematically guarantee the steadiness of the results. 3. Maximization of Propellant Loading Fraction: The maximization of the propellant loading fraction is achieved by optimization. 4.