# Frechet-Derivative-Based Global Sensitivity Analysis and Its Physical Meanings in Structural Design
## 5 Test Examples
### Example 4: A Dam Seepage Model  
The last example involves the sensitivity analysis of a steady-state confined seepage below a dam [18], as shown in Figure 12. During the analysis, it is assumed that the seepage is anisotropic and uniform. The permeability parameters of the silty gravel layer are denoted as \(T_{\mathrm{g},x}\) (in \(x\)-direction) and \(T_{\mathrm{g},y}\) (in \(y\)-direction), while those of the silty sand layers are defined by \(T_{\mathrm{s},x}\) (in \(x\)-direction) and \(T_{\mathrm{s},y}\) (in \(y\)-direction). Figure 11: GSIs for Borgonovo’s growing size model with a multiplicative structure (Example 3). The governing equation of the seepage problem in Figure 12 reads as follows:  
\[\left\{ \begin{aligned} & T_{\mathrm{g},x}\frac{\partial^{2}h_{ \mathrm{W}}}{\partial x^{2}}+T_{\mathrm{g},y}\frac{\partial^{2}h_{\mathrm{W}}}{ \partial y^{2}}=0,\\ & T_{\mathrm{s},x}\frac{\partial^{2}h_{\mathrm{W}}}{\partial x^{2 }}+T_{\mathrm{s},y}\frac{\partial^{2}h_{\mathrm{W}}}{\partial y^{2}}=0,\end{aligned} \right. \tag{45}\]  
and the seepage discharge \(q\) at the downstream side of the dam is calculated by  
\[q=-\int_{\Omega_{\mathrm{CD}}}T_{\mathrm{g},y}\frac{\partial h_{\mathrm{W}}(x, y)}{\partial y}\mathrm{d}x, \tag{46}\]  
where the hydraulic head \(h_{\mathrm{W}}\) on the AB segment is equal to \(20+h_{\mathrm{D}}\), and the failure event is defined when the seepage discharge \(q\) exceeds the threshold \(q_{\mathrm{lim}}=30\) (L/h/m). In the dam seepage model, the five input parameters \(\{T_{\mathrm{g},x},T_{\mathrm{g},y},T_{\mathrm{s},x},T_{\mathrm{s},y},h_{ \mathrm{D}}\}\) are considered independent random variables, where their probabilistic information is listed in Table 2. The deterministic analysis of Equation (45) is conducted by the finite element method with a total of 3413 nodes and 1628 quadratic triangular elements. Different GSIs of the first four parameters in Table 2 are calculated for the sensitivity analysis, including the variance-based GSI \(\mathcal{S}_{i}^{\mathrm{F}}\), the moment-independent GSI \(\mathcal{S}_{i}^{\delta}\), the importance measures \(\mathcal{S}_{i}^{\mathcal{F}}(\mu)\) based on the Fre-GSI (with respect to the mean value), and the failure-probability-based GSI \(\mathcal{S}_{i}^{\mathrm{F}}(\mu)\) (with respect to the mean value). The GSA results are presented in Figure 13, where \(\mathcal{S}_{i}^{\mathrm{S}}\) and \(\mathcal{S}_{i}^{\mathrm{F}}(\mu)\) are adopted from [18], respectively; \(\mathcal{S}_{i}^{\delta}\) is calculated by using \(10^{6}\) MCS samples, while \(\mathcal{S}_{i}^{\mathcal{F}}(\mu)\) is computed via the central difference method [10] by taking 10% of the mean value as the difference size. Note that the absolute value of \(\mathcal{S}_{i}^{\mathrm{F}}(\mu)\) is much higher than 1, since the mean value of the permeability parameter is very small. Hence, to facilitate discussion, all the results of \(\mathcal{S}_{i}^{\mathrm{F}}(\mu)\) shown in Figure 13a are after normalization, i.e., \(\mathcal{S}_{i}^{\mathrm{F}}(\mu)\rightarrow\mathcal{S}_{i}^{\mathrm{F}}( \mu)/\sum_{i=1}^{4}\mathcal{S}_{i}^{\mathrm{F}}(\mu)\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Distribution** & **(Mean, SD)** \\ \hline \(T_{\mathrm{g},x}\) & Lognormal & \((5\times 10^{-6},\ 5\times 10^{-6})\) (m/s) \\ \(T_{\mathrm{g},y}\) & Lognormal & \((2\times 10^{-6},\ 2\times 10^{-6})\) (m/s) \\ \(T_{\mathrm{s},x}\) & Lognormal & \((5\times 10^{-7},\ 5\times 10^{-7})\) (m/s) \\ \(T_{\mathrm{s},y}\) & Lognormal & \((2\times 10^{-7},\ 2\times 10^{-7})\) (m/s) \\ \(h_{\mathrm{D}}\) & Uniform & \((8.5,\sqrt{3}/2)\) (m) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Uncertain parameters in the dam seepage model (Example 4). Figure 12: Schematic representation of the dam seepage model (Example 4). From Figure (a)a, it can be seen that the results of \(\mathcal{S}_{i}^{\text{F}}(\mu)\) via EQ3 are very close to the ones from [18].