# Frechet-Derivative-Based Global Sensitivity Analysis and Its Physical Meanings in Structural Design
## 1 Introduction  
Sensitivity analysis, known as a powerful means for detecting which model inputs may have influential impacts on the model output(s), has played an important role in computational model-based structural design, evaluation, and optimization in various fields, e.g., structural engineering [1, 2, 3], wind engineering [4, 5], bridge engineering [6], and aerospace engineering [7, 8]. When the basic input variables are deterministic, the partial derivatives of the model output (quantity of interest, QoI), with respect to the input variables, are widely employed as sensitivity indices belonging to the local category [9]. However, if the QoI concern is stochastic due to the aleatory uncertainty of input parameters, the global sensitivity index (GSI), which can provide more quantitative information, should be seriously taken into account [10, 11]. On the definition of \"global sensitivity analysis\" given by [PERSON] et al. [12], wherein the whole variation domain of the inputs is considered, many GSIs have been established in the literature. Among them, perhaps the most notable and classical GSIs are variance-based GSI [13, 14], moment-independent GSI [15, 16], and failure-probability-based GSI [17, 18]. Note that the studied failure-probability-based GSI is also called the \"reliability-oriented sensitivity index\" [8, 19], or the \"reliability sensitivity index\" [20]. To avoid ambiguity, the term \"failure-probability-based GSI\" is adopted throughout this study. The variance-based GSI (a.k.a. [PERSON]' index) quantifies the impact of one input or the interaction effect of two or more inputs on the variance of the QoI. The moment-independent GSI (e.g., [PERSON]'s \(\delta\)-index) is based on the absolute difference between the conditional distribution and the unconditional one. For most engineering issues that adopt nonlinear and non-monotonic models, the \"size\" effect of sensitivity can be reflected by these two GSIs with non-negative values, while the \"direction\" effect of sensitivity can only be observed via data visualization [21; 22], e.g., in the form of 2D/3D scatter plots, box plots, cobweb plots, etc. In fact, the direction effect can be more essential for structural analysis, especially for reliability-based design optimization [11]. In this circumstance, the direction property of sensitivity is supposed to be integrated into the optimization process rather than having data visualized in each iteration. Hence, the failure probability-based GSI, which can reflect both the size and direction properties of the sensitivity, has attracted a lot of attention in reliability engineering. Nonetheless, the information, which failure probability can provide to quantify the uncertainty, is somewhat limited. In other words, the uncertainty of the QoI may not be completely characterized if only the probability of failure is taken into account. Similarly, [PERSON] [23] showed that non-informative results may be obtained when one relies on variance as the sole representative. In spite of widespread applications of the different GSIs mentioned above, there still lacks an informative GSI, which is supposed to perform the following: (1) quantitatively detect the most dominant model inputs (_importance measure_); (2) comprehensively describe the most influential directions (_direction indicator_); (3) intelligently reveal physical mechanisms (as a _probe_) of complex models with uncertain parameters. Most of the available GSIs satisfy the first feature, but only some of them hold the second feature, and they rarely posses the third one. For the above-mentioned requirements, the perturbed-law-based sensitivity index [24] (called DMBRSI in its first appearance [20]) can be an appropriate solution. It can measure how wide the change is in the probability of failure when shifting the mean values or the variances of input variables.