2.1 The framework of monitoring scheme optimization

The framework of monitoring scheme optimization is composed of three modules: (1) numerical simulation, (2) clustering analysis to determine typical positions, and (3) deduction of the overall stress distribution. In fact, the clustering and deduction analysis are processes of data clustering–reconstruction. The flowchart of the proposed model is shown in Figure 1.

The typical positions are defined as the points that can be used to describe the mechanical states of any position; thus, it is essential to obtain the information of the whole section. In order to realize this goal, a numerical model is developed first to constitute the data set of the mechanical behavior of the overall tunnel section. The tunnel section is divided into a number of elements, whose mechanical information is recorded for feature analysis of machine learning. Then, the constituted data set is fed into the clustering model to determine the optimized positions of sensors. The clustering algorithm is superior in terms of determining the key points of the mechanical behavior of the entire data set. By learning the variation features of the time series of each element, the clustering model divides all of the elements into different clusters. The mechanical behavior of elements in one cluster can be characterized by only one element, that is, a clustering centroid. Accordingly, it is determined as the typical positions of monitoring points. Subsequently, the data reconstruction analysis is conducted using supervised learning models to derive the mechanical behavior of the whole tunnel section based on the limited data from typical monitoring positions. By evaluating the characterization results derived by the data from different points, the optimal positions and numbers of points can be obtained. Finally, the optimal monitoring scheme is determined.

2.2 Data clustering–reconstruction method

The spectral algorithm is a clustering algorithm developed on the basis of graph theory. This algorithm has been widely used as it does not require many assumptions about the data structure (Alshammari et al., 2021). In addition, considering that the responses of the tunnel structure have strong spatial and temporal correlations, and linear regression (LR) is a classical model used to describe the relationship between time series (Tan, Chen, Wang, et al., 2020), the spectral algorithm and the LR model are used in this study for data clustering–reconstruction analysis. Specifically, the spectral algorithm is used to determine the typical positions for monitoring and LR mode is used to derive the information of those points without monitoring.

2.3 Evaluation indicators for model performance

The performance of the monitoring scheme can be evaluated by calculating the characterization error of a model on the overall section. Three evaluation indicators are adopted in this study, namely, root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) (Du, Li, et al., 2021; Du, Zhou, et al., 2021). The detailed descriptions of these three indicators are as follows.

3.1 Background for the case study

The Nanjing Yangtze River tunnel is located in Jiangsu, China, and became operational in January 2016. This tunnel is 7014 m long, including the shield-crossing portion of 3537 m. This tunnel is constructed with fabricated lining, and each lining ring consists of 10 segments, namely, one key segment, two adjacent segments, and seven standard segments, as shown in Figure 2a. The external diagram of tunnle segment is 14.5 m, and the width and thickness of the segment are 2.0 and 0.6 m, respectively. Consisting of two lines, the tunnel is designed in the form of two-tubing dual eight-lane expressways. The tunnel internal structure is shown in Figure 2b. The main geological layers to which the tunnel boring machine (TBM) is exposed are mucky silty clay, silty clay with silty sand, silty-fine sand, medium-coarse sand, gravelly sand, siltstone, and pebble. The highest water pressure that this tunnel subjected to is nearly 0.7 MPa. It is one of the biggest underwater shield tunnels in the world. In addition, the geological conditions to which this tunnel is exposed are also complicated compared to other such tunnels worldwide. More detailed information about this case study can be found in our previous studies (Tan, Chen, Wang, et al., 2020).

3.2 Numerical modeling

To determine the mechanical behaviors of a certain section of the tunnel structure, the finite element method performed on the second development was used for numerical modeling. The geological layers of this section are shown in Figure 3a, and the corresponding mechanical property parameters are presented in Table 1. Along this line, a numerical model is developed and is shown in Figure 3b. The numerical model of the tunnel segment was divided into 100 elements in the circumferential direction and eight elements in the longitudinal direction, namely, 800 elements in total.

3.3 Numerical results of tunnel lining stress

The water pressure applied to the tunnel varies periodically from year to year. It consists of two components: initial pressure and dynamic pressure. The former is determined by construction investigations, and is 0.437 MPa for this section, calculated according to Figure 3a. The latter varies with the season and is recorded by the Changjiang Maritime Safety Administration (specific water level data are available from the website https://cj.msa.gov.cn/). As an example, the variation of the water level of the Yangtze River tunnel over 1 year (366 days) was determined to calculate the external load applied on the tunnel structure, as shown in Figure 6. Therefore, the actual water pressure applied on the model is calculated by the sum of the dynamic pressure and the initial water pressure.

Pressure applied to the tunnel can be calculated using Equations (9)–(12), so the corresponding mechanical behavior of the tunnel can also be obtained. Each element records the stress variation in 1 year, so a time series containing 366 data included one element, expressed as ${\mathit{�}}_{�}=\{{�}_{�}^1,{�}_{�}^2,\mathrm{\cdots },{�}_{�}^{366}\}$ , and a data set $\mathit{�}\in {�}^{800\times 366}$ containing 800 time series of all elements was also prepared. For example, Figure 7 shows some numerical results of stress variation at the locations of arch crown, hance, and spandrel.

4.1 Model instantiation on data set

First, the clustering experiment was conducted. The number of clustering centroids was set to 4, 8, 10, and 20 in sequence to determine the reasonable number of monitoring points and the typical positions of segment stress variation. Based on the numerical results obtained for the tunnel, the cluster centroids were obtained using spectral clustering algorithms, as shown in Figure 8. For comparison, the monitoring scheme with different numbers of points was uniformly distributed on a tunnel surface. The red points represent the positions of clustering centroids when $�=4$ , while the blue points represent the positions of the four added clustering centers relative to the first four clustering centers when $�=8$ ; the purple and black points also have similar meanings. Newly added clustering centroids and previous ones are separated by a circle. The place of the monitoring points in the segment is represented by the distance from the clustering centroids to the center of the circle. Specifically, the smaller the distance, the closer it is to the inside of the segment.

Based on the clustering analysis results, the data reconstruction experiment was conducted. The data at the clustering centroids are regarded as the input to be fed into the LR model to derive the stress information of all sections. As a result, the characterization ability of the model under different monitoring conditions was evaluated by three indicators, as shown in Figure 9.

4.2 Discussion of experimental results

According to the monitoring scheme shown in Figure 8, it can be found that when $�=4$ , the clustering centroids are mainly distributed in the spandrel and hance of the tunnel. With increasing monitoring points, some clustering centroids are added without changing previous ones and distributed close to previous ones in the space. Furthermore, if there are too many additional measurement points, the monitoring points show a clustering distribution feature, such as the distribution when $�=20$ . Based on the analysis of model reconstruction results shown in Figure 9, it can be seen that the characterization ability of the model is not directly proportional to the increase in the clustering center number. For example, in contrast to the condition with four monitoring points, although the number of clustering centers under the condition with eight monitoring points is doubled, the characterization ability of the model is not significantly enhanced. In addition, the characterization ability of the LR model when $�=8$ is stronger than when $�=10$ . Therefore, a larger number of monitoring points in practical engineering does not mean a better choice because too many monitoring points can result in information redundancy and resource wastage.

To this end, for the case study in this paper, it was suggested to set up eight monitoring points in the field to capture the stress variations of the whole tunnel section after considering the influence of equipment life and other factors on the data quality. In general, spandrel and hance are critically important locations for sensing the whole surface behavior of the tunnel. Then, some monitoring points can be added to the arch crown and inch arch to monitor more mechanical behaviors. Based on the optimized monitoring scheme, the elements randomly selected from the arch crown, spandrel, and hance are derived as examples, which are shown in Figure 10. As shown, the deduction results are in good agreement with the numerical results. This flow can deduce the mechanical behavior of the overall section with limited sensors and provide a deduction accuracy of up to 98% (100%–1.091%). Therefore, the optimal monitoring scheme is reasonable.

4.3 Verification of the proposed model

In order to verify the reliability of the proposed monitoring scheme optimization model, other widely used methods are adopted to conduct data clustering and reconstruction experiments. Clustering algorithms used in this study included density peaks clustering algorithm (DPCA), K-means, and fuzzy C-means (FCM). Also, the support vector machine (SVM) model was adopted for data reconstruction experiments (Alex & Alessandro, 2014; Gao et al., 2017; Tan et al., 2022). These clustering models are instantized on the prepared numerical data set and the key points of segment stress variation when $�=8$ are determined, as shown in Figure 11.

As shown in Figure 11, although there are some differences in the distribution of clustering centers determined by different methods, the distribution positions are similar, that is, spandrel, hance, inch arch, and arch crown are key positions of stress variation. Furthermore, the data at key positions determined by different clustering models were fed into SVM model and the LR model to characterize the stress distribution of the overall section. The deduction ability of different models is evaluated, as shown in Figure 12.

The performances of the LR model and the SVM model are compared in terms of three indicators: RMSE, MAE, and MAPE. Comparison results show that the values of all three evaluation indicators of the LR model are smaller than those of the SVM model, indicating that the LR model is superior to the SVM model in terms of tunnel mechanical behavior deduction. In addition, among all clustering algorithms, the spectral algorithm is the most suitable one to cluster the data of tunnel mechanical behaviors. The error between the numerical simulation results and the deduction results obtained from the combined application of spectral and LR models is the smallest among all models. To this end, it can be concluded that the combined application of spectral and LR models is reliable to characterize the stress distribution of the overall tunnel section. The proposed model of monitoring scheme optimization is reliable.