Comparison Between Single And Multifidelity Algorithm

165

8.12. Comparison Between Single and Multifidelity

166

In Fig. 62 a comparison is presented between the value of the objective function assumed for each sample in the case of MFEI and SFEI algorithms. In particular, the performance of the SFEI algorithm shows, in the first calculations performed, an exploration phase characterized by a strong variability of the objective function.

However, with increasing iterations the exploitation towards the minimum shows a reduced fluctuation of the objective.

This is proven by the trends of the global minima with the number of iterations. The SFEI algorithm shows in Fig. 63 a decreasing trend that is not excessively accentuated, if compared to the trend of the MFEI algorithm. However, as in the multifidelity case, it can be assumed that by increasing the number of iterations, the minimum can further decrease.

The sampling carried out in the iterations are shown in Fig. 64. The SFEI algorithm shows a sampling characterized by a greater variability of the thickness of the TPS.

Furthermore, the values themselves are also significantly higher. This can be interpreted by the fact that the low fidelity aerothermodynamic model has in output higher thermal loads than the high-fidelity model. To cope with the higher thermal loads, the thickness of the TPS must necessarily increase. In Fig. 65-66 what has just been said can be noted. Although both the mass of the TPS and the structural temperature have an average decreasing trend, the values reached are significantly higher than the results obtained with the multifidelity algorithm. In Fig. 67 the propellant mass computed performing the SFEI algorithm shows the same trend.

In Fig.68 are reported the outcomes of the trajectory model to characterize the optimum re-entry profile.

The comparisons between the single fidelity and multifidelity trends of pressure, heat flux and gas side wall temperature at the point of stagnation are shown in Fig.

69-70-71 respectively. Focusing on the heat flux, the single fidelity result shows a peak of 6.333 ∙ 10⁵ 𝑊/𝑚² at the altitude of 5.771 ∙ 10⁴ 𝑚 while the multifidelity output shows a maximum heat flux of 5.541 ∙ 10⁵ 𝑊/𝑚² at the altitude of 5.901 ∙ 10⁴ m. Considering the wall temperature, on average the values obtained with the SFEI algorithm are greater than the one obtained with MFEI strategy, confirming what has been said earlier about the single fidelity optimization result. The same result can be observed in the stagnation point pressure trend.

In conclusion, the single fidelity algorithm leverages effectively the low-fidelity aerothermodynamic models, allowing design exploration to be carried out in a short time given the limited calculation expense. However, the global minimum value of the objective function is above 1 (mean reference value for atmospheric re-entry vehicles). This suggests that aerothermodynamic physical models are conservative, because tend to overestimate thermodynamic variables such as heat flux or wall temperature. Consequently, the optimum design obtained with the SFEI algorithm overestimates the structural mass, the temperature of the TPS and the mass of

167

propellant burned. By contrast, the multifidelity algorithm is computationally less efficient given the much longer computation times. However, the information coming from high fidelity numerical simulations allows to enrich the knowledge of the objective function, leading to a better optimal design.

Some considerations can be made on the implementation of a SFEI algorithm with high fidelity model. In this work, suitable tests were not done due to a high time required and computational cost. However, some considerations can be made. The model that allows the computation of the trajectory has been set in order to have 23 points in the range of heights of interest for the problem. Consequently, an iteration of the SFEI or MFEI algorithm involves evaluating the aerothermodynamic model 23 times. Based on this consideration and the computational performance of the MFEI algorithm known, the time required to perform 100 iterations by implementing the multifidelity algorithm corresponds to approximately the time required to perform 6 iterations by implementing the single high-fidelity model.

Therefore, two important implications can be outlined: the multifidelity algorithm is computationally more efficient than the single high-fidelity algorithm and allows to improve the solution obtained with the single low-fidelity algorithm; While it can be said with some confidence that 6 single high-fidelity iterations are not enough to achieve convergence, nothing can be said about the level of accuracy achievable by performing 100 iterations with the high-fidelity model.

168

Figure 62: 𝑓(𝑥)-iterations. The graph above is the result of the MFEI algorithm.

The lower one of the SFEI algorithm.

169

Figure 63: 𝑓(𝑥) overall minimum-iterations. The graph above is the result of the MFEI algorithm. The lower one of the SFEI algorithm.

170

Figure 64: sample points selected during iterations. In red is reported the sample corresponding to the overall minimum of the objective function. The graph above is the result of the MFEI algorithm. The lower one of the SFEI algorithm.

171

Figure 65: TPS mass-iterations. The graph above is the result of the MFEI algorithm. The lower one of the SFEI algorithm.

172

Figure 66: TPS maximum temperature-iterations. The graph above is the result of the MFEI algorithm. The lower one of the SFEI algorithm.

173

Figure 67: Propellant mass-iterations. The graph above is the result of the MFEI algorithm. The lower one of the SFEI algorithm.

174

Figure 68: trajectory model outcomes.

175

Figure 69: Stagnation point pressure-altitude. The graph above is the result of the MFEI algorithm. The lower one of the SFEI algorithm.

176

Figure 70: Stagnation point heat flux-altitude. The graph above is the result of the MFEI algorithm. The lower one of the SFEI algorithm.

177

Figure 71: Stagnation point wall temperature gas side-altitude. The graph above is the result of the MFEI algorithm. The lower one of the SFEI algorithm

178

179