A note on degenerate derangement polynomials and numbers

1.   Introduction

With the continuous development of our country's economy and society and continuous urban construction, above-ground resources can no longer meet people's needs and urban construction has gradually turned to underground development ^[1]. Underground engineering continues to expand in the direction of "long, large and deep", and the construction scale and difficulty of foundation pit engineering are increasing daily ^[2]. The construction of an urban foundation pit often has a complex surrounding environment and the excavation construction significantly impacts the surrounding environment ^[3]. Foundation pit engineering has the characteristics of solid regionalization and the monitoring and warning value varies significantly in different regions. For safe construction, it is significant to carry out construction control and deformation prediction for foundation pits [4−6].

In recent years, with the continuous development of computer science, finite element calculation and artificial neural networks have also been rapidly developed. They are widely used in the fields of mining and civil engineering. Machine learning is often used to predict flying rock distance and ground vibration during blasting, the ultimate bearing capacity of the single-driven pile and the performance of tunnel boring machines, etc. ^[7,8,9,10]. Armaghani et al. ^[11] used particle swarm optimization algorithm (PSO) -artificial neural network (ANN) and imperialist competitive algorithm (ICA)-artificial neural network and ordinary artificial neural network to predict the penetration rate of tunnel boring machine (TBM). It is found that ICA-ANN and PSO-ANN hybrid models are better than ordinary neural network prediction. With the continuous development of computer science, many advanced models are often used in time series prediction, such as ELM-PSOGWO, ANN-EMPA, LSTM-INFO, ELM-SAMOA, RVM-IMRFO ^{[12,13,14,15,16]}. Machine learning and optimization models have become powerful tools to approach and solve engineering problems due to the flexibility of technology. Because of the high complexity and nonlinearity of deep foundation pit excavation deformation prediction, a neural network is often used to research deep foundation pit excavation deformation prediction. Hong et al. ^[17] used convolutional neural networks and long short-term memory neural networks to integrate spatial features. They found that the accuracy and reliability of the combined model considering spatial-temporal correlation were higher than that of the LSTM model considering only temporal correlation. Niu et al. ^[18] adopted an ARIMA-NAR neural network combination model based on Kalman denoising for prediction analysis. Jiang et al. ^[19] proposed a (PSO-VMD)-NARX-GRU prediction model for landslide displacement, whose accuracy was significantly higher than that of BP, SVM and ARIMA prediction models in static models. Zhang et al. ^[20] optimized and modified the wavelet network model using weight parameters and gradient descent method and proved by engineering examples that the improved model could make the predicted value close to the measured value. Meng et al. ^[21] analyzed the horizontal displacement of containment through multi-step rolling prediction based on the BP neural network and found that the method was efficient and reliable. Based on the background of deep foundation pit engineering of a subway station in Suzhou, Zhao et al. ^[22] adopted the 10-break cross-validation method based on the BP neural network and LSTM deep network to test the accuracy of the dynamic prediction model. They found that it has high reliability and generalization ability, which can provide a reference for construction information management.

In summary, many researchers have studied the deformation prediction method of deep foundation pits utilizing numerical simulation and machine learning neural networks. However, due to various errors, the prediction of environmental deformation around deep foundation pit excavation has certain limitations and the problems of large data dimension, slow model learning speed and low accuracy exist in the above prediction methods. This paper establishes a finite element simulation model based on engineering practice, the Grey Wolf optimization algorithm optimizes the extreme learning machine neural network and the GWO-ELM deep foundation pit excavation deformation prediction model is proposed. The ELM algorithm model is used to predict the excavation deformation of the foundation pit and the defects of slow learning speed and poor generalization performance of traditional neural networks are improved. GWO algorithm is used to optimize the ELM model to obtain the optimal input weight matrix and hidden layer deviation of the ELM algorithm, further improve the model's accuracy and provide specific theoretical reference and application value for the prediction of deep foundation pit engineering.

2.   Project overview

2.1.   Engineering background

The research relies on the construction of a new rainwater and sewage pipeline laying project in Baoding Science and Technology Park. The size of the top pipe working well for the entire pipeline construction is a rectangular well with an internal clearance of 8.5 × 6 m and the receiving well is a rectangular well with an internal clearance of 5 × 5 m, with a depth of approximately 6.16–8.77 m. The foundation pit support is soil nail wall support and the safety level of the side of the foundation pit is Level 2.

2.2.   Geological conditions

The soil layer parameter indicators and their distribution of this project are shown in Table 1.

2.3.   Monitoring program

The monitoring level of the foundation pit of this project is Level 1 and the soil nail support is Level 2. The impact of construction on the ground and underground pipelines, the safety of surrounding buildings, the stability monitoring of open caisson and the impact of construction on the stability of surrounding slopes are the key points of this monitoring plan. The specific monitoring point layout is shown in Figure 2.

3.   GWO-ELM neural network model

3.1.   Extreme learning machine model

ELM is a new single hidden layer feedforward neural network algorithm ^{[23,24,25,26]}. Its biggest advantage is that the connection weights of the input layer and the hidden layer and the threshold of the hidden layer are randomly generated in the model training process. It does not need to be adjusted again after setting. It has the characteristics of simple structure and good generalization ability and has a fast rate.

Given a different training sample, $\left({{x_i}, {t_i}} \right) \in {R^n} \times {R^m}\left({i = 1, 2, \ldots, N} \right)$, SLFN output with $\tilde N$ hidden nodes is available.

where ${o_j}$ is the output of the network.

${a_i} = {\left[{{a_{i1}}, {a_{i2}}, \ldots, {a_{in}}} \right]^T}$ and ${b_i}$ are the learning parameters randomly generated by the JTH hidden node, respectively. ${\beta _i} = {\left[{{\beta _{i1}}, {\beta _{i2}}, \ldots, {\beta _{im}}} \right]^T}$ is the output weight of the output node; $f\left({{x_j}; {a_i}, {b_i}} \right)$ is the activation function of the original ELM. Let ${a_i} \cdot {x_j}$ be the inner product of ${a_i}$ and ${x_j}$. Equation (1) can be succinctly written as Eq (2).

where:

where $H$ is the output matrix of hidden layer nodes; β is the output weight; O is the training set objective matrix.

Therefore, the three-step ELM algorithm can be summarized as follows.

Input: Training set $\left({{x_i}, {t_i}} \right) \in {R^n} \times {R^m}\left({i = 1, 2, \ldots, N} \right)$, activation function $f$, hidden node number $\tilde N$.

Output: Output weight $\beta$.

1) Randomly assign hidden node parameter $\left({{a_i}, {b_i}} \right)$, $i = 1, \ldots, \tilde N$.

2) Calculate the output matrix of hidden layer $H$.

3) Calculate the output weight $\beta$, $\beta = {H^ + }T$.

3.2.   Grey Wolf optimization algorithm

The GWO is a new meta heuristic algorithm with strong search capabilities proposed by Negi et al. ^[27]. This algorithm is based on the framework of systematization, recursive evolution and hierarchy. Due to its strong convergence performance and fewer parameters, its optimization performance is more reliable than other biomimetic algorithms ^[28,29].

The GWO optimization algorithm includes hunting and trapping of gray wolves and the hunting of prey is achieved through generation after generation. The main definitions of the algorithm are as follows:

a. The direct distance between the prey and the Wolf $\vec D$ is determined before the prey is eaten.

where t is the current number of iterations; $\vec C$ is the oscillation coefficient, $\vec C = 2{\vec r_1}$; ${\vec r_1}$ is the spatial distance coefficient ${\vec r_1} = random\left({0, 1} \right)$; ${\vec X_p}\left(t \right)$ is the position vector of the t-generation gray wolf prey; $\vec X\left(t \right)$ is the current position vector of the t-generation gray wolf.

b. The position of the next generation of gray wolves is constantly updated to a shortened $\vec D$.

where $\vec X\left({t + 1} \right)$ is the position vector of the t + 1 generation gray wolf; $\vec \mu$ is the coefficient; ${\vec r_2}$ is a random number of [0, 1]; During the entire iteration process, $\vec a$ decreased from 2 to 0.

c. During predation, $\alpha$ wolves, $\beta$ wolves and $\delta$ wolves are closest to the prey. The distance between the other gray wolves of the t generation and these three wolves can be determined by the equation.

Predation direction is determined according to the following formula:

where $k = \alpha, \beta, \delta$ and $i = 1, 2, 3$.

4.   Engineering application

4.1.   Finite element simulation

Use MIDAS GTS NX software to establish a three-dimensional finite element calculation model for deep foundation pit excavation and analyze and study the surface deformation and settlement during the excavation process. The model size is the same as the physical engineering size, with a foundation pit side length of 18.5 m. According to engineering experience, the horizontal influence range is 4–6 times the physical engineering side length and the vertical influence range is 3–5 times the depth. The outer model has a side length of 60 m and a depth of 40 m. The foundation pit support adopts the method of spray anchor support. The boundary constraint adopts automatic constraints to constrain the translational degrees of freedom of the side and bottom nodes. Limit the horizontal displacement of the side, limit the horizontal displacement and normal displacement of the bottom.

The modified Moore-Coulomb model is selected as the soil constitutive model ^[30]. Compared with the ordinary Moore-Coulomb model, the simulated results of the modified Moore-Coulomb model are more appropriate to the unloading process of soil in deep foundation pit excavation. The results of model extraction are mainly concentrated in the vicinity of the foundation pit model, so the grid division of this part is relatively fine. The number of units in this calculation model is 12460 and the number of nodes is 10718. The model is shown in Figure 3.

Based on the excavation and support sequence at the construction site, the simulation of the foundation pit is divided into ten working conditions. Before excavation, balance the initial geostress and reset the displacement in the Z direction to zero.

Condition 1: Excavate the soil to –2 m.

Condition 2: Drive the first row of soil nails into the soil and spray concrete on the grid.

Condition 3: Excavate the soil to –4 m.

Working condition 4: Drive the second row of soil nails into the soil and spray concrete on the grid.

Working condition 5: Excavate the soil to –6 m.

Working condition 6: Drive the third row of soil nails into the soil and spray concrete on the grid.

Condition 7: Excavate the soil to a depth of –8 m.

Working condition 8: Drive the fourth row of soil nails into the soil and spray concrete on the grid.

Condition 9: Excavate to the bottom of the pit.

Working condition 10: Drive the fifth row of soil nails into the soil and spray concrete on the grid.

4.2.   Settlement analysis of surrounding buildings

Figure 4 Vertical displacement cloud map of some excavation conditions and Figure 5 contrast curve between measured and simulated settlement values of surrounding buildings.

From Figures 4 and 5, it can be seen that as the construction progresses, the settlement of the garage on the northwest side of the foundation pit gradually increases. The shape of the actual value change curve is roughly the same as that of the simulated value change curve, but there is still a certain deviation. Overall, the simulated value is slightly larger than the measured value, which may be due to the software having a certain warning and protection effect. The measured value of monitoring point CK-1 is slightly greater than the simulated value because the monitoring point CK-1 is relatively close to the foundation pit and is greatly affected by the construction of the foundation pit. Some nearby loads and the impact of surrounding construction are not considered in the software. The maximum value monitored on site was 7.93 mm and the maximum value simulated by finite element analysis was 7.8709 mm, both of which did not exceed the warning value of 15 mm, indicating that the support structure was applied in a timely and effective manner.

4.3.   Deep horizontal displacement analysis

Figure 6 shows the cloud map of the deep horizontal displacement of the soil during the excavation of the foundation pit. Figure 7 shows the comparison curve between the monitoring values of the deep horizontal displacement of the soil after excavation and the finite element numerical simulation values.

As can be seen from Figure 6, with the excavation of the foundation pit, each point on the side wall of the foundation pit moves toward the interior of the foundation pit and the horizontal displacement gradually increases. The horizontal displacement difference of each point in the same working condition does not exceed 10 mm. The maximum horizontal displacement of the side wall of the foundation pit is located in the middle of the slope. With the excavation of the foundation pit, the horizontal displacement reaches the maximum value. As can be seen from Figure 7, the overall trend of the measured horizontal displacement of deep foundation pit is roughly the same as that of the simulated value, which increases first and then decreases with the increase of the depth of foundation pit. However, there is still a deviation between the simulated value and the measured value and the maximum difference is 3.0887 mm. The simulated value is smaller than the measured value because mechanical vibration and vehicle flow are not taken into account in the simulation process.

To sum up, the measured value and the simulated value are compared and analyzed from two aspects: the settlement of surrounding buildings and the horizontal displacement in deep soil layer. From the results, there is still a certain error between the simulated value and the measured value, but the error is small. It can be used as the input data of the neural network.

4.4.   Surface settlement prediction

The monitoring point DB-2 in the north of the foundation pit was selected as the surface settlement prediction point, the excavation depth, the number of soil nails and the settlement of the building were taken as the input factors and the monitoring point DB-2 settlement value was taken as the output factors. The specific forecasting process is as follows:

(1) Weighted smoothing processing. In order to avoid the influence of accidental data on the prediction accuracy of the model, all the sample data are weighted with smooth noise reduction and balanced sample data.

(2) Data normalization processing. In order to eliminate the influence of different data dimensions, the prediction of excavation depth, number of soil nails and settlement value is normalized to the interval [0, 1].

(3) Model parameter setting. The collected 60 data sets are sorted, the first 45 groups are selected as the training set, and the last 15 groups are selected as the test set. The network topology is 3 × 20 × 1. The number of iterations is 1000 and the learning rate is 0.01. The depth of excavation, the number of soil nails and the settlement of the building are selected as the influencing factors, so the input layer node is set to 3. The model output is the surface settlement value, so the number of nodes in the output layer is set to 1. The number of neuron nodes in the hidden layer is set to 20. The Grey Wolf optimization algorithm optimizes the input weight and hidden layer threshold in ELM neural network. The size of the wolves is 40 and the number of iterations is 100. At the same time, the results are compared with those of the unoptimized ELM prediction model under the same conditions.

(4) Analysis of model prediction results. The predicted value of land surface settlement was obtained using the GWO-ELM prediction model and the test set results were shown in Table 2. The comparison diagram of DB-2 predicted value and error curve at monitoring points is shown in Figures 8 and 9. The prediction accuracy of different models is shown in Table 3.

It can be seen from Figures 8 and 9 and Table 3 that the obtained root mean square error and average absolute error are 0.31258 and 0.26145, respectively, which are closer to the actual surface settlement and have smaller errors than the ELM prediction model. The determination coefficient R² = 0.98725 is higher than the ELM prediction model and the surface fitting effect is better, the prediction accuracy is higher and it has relative reliability.

5.   Limitations and future works

The GWO-ELM and ELM prediction models adopted in this paper are used to predict the data in foundation pit excavation projects. These can provide a reference for the informatization construction of deep foundation pit excavation. Then, various unique data monitoring points can be further considered, monitored and predicted, such as vehicle load vibration, unpredicted rainfall and snow amount of continuous rainfall and snowfall, earthquake impact, etc. This paper's monitoring and prediction project is based on the deep foundation pit project of Baoding Automobile Science and Technology Industrial Park. In the future, different foundation pit projects can be considered to be added or the engineering situation of all foundation pit projects in a particular area can be sorted out and the influencing factors and deformation causes of foundation pits can be comprehensively analyzed.

6.   Conclusions

This article takes the excavation depth of the finite element model, the number of soil nails and the settlement of surrounding buildings as input factors and the surface settlement as output factors. Unlike using actual monitoring data as samples, it can achieve advanced prediction and is an effective analysis method.

The GWO-ELM based deformation prediction model for deep foundation pit construction excavation is significantly better than the ELM model and has achieved good prediction results. The model has good generalization ability and reduces human interference during the modeling process, making it an effective prediction method.

The mean absolute error of the extreme learning machine neural network optimized by gray wolf is 0.26145, the mean squared error is 0.31258 and the R² is 0.98725, which are better than the extreme learning machine neural network non-optimized.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was supported by the Natural Science Foundation of Hebei Province (E2018201106), the High-Level Talents Project of Hebei Province (B2017005024).

Conflict of interest

The authors declare that there is no conflict of interest in the publication of this article.