# Design and Optimization of Modular Solid Rocket Grain Matching Multi-Thrust Performance Curve
## 3 Results and Discussions
### Single-Thrust Case
The calculated \(Po\) value of 60.2 (well below the critical threshold of 100) suggests that erosive burning is unlikely to occur for typical double-base propellants [22]. Notably, Table 3 reveals an interesting observation that the burning rate and web thickness of star grains and slot grains are nearly identical. This suggests that both types of grains burn out simultaneously, making a more stable chamber pressure and a shorter tail-off period. To validate the chamber pressure obtained, the Poisson equation--Eikonal equation--finite element (PEF) method is employed. This method involves solving a Poisson equation using the finite element method to approximate the viscous solution of the Eikonal equation. By transforming the burn-back analysis problem into a specialized stationary thermal conduction problem, the PEF method can be implemented by COMSOL software (version 5.6). It is particularly useful for analyzing the burn-back of three-dimensional grains with irregular geometry and a complex burning rate distribution. The internal ballistics model assumes a zero-dimensional quasi-steady state, and further details can be found in [PERSON]'s work [20]. The burning surface regression is shown in Figure 7, and the pressure-time curve is shown in Figure 8.  
Figure 8: Internal ballistic of the single-thrust grain. Figure 7: Burning surface regression of the single-thrust grain. The obtained results demonstrate a good match between the chamber pressure and the desired target, with a maximum deviation of 3.3% during the pressure platform. The relative mean squared error (RMSE) between two curves is 0.003685. Notably, both the star grain and slot grain exhibit a simultaneous burnout, resulting in a shorter tail-off. The performance matching degree can also be given by linear regression as follows [14]:  
\[r^{2}=1-\frac{\left(\mathbf{p}_{\text{T}}-k\mathbf{p}_{\text{D}}\right)\cdot\left(\mathbf{p }_{\text{T}}-k\mathbf{p}_{\text{D}}\right)}{\left(\mathbf{p}_{\text{T}}-\mathbf{p}_{\text{ T}}\right)\cdot\left(\mathbf{p}_{\text{T}}-\mathbf{p}_{\text{T}}\right)},\;k=\frac{\mathbf{p}_{ \text{D}}\cdot\mathbf{p}_{\text{T}}}{\mathbf{p}_{\text{D}}\cdot\mathbf{p}_{\text{D}}} \tag{34}\]  
where \(\mathbf{p}_{\text{D}}\) and \(\mathbf{p}_{\text{T}}\) are the discretized form of designed pressure-time curve and target pressure-time curve, respectively. \(-\)\(\mathbf{p}_{\text{T}}\) is a vector consisting of the mean value of \(\mathbf{p}_{\text{T}}\). Note that \(r^{2}\) reflects an important aspect of the performance-matching degree. The closer \(r^{2}\) is to 1, the more accurately the curve shape matches the target. In engineering practice, an \(r^{2}>0.95\) demonstrates that the design captures 95% of pressure fluctuations, representing an accurate match to the target performance curve. In our single-thrust case, \(r^{2}\) is 0.9764 (>0.95), suggesting a very accurate match to the target performance curve. The parametric sensitivity analysis is summarized in Table 4, where each parameter is increased by 1% to evaluate its effect on RMSE and \(r^{2}\). The results clearly indicate that \(R\), \(l_{j}\), \(L_{\text{st},j}\), and \(r_{\text{Ref, st},j}\) are the most influential parameters affecting the performance curve. These key parameters should be carefully controlled during the manufacturing process to ensure optimal performance. In contrast, the remaining parameters show a minor impact on the results. It is worth noting that the slot-grain design presents certain manufacturing challenges due to the separation of the grain into multiple independent parts by the slots. To address this issue, engineering considerations suggest that the slots should not completely penetrate the radius of the grain. Figure 9 illustrates the modified geometry of the slot grain.