# Frechet-Derivative-Based Global Sensitivity Analysis and Its Physical Meanings in Structural Design
## 6 Conclusions  
In this work, three classical global sensitivity indices (GSIs), i.e., variance-based GSI, moment-independent GSI, and failure-probability-based GSI, are revisited, and some properties of a newly developed Frechet-derivative-based GSI (Fre-GSI) are studied. A functional perspective of the stochastic parametric model of complex systems is proposed for revealing the physical meanings of the four GSIs in a unified description. Some practical links of the Fre-GSI with the other three GSIs are derived. Test examples are studied to verify the proposed links, and specific performances of the four GSIs are investigated as well. The main findings and conclusions are as follows:  
1. The importance measure and direction indicator are obtained from the Fre-GSI, whose basic properties are studied to illustrate the ability of Fre-GSI for global sensitivity analysis.
2. The four GSIs can be clearly comprehended through a functional description, wherein the distinctions among the four GSIs stem from the definitions of stochastic distances computed by different operators.
3. Some practical links of the Fre-GSI with other GSIs are analytically derived and verified numerically.
4. The complementary nature of the four GSIs is revealed. The variance-based GSI can effectively reflect the model structure, while the moment-independent GSI can reflect the uncertainty in distribution. The failure-probability-based GSI is useful for quantifying the impact of different failure criteria, while the Fre-GSI has robust capabilities on the importance measure and the direction of sensitivity.  
It is noteworthy to mention that while the selected examples are of a simplistic nature, they can serve as effective benchmarks for readers seeking a concise comprehension of the basic features of the four GSIs. Open issues remain to be studied in the future, e.g., the stronger connections between the Fre-GSI and the other GSIs (but not limited to the three  
Figure 13: Results of GSA (Example 4).  
types studied), especially for complex engineering structures involving high-dimensional and/or dependent inputs, as well as multivariate and/or time-dependent outputs.  
Conceptualization, Z.W.; methodology, Z.W.; validation, Z.W. and W.T.; formal analysis, Z.W. and X.W.; investigation, W.T.; writing--original draft preparation, Z.W. and X.W.; writing--review and editing, W.T. and X.W.; visualization, X.W.; funding acquisition, Z.W. and W.T. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (NSFC Grant No. 52208206), the Fundamental Research Funds for the Central Universities (Grant Nos. G2022 KY05103, G2021 KY05103), and the Natural Science Foundation of Shaanxi Province, China (Grant No. 2022)Q-513).  
Not applicable.  
The data that support the findings of this study are available from the corresponding author upon reasonable request. The data are not publicly available due to privacy.  
Author [PERSON] was employed by Forschungszentrum Julich GmbH. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} CRY & change of random variables \\ Fre-GSI & Frechet-derivative-based global sensitivity index \\ GSI & global sensitivity index \\ KDE & kernel density estimation \\ MCS & Monte Carlo simulation \\ PDF & probability density function \\ QoI & quantity of interest \\ SD & standard deviation \\ UP & uncertainty propagation \\ \end{tabular}