# Frechet-Derivative-Based Global Sensitivity Analysis and Its Physical Meanings in Structural Design
## 5 Test Examples
### Example 2: An Analytical and Nonlinear Model  
Consider an analytical model written by [7]  
\[X=\Theta_{1}+\Theta_{2}^{2}, \tag{35}\]  
where \(\Theta_{1}\) and \(\Theta_{2}\) are i.i.d. normal random variables with mean values \(\mu_{1}=\mu_{2}=0\) and standard deviations \(\sigma_{1}=\sigma_{2}=1\), respectively. Then \(X|\Theta_{1}\) follows a \(\chi^{2}\)-distribution shifted by \(\Theta_{1}\) (\(X>\Theta_{1}\)), and \(X|\Theta_{2}\) follows a normal distribution with mean value \(\mu_{1}+\Theta_{2}^{2}\) and standard deviation \(\sigma_{1}\), i.e.,  
\[ \begin{cases}p_{X|\Theta_{1}}(x|\Theta_{1})=\dfrac{1}{2\sqrt{2\pi}\sqrt{x- \Theta_{1}}}\left(\exp\left\{-\dfrac{(\sqrt{x-\Theta_{1}}-\mu_{2})^{2}}{2 \sigma_{2}^{2}}\right\}+\exp\left\{-\dfrac{(\sqrt{x-\Theta_{1}}+\mu_{2})^{2}} {2\sigma_{2}^{2}}\right\}\right),\\ p_{X|\Theta_{2}}(x|\Theta_{2})=\dfrac{1}{\sqrt{2\pi}\sigma_{1}}\exp\left\{- \dfrac{(x-\mu_{1}-\Theta_{2}^{2})^{2}}{2\sigma_{1}^{2}}\right\},\\ \text{and the unconditional PDF of }X\text{ is given by }\\ p_{X}(x)=\int_{-\infty}^{+\infty}\dfrac{1}{2\pi\sigma_{1}\sigma_{2}}\exp\left\{ -\dfrac{(x-\mu_{1}-\theta_{2}^{2})^{2}}{2\sigma_{1}^{2}}-\dfrac{(\theta_{2}- \mu_{2})^{2}}{2\sigma_{2}^{2}}\right\}\text{d}\theta_{2}.\end{cases}  \tag{36}\]  
Figure 5: Fre-GSIs of Equations (10) and (34) (Example 1). The derivatives of \(p_{X}(x)\) with respect to the distribution parameters can be computed numerically by  
\[ \begin{cases}\dfrac{\partial p_{X}(x)}{\partial\mu_{1}}=\int_{-\infty}^{+\infty} \dfrac{x-\mu_{1}-\theta_{2}^{2}}{2\pi\sigma_{1}^{3}\sigma_{2}}\exp\Biggl{\{}- \dfrac{(x-\mu_{1}-\theta_{2}^{2})^{2}}{2\sigma_{1}^{2}}-\dfrac{(\theta_{2}-\mu_ {2})^{2}}{2\sigma_{2}^{2}}\Biggr{\}}\mathrm{d}\theta_{2},\\ \dfrac{\partial p_{X}(x)}{\partial\sigma_{1}}=\int_{-\infty}^{+\infty} \dfrac{(x-\mu_{1}-\theta_{2}^{2})^{2}-\sigma_{1}^{2}}{2\pi\sigma_{1}^{4}\sigma _{2}}\exp\Biggl{\{}-\dfrac{(x-\mu_{1}-\theta_{2}^{2})^{2}}{2\sigma_{1}^{2}}- \dfrac{(\theta_{2}-\mu_{2})^{2}}{2\sigma_{2}^{2}}\[PERSON]{\}}\mathrm{d}\theta_{ 2},\\ \dfrac{\partial p_{X}(x)}{\partial\mu_{2}}=\int_{-\infty}^{+\infty} \dfrac{\theta_{2}-\mu_{2}}{2\pi\sigma_{1}\sigma_{2}^{3}}\exp\[PERSON]{\{}- \dfrac{(x-\mu_{1}-\theta_{2}^{2})^{2}}{2\sigma_{1}^{2}}-\dfrac{(\theta_{2}- \mu_{2})^{2}}{2\sigma_{2}^{2}}\Biggr{\}}\mathrm{d}\theta_{2},\\ \dfrac{\partial p_{X}(x)}{\partial\sigma_{2}}=\int_{-\infty}^{+\infty} \dfrac{(\theta_{2}-\mu_{2})^{2}-\sigma_{2}^{2}}{2\pi\sigma_{1}\sigma_{2}^{4}} \exp\Biggl{\{}-\dfrac{(x-\mu_{1}-\theta_{2}^{2})^{2}}{2\sigma_{1}^{2}}-\dfrac{ (\theta_{2}-\mu_{2})^{2}}{2\sigma_{2}^{2}}\Biggr{\}}\mathrm{d}\theta_{2}.\end{cases}  \tag{38}\]  
The analytical expressions of PDF and Fre-GSIs are shown in Figure 6. With respect to \(\Theta_{1}\) and \(\Theta_{2}\), the variance-based GSIs (\(\mathcal{S}_{i}^{\mathrm{S}}\)) are 0.3333 and 0.6666; the moment-independent GSIs (\(\mathcal{S}_{i}^{\delta}\)) are 0.4933 and 0.3049. With respect to \(\mu_{1}\), \(\mu_{2}\), \(\sigma_{1}\), and \(\sigma_{2}\), the failure-probability-based GSIs (\(\mathcal{S}_{i,j}^{\mathrm{F}}\)) for \(\Omega_{\mathrm{F}}:x>5\) (\(\mathrm{F}_{\mathrm{F}}=0.0303\)) are \(-0.0179\), 0.0000, 0.0111, and 0.1564, and those for \(\Omega_{\mathrm{F}}:x>15\) (\(\mathrm{F}_{\mathrm{F}}=1.23\ \times\ 10^{-4}\)) are \(6.55\ \times\ 10^{-5}\), 0.0000, \(3.49\ \times\ 10^{-5}\), and \(1.89\ \times\ 10^{-3}\); the IMs from the Fre-GSIs (\(\mathcal{S}_{i,j}^{\mathrm{F}}\)) are 0.7592, 0.0000, 0.6199, and 0.7987. With respect to \(\mu_{1}\), \(\mu_{2}\), \(\sigma_{1}\), and \(\sigma_{2}\), the values of \(\partial\mathcal{S}_{i}^{\mathrm{S}}/\partial\mathcal{S}_{i,j}^{\mathrm{F}}\) are \(0.0170\), 0.0000, 0.4420, and 0.8875; the values of \(\partial\mathcal{S}_{i}^{\delta}/\partial\mathcal{S}_{i,j}^{\mathrm{F}}\) are \(-0.0035\), 0.0000, 0.1755, and 0.4130. It should be emphasized that the above results are computed via the central difference scheme with a difference size of 0.1.