# Design and Optimization of Modular Solid Rocket Grain Matching Multi-Thrust Performance Curve
## 4 Conclusions  
This investigation explores the design and optimization of modular solid rocket grains matching multi-thrust performance curves. Several conclusions can be drawn from this study.  
1. The concept of modular grain is introduced, consisting of star grain, slot grain, and end-burning grain as the fundamental templates (with different materials or burning rates). By combining these templates, an arbitrary multi-thrust performance curve can be achieved mathematically.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Optimization Algorithm** & **Maximum Loading Fraction** & **Time Cost/s** \\ \hline Nelder–Mead & 87.88\% & 85.0 \\ \hline Simulated annealing & 87.20\% & 59.5 \\ \hline Random search & 80.15\% & 37.6 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Time cost of the triple-thrust case using different algorithms (the termination criterion is the change of the objective function value of less than \(10^{-3}\)).  
Figure 19: The von Mise stress of the modified slot grain with different burned web thicknesses (half-star angle only).  
2. A governing equation system with multiple algebraic equations suitable for modular grains is developed in order to match the desired multi-thrust performance curve.
3. Optimization techniques are utilized to determine the best configuration of the modular grains in order to achieve the highest propellant loading fraction. To achieve optimal results, it is essential that each segment have the same maximum web thickness, while also ensuring that the throat-to-port ratio reaches its upper limit. The propellant loading fraction for the cases of single-thrust, dual-thrust, and triple-thrust configurations reaches 81.37%, 81.37%, and 87.88% respectively.
4. Practical implementation of our method has successfully produced single-thrust, dual-thrust, and triple-thrust grains. These grains have demonstrated the ability to maintain steady pressure platforms, making them competitive alternatives to traditional multi-thrust grains.  
This theoretical work significantly simplifies the design of motor grains, especially for multi-thrust grains. By utilizing the combination of basic grain configurations, namely star, slot, and end-burning grains, complex performance curves can be mathematically guaranteed by governing equations. Future research should focus on three key aspects: (1) multi-objective optimization framework incorporating structural integrity, manufacturing constraints, and thrust stability trade-offs; (2) practical implementation of modular grain designs, focusing on manufacturability and structural integrity; and (3) the exploration of additional candidate grain (even 3D grain) configurations to serve as alternative grain templates.  
Conceptualization, W.L. and Y.H.; methodology, W.L. and Y.Z.; software, W.L. and Y.Z.; validation, W.L. and Y.Z.; formal analysis, W.L.; investigation, W.L.; resources, G.L.; data curation, W.L.; writing--original draft preparation, W.L.; writing--review and editing, Y.H. and G.L.; visualization, W.L.; supervision, Y.H. and G.L.; project administration, Y.H. and G.L.; funding acquisition, Y.H. and G.L. All authors have read and agreed to the published version of the manuscript.  
This research received no direct external fundings.  
Not applicable.  
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author(s).  
The authors declare no conflicts of interest.