## 4 Examples of Engineering Applications
### Comparative Study  
In order to illustrate the science and rationality of the method, this paper adopts the matter-element extension theory to evaluate the construction quality of hydraulic tunnels, which is used as a comparative study with the above. No further explanation about the matter-element extension theory is given, and specific reference can be made to the literature [33; 34]. The focus here is on the calculation of the correlation function value of the matter-element grade of hydraulic tunnels to be evaluated. (1) Correlation of evaluation indicators  
The correlation of the construction quality level \(t\) of the hydraulic tunnel to be evaluated is shown in Equation (24):  
\[K_{t}(x_{i})= \begin{cases}\frac{\rho(x_{i},x_{it})}{\rho(x_{i},x_{it})-\rho(x_ {i},x_{it})},x_{i}\
otin x_{it}\\ \frac{-\rho(x_{i},x_{it})}{|x_{it}|},x_{i}\in x_{it}\end{cases}  \tag{24}\]  
where  
\[\rho(x_{i},x_{it})=\left|x_{i}-\frac{a_{it}^{+}+a_{it}^{-}}{2}\right|-\frac{a_ {it}^{+}-a_{it}^{-}}{2} \tag{25}\]  
\[\rho(x_{i},x_{ip})=\left|x_{i}-\frac{a_{ip}^{+}+a_{ip}^{-}}{2}\right|-\frac{a_ {ip}^{+}-a_{ip}^{-}}{2} \tag{26}\]  
where \(x_{i}\) is the \(i\)th construction quality evaluation index; \(x_{it}=\left(a_{it}^{+},a_{it}^{-}\right)\) is the magnitude range of the \(i\)th index divided by the evaluation grade \(U_{t}\); \(x_{ip}=\left(a_{ip}^{+},a_{ip}^{-}\right)\) is the magnitude range of the \(i\)th index divided under each level; and \(|x_{it}|\) is the interval length. (2) Determine the construction quality evaluation grade of the hydraulic tunnel  
By combining the weights of the indicators and the value of the correlation function, the comprehensive correlation of the evaluation object can be calculated; see Equation (27). If \(K_{t_{0}}(T_{0})=maxK_{t}(T_{0})\), the evaluation object \(T_{0}\) belongs to level \(t_{0}\). \[K_{t}(T_{0})=\sum_{i=1}^{n}\omega_{i}K_{t}(x_{i}) \tag{27}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Stage** & **Construction** & **Tunnel** & **Support** & **Anti-Drainage** & **Secondary Lining** & **Tunnel** & **Tunnel** \\
**Measurement** & **Excavation** & **Construction** & **Construction** & **Concrete** & **Construction** & **Grouting** & **Whole** \\ \hline Evaluation & & & & & & & & \\ grade & III & II & II & III & II & III & III \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Construction quality evaluation grade of concrete structures in hydraulic tunnels. where \(K_{t}(T_{0})\) is the comprehensive correlation degree of \(T_{0}\) with respect to grade \(t\); and \(\omega_{i}\) takes the objective weight value \(\beta_{i}\) of the CRITIC method. According to the grading standards of each evaluation index in Section 4.3, the classical domain and node domain of the construction quality evaluation index of the hydraulic tunnel can be determined, as shown in Table 12.  
Based on the expected value of each evaluation indicator in Table 11, the correlation degree matrix of the evaluation indicators of hydraulic tunnel construction quality can be obtained from Equations (24)-(26):  
\[ \begin{bmatrix}-0.4120&-0.2010&0.4400&-0.1650&-0.4433\\ -0.4046&-0.1658&0.4133&-0.2200&-0.4800\\ -0.4172&-0.2247&0.3200&-0.1200&-0.4133\\ -0.3684&0.0667&-0.0400&-0.4000&-0.6000\\ -0.4178&-0.2271&0.3067&-0.1150&-0.4100\\ -0.2768&0.4133&-0.3520&-0.5950&-0.7300\\ -0.3188&0.4133&-0.2480&-0.5300&-0.6867\\ -0.3641&0.1067&-0.0640&-0.4150&-0.6100\\ -0.2340&0.2933&-0.4240&-0.6400&-0.7600\\ -0.3950&-0.1166&0.2533&-0.2800&-0.5200\\ -0.3418&0.2800&-0.1680&-0.4800&-0.6533\\ -0.4031&-0.1585&0.3867&-0.2300&-0.4867\\ -0.4219&-0.2449&0.2000&-0.0750&-0.3833\\ -0.3418&0.2800&-0.1680&-0.4800&-0.6533\\ -0.3264&0.3733&-0.2240&-0.5150&-0.6767\\ -0.3134&0.4400&-0.2640&-0.5400&-0.6933\\ -0.4194&-0.2340&0.2667&-0.1000&-0.4000\\ -0.3786&-0.0229&0.0400&-0.3600&-0.5733\\ -0.3984&-0.1345&0.3067&-0.2600&-0.5067\\ -0.4194&-0.2340&0.2667&-0.1000&-0.4000\\ -0.4081&-0.1827&0.4800&-0.1950&-0.4633\\ \end{bmatrix}