Real-Time Construction Simulation Coupling a Concrete Temperature Field Interval Prediction Model with Optimized Hybrid-Kernel RVM for Arch Dams

: Joint grouting simulation is an important aspect of arch dam construction simulation. However, the current construction simulation model simpliﬁes the temperature factors in joint grouting simulation, which leads to the di ﬀ erence between the simulation results and the actual construction schedule. Furthermore, the majority of existing temperature prediction research is based on deterministic point predictions, which cannot quantify the uncertainties of the prediction values. Thus, this study presents a real-time construction simulation method coupling a concrete temperature ﬁeld interval prediction model to address these problems. First, a real-time construction simulation model is established. Secondly, this paper proposes a concrete temperature interval prediction method based on the hybrid-kernel relevance vector machine (HK-RVM) with the improved grasshopper optimization algorithm (IGOA). The hybrid-kernel method is adopted to ensure the prediction accuracy and generalization ability of the model. Additionally, the improved grasshopper optimization algorithm (IGOA), which utilizes the tent chaotic map and cosine adaptive method to improve the algorithm performance, is developed for the parameter optimization of HK-RVM. Thirdly, concept drift detection based on variable window technology is proposed to update the prediction model. Finally, an arch dam project in China is used as a case study, by which the superiority and applicability of the proposed method are proven.


Introduction
Arch dams have the advantages of strong overload capacity, good seismic performance, and saving construction materials, which have become the popular dam type for high dam construction in China [1]. In order to prevent temperature cracks, an arch dam is usually divided into several dam monoliths along the dam axis direction during the construction period. The surface between adjacent dam monoliths is called the transverse joint, which is where the grouting system is embedded [2]. For the sake of the dam's integrity and safety, it is essential to grout on the transverse joint surface during the construction period. Then, the independent dam body will be cemented as a whole. Therefore, joint grouting is an essential procedure in arch dam construction. In addition, if the joint grouting lags behind, the dam concrete pouring will be influenced due to the cantilever height constraint, which will lead to a delay in the overall construction progress of the dam. Therefore, joint grouting schedule management is an important aspect of dam construction schedule management, which has a crucial influence on the safety and the construction schedule of a dam project.
is an effective approach for interval prediction, and has been widely used [38][39][40]. RVM is based on a probabilistic Bayesian learning framework, which can provide probabilistic predictions. Besides, the prediction effect of the RVM model depends on the selection of the kernel functions and the kernel parameters [41]. A single kernel function cannot have a strong learning ability and generalization ability at the same time. Hybrid-kernel learning is a good solution [42]. Based on specific rules, multiple single kernel functions are combined into a new kernel function. For the problem of parameter optimization, intelligent optimization algorithms are introduced to determine the optimal parameters. Many scholars adopted different intelligent optimization algorithms to optimize the parameters of RVM, including the ant lion optimizer [43], whale optimization algorithm [44], imperialist competitive algorithm [45], cuckoo search optimization [46], particle swarm optimization [47], gravitational search optimization [48], and quantum-inspired gravitational search [49]. The grasshopper optimization algorithm (GOA), proposed by Saremi, is a new swarm intelligence algorithm [50]. The basic idea was inspired by the swarm behavior of grasshoppers. This algorithm has the advantages of simple structure, strong stability, fewer parameters, and faster convergence, which has been widely used in parameter optimization [51].
The temperature data in the construction period are a group of dynamic flow data that grows rapidly over time, and the environmental temperature and water-cooling stage transition in the dam construction period change with time. These data characteristics may cause the prediction performance of the prediction model to decline significantly over time. This is called concept drift. The study of concept drift detection methods for the prediction model is necessary to realize the dynamic updating of the prediction model. For concept drift detection methods, window-based detection methods are effective [52,53], such as efficient concept-adapting very fast decision tree (E-CVFDT) [54] and Adwin [55]. The window size of these methods is sensitive to concept drift detection. A fixed window can improve the analysis speed; however, it is difficult to determine the size of the window. In addition, it is not easy to recognize the concept drift when the window size is too large, and it is difficult to update the model when the window size is too small due to the small amount of data. In order to deal with this problem, this paper proposes a concept drift detection method based on the variable window technique.
In summary, the current construction simulation studies simplified the influence of temperature factors on joint grouting. Additionally, the majority of existing temperature prediction research is based on point prediction, which cannot effectively quantify the uncertainty of the concrete temperature prediction value. Therefore, the aim of this research was to propose a real-time construction simulation method coupling a concrete temperature interval prediction model with the improved grasshopper optimization algorithm. The main aim of this paper is to: Establish a real-time construction simulation model that couples the concrete temperature field to realize a more accurate simulation.

2.
Propose a concrete temperature interval prediction model based on the hybrid-kernel relevance vector machine (HK-RVM) and the improved grasshopper optimization algorithm (IGOA) to obtain reliable prediction intervals of concrete temperature.

3.
Develop a concept drift detection method based on the variable window technology to update the prediction model.
The remainder of this paper is organized as follows: The framework of this paper is shown in Section 2. In Section 3, a detailed description of the proposed methodologies is presented. In Section 4, we present a case study to verify the applicability of the proposed method. In Section 5, we provide a discussion to investigate the superiority of the proposed method. Section 6 outlines the conclusions and future works.

Research Framework
In this work, a real-time construction simulation method coupling the concrete temperature field interval prediction model was proposed. The research framework is composed of three levels: the construction simulation model, the temperature interval prediction model, and a case study, as shown in Figure 1. At the construction simulation model level, the real-time construction simulation model was divided into the dam block construction simulation and the joint grouting simulation. The concrete age, cover weight, and temperature field are the key parameters for the joint grouting simulation, and these were obtained from the dam block construction simulation results. The temperature field can be predicted based on the interval prediction model.
At the concrete temperature field prediction model level, it contains the concrete temperature interval prediction model and the concept drift detection for model updating. The details are as follows: (1) In the sublevel of temperature interval prediction model, firstly, the tent chaotic map and cosine adaptive methods are adopted to improve the performance of the grasshopper optimization algorithm (GOA). Secondly, the IGOA is used for parameter optimization of the HK-RVM prediction model. (2) In the sublevel of concept drift detection for model updating, a concept drift detection based on the variable window technique is proposed to update the prediction model to avoid prediction performance degradation. Firstly, the error of new data is analyzed based on the established temperature interval prediction model. Secondly, the sliding window size is determined according to the error analysis results. Thirdly, whether concept drift occurs depends on the change of the sliding window size. If the concept drift occurs, the temperature interval prediction model will be retrained. Finally, a dynamic concrete temperature interval prediction model can be established.
At the case study level, the proposed model was applied to an arch dam in China. The temperature field prediction results and the construction simulation scheme were obtained. Compared with the relevance vector machine (RVM) model with a single kernel function and Gaussian process regression (GPR) model, the prediction accuracy and generalization ability of the proposed model were verified. Compared with the traditional simulation model without considering the temperature field, the simulation results with a considerable temperature field were closer to the actual situation.

Problem Formulation
According to the construction method of arch dams, the construction simulation of arch dams can be simplified into two parts: the dam block construction simulation and joint grouting simulation. According to the requirements of construction specifications, the joint grouting should consider not only the concrete internal temperature in the temperature control area during the construction period but also the concrete age and cover weight. The concrete age is related to the pouring time of the dam block, and the concrete cover weight is related to the current dam pouring face, and these are all obtained by the dam block simulation. The concrete temperature field is predicted by the HK-RVM interval prediction model. In addition, dam construction progress and joint grouting progress are mutually restricted due to the cantilever height constraint. The real-time construction simulation mathematical model is expressed in Figure 2. The model defines the objective function of real-time construction simulation, dam block construction simulation, joint grouting simulation, and the relationship between dam block construction simulation and joint grouting simulation. The detailed descriptions of the symbols are shown in Appendix A.

Hybrid-Kernel Relevance Vector Machine
Given a training sample set x i , y i N i=1 x i ∈ R d , y i ∈ R , the relevance vector machine model can be defined as follows: where N is the number of data samples, w i are the model weights, ε ∼ N(0, σ 2 ) is the independent additive Gaussian noise, and K(x, x i ) is a nonlinear kernel function. The selection of the kernel function is the key to the model performance. Among many kernel functions, the polynomial kernel function is a typical global kernel function with a strong generalization ability [42]. The gauss kernel function is a type of local kernel function with a good local learning ability [42]. The concrete temperature data are dynamic and nonlinear. The concrete temperature is influenced by the continuous change of the hydration heat, as well as the random fluctuation of the external conditions. A single kernel function cannot meet the temperature prediction demand [42]. In order to ensure the global generalization ability and local learning ability of the model, the hybrid-kernel method is employed in this paper. The expression is shown as follows: where the first term on the right side of the equation is the gauss kernel function, γ is the kernel parameter of the gauss kernel function. The second term is the polynomial kernel function, c and d are the kernel parameters of the polynomial kernel function. λ is the adjustment coefficient. Suppose the target value y i is independent, then the likelihood function can be described as According to the Bayesian criterion and multivariate conditional Gaussian distribution, the posterior distribution can be written as . At last, the hyperparameter α and the variance σ 2 can be estimated using the maximum likelihood method. For new data x * , the prediction value y * and variance σ 2 * can be described as follows: Given the 1 − ζ confidence level (CI), the confidence interval of prediction value y * can be described as where L b ξ is the lower bound of the prediction intervals, and U ξ b is the upper bound of the prediction intervals. z ξ/2 is the bilateral quantiles of a standard normal distribution.
The prediction accuracy of the HK-RVM model depends on the selection of kernel function parameters. According to the HK-RVM model, the parameters involved can be expressed as (λ, γ, c, d).
In the next section, the IGOA algorithm will be introduced to optimize the model parameters.

Improved Grasshopper Optimization Algorithm (IGOA)
The grasshopper optimization algorithm (GOA) is a new metaheuristic optimization algorithm [50]. The GOA mimics the swarming behavior of grasshoppers. The grasshoppers are affected by social interaction, gravity and wind advection in the process of migration and foraging. For the optimization problem, the influence of gravity is usually ignored, and the wind direction is assumed to always point to the target position. Then, the mathematical model can be represented as follow: where c is a decreasing coefficient,ub d and lb d , respectively, represent the upper bound and lower bound in the d th dimension, T d is the best solution so far in the dth dimension, and s is a function to define the strength of social forces, as shown in Equation (8).
where f denotes the attraction intensity between grasshoppers, l is the length scale of attraction, and r is the range of comfortable areas. The force between grasshoppers can be divided into attraction and repulsion according to the strength of the social forces.
In the standard GOA algorithm, the control parameter c usually adopts the linear method to dynamically adjust. It means that the parameter c decreases at a constant rate. In order to effectively balance the exploration and exploitation ability of the algorithm, the cosine adaptive method is introduced to dynamically adjust parameter c. In the early stage, the parameter c decreases more slowly than the decline rate with the linear method, which can expand the search space and increase the global searching ability. With the increase of iteration times, grasshoppers will converge near the target position. The parameter c decreases more quickly than the decline rate with the linear method in the middle stage. In the later stage, the parameter c decreases more slowly than the decline rate with the linear method, which can reduce the search space and increase the local exploration ability. The expression is shown as follows: where L shows the maximum number of iterations, l is the current iteration, c max = 1, c min = 10 −5 .
In order to avoid the algorithm falling into the local optimal solution, the skew tent chaotic mapping was adopted to improve the search efficiency of GOA. Compared with the logistic chaotic map, the chaos sequence generated by the skew tent chaotic map had better randomness, more uniform distribution, and was not sensitive to the initial value. This map can effectively improve the search space of GOA and accelerate the convergence of the algorithm. The expression is shown as follows: where x n is the nth number in the sequence, β is the control parameter of the chaotic map.
In order to show the performance of improved grasshopper optimization algorithm (IGOA), the statistical analysis of improved grasshopper optimization algorithm (IGOA) was tested on 6 unimodal benchmark functions and 4 multimodal benchmark functions. The benchmark functions used are available in Appendix B. The unimodal functions (F1-F6) have only one optimal solution, so the convergence speed and exploitation ability of the improved grasshopper optimization algorithm (IGOA) can be tested. The multimodal functions (F7-F10) have multiple local optimal solutions, so we can test the local optimization and exploration ability of the improved grasshopper optimization algorithm (IGOA) using the multimodal functions.
For solving the test functions, 50 search agents and 500 iterations were employed. Firstly, the convergence rate and global optimization ability are tested on the unimodal functions (F2) and the multimodal functions function (F7).  Table 1. The IGOA is obviously superior to GOA in both unimodal and multimodal function optimization problems.  For the concrete temperature data stream, the data arrive one by one, and each temperature data point is independent of each other. We can obtain the prediction mean value and variance for each data point by the HK-RVM prediction model. According to the normal distribution characteristics, if the target value exceeds the prediction intervals at the specified confidence level, then the new monitoring data is considered unsuitable for the current model.
As mentioned before, the window size is essential for concept drift detection. In the variable window technique, the size of the sliding window is determined by the number of target values exceeding the prediction interval at a specified confidence level. When the number reaches the specified threshold, the sliding window is closed. Then, the size of the data in the window is taken as the width of the sliding window. If there is no concept drift in the data stream, the width of the sliding window will remain stable. Therefore, we can judge whether concept drift occurs by observing the size of the sliding window. In practical applications, when concept drift is detected, the HK-RVM model will be retrained using the data in the sliding window, and then the model is updated.

Case Study
In this section, the proposed method in this paper is applied to an arch dam in southwestern China. The dam is a concrete double curvature arch dam. The maximum dam height is 270.0 m, and the total concrete volume is 2.80 million m 3 . The dam is divided into 15 dam monoliths with 14 transverse joints horizontally. The dam has 29 grouting areas vertically. Joint grouting is arranged at the transverse joints between dam monoliths. In this section, we describe the historical temperature data, establish the interval prediction model of the temperature field, and calculate the simulation construction schedule coupling temperature field interval prediction model.

Temperature Data Description
By integrating the meteorological, water cooling, and thermometer data, the sample dataset was obtained. The integrated data include the environmental factors (air temperature), water-cooling factors (water temperature and water flow), heat conduction factor (concrete age, and upper and lower dam block concrete temperatures), as well as the target temperature value (dam block concrete temperature value). We collected about 4800 pieces of data in a dam block. Then, 200 pieces of data were selected as the sample dataset to establish the model, and the rest data were chosen as the dataset of concept drift detection. The model establishment is shown in Section 4.1.2.

Model Establishment
The IGOA optimized HK-RVM was adopted to establish the concrete temperature interval prediction model in this section. The dataset was divided into a training dataset and a testing dataset using a five-fold cross-validation. Then, 20% of the data were used as the testing dataset, and the remaining 80% were used as the training dataset. For the parameters involved, the control parameter of tent chaotic map was set as 0.5, the population number of IGOA was set as 20, and the maximum number of iterations was set as 200, the attraction intensity between grasshoppers was set as 0.5, and the length scale of attraction was set as 1.5.
The convergence curves for the five models are shown in Figure 4. The majority of the modeling converged smoothly around 40-50 iterations. The fitness values of each model were almost the same when the model converged. Then, the temperature prediction intervals model was established. The prediction intervals are shown in Figure 5. The target values are basically within the prediction intervals at a 95% confidence level. The model performance (prediction accuracy and generation ability) will be stated in the Discussion section.  With the progress of the project, the temperature data are increasing quickly, which may exist concept drift. This will lead to the decline of the model prediction performance. Here, 4800 pieces of temperature data were collected, and then these data would be input one by one into the established HK-RVM model for concept drift detection. As mentioned in Section 3.2.3, we determine whether concept drift occurs by observing the size of the sliding window. Here, the basic window size was set as 200, and the specified threshold (the number of target values exceeding the prediction interval) was set as 10. Then, the size of the sliding window would be obtained, as shown in Figure 6. The size of each window fluctuated around 200, but the size of the 6th and 12th windows changed significantly. It shows that there is a sudden concept drift and the model needs to be retrained.

Construction Simulation Coupling Temperature Field Interval Predictions
In the real-time construction simulation mathematical model, the joint grouting simulation included three important parameters: the concrete age, the cover weight, and the temperature field. The concrete age and cover weight can be obtained using a dam block construction simulation. The temperature field is predicted by HK-RVM, as described in Section 4.1. According to the construction technical requirements, the concrete age in the grouting area should be more than 120 days old, and the cover weight of the grouting area should be more than 27.0 m. Besides, a cooling area, a transitional cooling area, and a covering area must be set above the grouting area. Generally, the water-cooling process is divided into the first cooling stage, the midterm cooling stage, and the second cooling stage. When joint grouting, the temperature of the grouting area and the cooling area should reach the target temperature of the second cooling stage (also called closure temperature), and the temperature of the transition area should reach the target temperature of the midterm cooling stage. In this case study, the target temperature of the first cooling stage was 22-24 • C, the target temperature of the midterm cooling stage was 18-19 • C, and the target temperature of the second cooling stage was 12-13.5 • C.
Based on the established interval prediction model in Section 4.1, it was easy to obtain the temperature change curve of all dam blocks in the grouting area, and then the temperature field at a certain time can be obtained. The concrete temperature filed 3D cloud picture is shown in Figure 7 According to the temperature field results and joint grouting requirements, the recommended grouting time is 6 October in the second year.

Prediction Model Comparison
In this section, the prediction accuracy and generalization ability of the model were evaluated to illustrate the superiority of the proposed model. To illustrate, the training dataset is used to measure the model prediction accuracy, while the testing dataset is used to test the generalization ability. Then, the RVM model with different kernel functions (polynomial kernel, gauss kernel, and hybrid-kernel), and Gaussian process regression (GPR) were employed to compare the model performance. Three measurements were employed to compare the model performance: prediction interval coverage probability (PICP), prediction intervals normalized average width (PINAW), and coverage width-based criterion (CWC). The PICP represents the probability of the target value falling into the constructed confidence interval. The PINAW is applied to measure the width degree of the prediction interval. Generally speaking, the larger the PICP is, the more reliable the prediction interval is. The smaller PINAW is, the more accurate the prediction interval is. However, the reliability and accuracy of the model cannot be satisfied at the same time, so PICP and PINAW are contradictory indexes. The CWC can consider comprehensively the reliability and accuracy of prediction intervals. When the CWC is smaller, the prediction interval performs better in reliability and accuracy. The formulas for these measurements are presented in [56]. A detailed discussion is presented below. •

Model prediction accuracy
The prediction accuracy comparison of different interval prediction models was analyzed, as shown in Table 2. According to the comparison results, the PICP of different prediction models has little difference. The PINAW of IGOA-HKRVM is the smallest among the models, thus a narrower prediction range can be obtained. The CWC of IGOA-HKRVM is the smallest among all models. It reveals that the models proposed have higher coverage probability and narrower prediction intervals width than other models. Therefore, the superiority of the model prediction accuracy was proved. •

Model generalization ability
The generalization ability comparison of different interval prediction models was analyzed, as shown in Figure 8. According to the comparison results, the PICP, PINAW, and CWC values of IGOA-HKRVM are 0.975, 0.024, and 0.062, which are the best solution among the models. In addition, IGOA-RVM gauss model performs well in the training set, but it is inferior in the testing set. However, IGOA-RVM poly model performs well in the testing set. It shows that the generalization ability of the polynomial kernel function is better than the Gaussian kernel function, but the learning ability of the polynomial kernel function is inferior to the Gaussian kernel function. The hybrid-kernel method can effectively combine the advantages of polynomial kernel function and Gaussian kernel function, and the comparison results also indicate the superiority of the model generalization ability.

Simulation Results Comparison
The traditional simulation method ignores the influence of the temperature field on the joint grouting. In order to prove the advancement and applicability of the proposed method, we set the simulation results without considering the temperature field as the comparison scheme. Firstly, the joint grouting time comparison between the proposed method and the comparison scheme is carried out. The comparison between the joint grouting time with and without the temperature field and the actual grouting time is shown in Figure 9. The vertical coordinate shows the joint grouting time for each grouting area (take 15 March in the first year as time 0). According to the simulation results of the 1st to 11th grouting area, the joint grouting progress lags behind the comparison scheme, but is closely in accordance with the actual grouting time. It shows that the joint grouting simulation considering the temperature field can obviously improve the simulation accuracy. Based on the construction simulation model and established temperature field prediction model, the grouting time of the other grouting areas can be obtained. It can provide guidance for the subsequent dam joint grouting construction. Then, the dam construction schedule comparison between the proposed method and the comparison scheme is carried out. According to the simulation results, the dam will be completed in July in the fourth year when the temperature field is considered, and the dam will be completed in June in the fourth year when the temperature field is not considered. The dam monthly construction intensity can also be obtained by construction simulation, which can provide a reference for the production and allocation of concrete resources. The comparison of the construction intensity with and without the temperature field is shown in Figure 10.

Conclusions
Joint grouting simulation is an important aspect of dam construction schedule management, which has a crucial influence on the safety and construction schedule of the dam project. However, the current construction simulation model simplified the temperature factors in joint grouting simulation, which led to a deviation between the simulation schedule and the actual situation. Meanwhile, the majority of existing temperature prediction studies were based on deterministic point prediction, which cannot quantify the uncertainties of the prediction value. In order to solve these problems, this paper proposed a real-time construction simulation method coupling concrete temperature field interval prediction model. The main contributions of this paper are as follows:

1.
A real-time construction simulation model was proposed. The model coupled the dam block simulation and joint grouting simulation, and analyzes the relationship between them.

2.
A high-precision, high-generalization capability interval prediction model was established for concrete temperature prediction. The interval prediction model was constructed based on the hybrid-kernel relevance vector machine (HK-RVM) and the improved grasshopper optimization algorithm (IGOA). Compared with the unmodified models and the common models, the proposed model demonstrates superiority with regard to the accuracy, generalization ability.

3.
A concept drift detection method based on the variable window technology was developed for updating the prediction model.
In the future, further study on the influence of the temperature field on dam construction schedules is necessary. Based on the concrete temperature field prediction model, the placing intermittent time of the concrete dam block can be optimized to realize the optimization of the entire dam construction schedule.