# A Multi-Fidelity Successive Response Surface Method for Crashworthiness Optimization Problems
## 7 Conclusions  
In this paper, we introduced the MF-SRS method, a novel optimization approach that leverages Gaussian processes to better handle the inherent complexity of crashworthiness problems.  
While the traditional SRSM is effective in some aspects, it has been found to have significant limitations, particularly in the prohibitive use of costly function evaluations and the shortcomings associated with the PSR method. The proposed MF-SRS method addresses these challenges by employing a multi-fidelity approach that captures in-depth information from both high-fidelity and low-fidelity models, wisely reuses information from previous iterations, and queries new data at unexplored locations via adaptive infill seed criteria. This combination results in a more robust and versatile framework compared to SRSM.  
Our experiments, ranging from benchmark functions to a real-world crashworthiness application, yielded promising results. MF-SRS consistently demonstrated improved performance over both SRSM and GP-SRS, despite the initial additional cost of low-fidelity evaluations. Specifically, it achieved a remarkable 14.1% improvement in the optimal value of specific energy absorption over SRSM, underscoring the stability and precision of the optimization process. This efficiency is even more pronounced when considering its ability to achieve optimal values with less computational burden. The extent of such burden depends on several factors, including available computational power and parallel job logic on an HPC. Moreover, in a scenario characterized by a low-fidelity model with a coarse mesh and no element erosion, MF-SRS successfully detected intricate relationships between different fidelities, reinforcing its effectiveness and efficiency in various application contexts.  
In our future research, we plan to primarily investigate the optimal ratio of high- and low-fidelity points per iteration and within the initial DoE, with the aim of adaptively adjusting these numbers based on the given problem. While NARGP stands out as a promising multi-fidelity approach, this rapidly evolving field introduces exciting alternatives at a rapid pace. Of particular interest are Multi-Fidelity Bayesian Neural Network approaches, which offer the potential to address uncertainties in sampling and to capture intricate correlations between fidelities thanks to the Neural Network framework. Finally, tailoring specific covariance functions relevant to certain iterations within the GP-SRS method offers another intriguing avenue of exploration.  
Figure 19: The job schedule of the multi-fidelity DoE (**top row**) and the convergence curves over the cumulative cost (**bottom row**) in the case of 5, 7, and 8 nodes available on the HPC.  
**Author Contributions:** Conceptualization, P.L. and R.S.; methodology, P.L.; software, P.L.; validation, P.L.; formal analysis, P.L., R.S., and T.S.; investigation, P.L.; resources, P.L. and R.S.; data curation, P.L.; writing--original draft preparation, P.L. and R.S.; writing--review and editing, R.S. and T.S.; visualization, P.L.; supervision, R.S. and T.S.; project administration, R.S.; funding acquisition, R.S. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Data Availability Statement:** The data used to support the findings of this study are available from the corresponding author upon request.  
**Acknowledgments:** We would like to express our sincere gratitude to [PERSON], [PERSON], and [PERSON] for their valuable contributions.  
**Conflicts of Interest:** The authors declare no conflict of interest.