# A Multi-Fidelity Successive Response Surface Method for Crashworthiness Optimization Problems
## 6 Results and Discussion
### Results of the Engineering Use Case  
In this section, we present the results of the crash box optimization, comparing the three approaches introduced so far: SRSM, GP-SRS, and MF-SRS. For the MF-SRS approach, we use a high-fidelity DoE consisting of 60 points, with an addition of four more high-fidelity points at each iteration. Since the \"cost\" of low-fidelity simulations is less than one-sixth of the high-fidelity ones, we set the 2:1 ratio that has shown great results so far. In contrast, the GP-SRS approach uses a slightly larger DoE of 70 points and adds five high-fidelity points to each iteration. The SRSM method also starts with a 70-point DoE and iteratively adds eight high-fidelity points at each iteration based on the linear polynomial criterion. To achieve faster and more robust convergence, at each iteration, we record the best feasible value using the criterion given in Formula (24) based on \(n_{ARFT}=4\). If this value does not improve, we retain the best feasible value achieved in prior iterations. If all designs at a specific iteration fail to meet the constraints, we mark the point in red. From the results shown in Figure 16, the performance of MF-SRS stands out. It is clearly the most effective approach, achieving a 13.1% and a 14.1% SEA improvement over GP-SRS and SRSM, respectively. Alongside the improved optimum, the fast convergence and the ability to accurately predict the feasibility limits of the domain are remarkable. This is particularly evident in the result of the first iteration where, despite the deficit of 10 high-fidelity points, MF-SRS still manages to identify an optimum that outperforms the other approaches. The performance of GP-SRS is solid, although with a slower progression than expected. We believe that with more relaxed convergence criteria, it may eventually surpass the maximum obtained by MF-SRS, albeit at a significantly higher computational cost. Conversely, SRSM appears to be the least robust approach. Despite the rapid improvement in the fourth iteration, the best-reported point often does not match with the verification point, which is a discrepancy that is curious considering the predictive potential offered by the eight new high-fidelity samples added at each iteration. From a computational point of view, SRSM seems to be the most \"expensive\" and, in terms of prediction, the most unreliable of the methods investigated. Further details of the optimal designs of response values and input variables are provided in Table 2 and Appendix C.4, respectively. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Method** & \(\mathbf{n_{iter}}\) & \(\mathbf{SEA}\) **(J/kg)** & \(\mathbf{F_{max}}\) **(kN)** & \(\mathbf{F_{in}}\) **(kN)** & \(\mathbf{EA_{B}}\) **(j)** & \(\mathbf{CFE}\) & \(\mathbf{ULC}\) \\ \hline SRSM & 19 & 8159.8 & 51.9 & 26.9 & 271.9 & 0.52 & 0.48 \\ GP-SRS & 17 & 8233.2 & 59.0 & 28.5 & 260.8 & 0.51 & 0.48 \\ MF-SRS & 15 & **9313.5** & 61.5 & 31.9 & 244.2 & 0.51 & 0.49 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the response functions of the optimal designs of each method. The best achievable SEA value among the three methods is highlighted in bold. Figure 15: High-fidelity model: 23,548 shell elements with element erosion (**a**) and low-fidelity model: 5940 shell elements without element erosion (**b**). Further comprehensive details about these models are provided in Appendix C.2. The iteration by iteration evolution of the MF-SRS method is shown in the parallel coordinates plot in Figure 17. The light gray lines represent designs that violate the constraints and are, therefore, unfeasible, while the blue lines denote feasible designs. It is clear that the density of the darker lines increases around the optimal design, which is shown in orange. This provides further evidence of the robust convergence of the MF-SRS method.