Crack-Considered Elastic Net Monitoring Model of Concrete Dam Displacement

College of Water Conservancy and Environmental Engineering, Zhejiang University of Water Resources and Electric Power, Hangzhou 310018, China State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China Key Laboratory for Technology, Rural Water Management of Zhejiang Province, Zhejiang University of Water Resources and Electric Power, Hangzhou 310018, China


Introduction
Structural safety monitoring of concrete dams depends on visual observation and monitoring data analysis with the purpose of, as early as possible, identifying eventual anomalous behavior which could deteriorate the dam. e analysis results can help the dam administrative department take corresponding measures to eliminate the impact of abnormal behavior [1]. Displacement monitoring data have been widely acknowledged as a significant source for the performance and health condition assessments of concrete dams [2] because it directly reflects the structural behavior. erefore, various monitoring models have been built to explain and predict the dam displacement which is influenced by many factors such as hydrostatic pressure, concrete temperature, and time effect [3]. e hydrostatic-seasontime (HST) model is commonly used to interpret dam deformation [4]. Wu et al. [5] discussed the factor selection of displacement statistical models for concrete dams. Rocha [6] introduced the multiple linear regression (MLR) model to analyze the deformation behavior of concrete dams. Based on the assumption that the residual error sequence of MLR models is uncorrelated and that multicollinearity does not exist among those explanatory variables, the optimal unbiased estimates of MLR model coefficients can be obtained using the least-squares method [7]. If some explanatory variables are linearly related, the stepwise least square regression (SLSR) method can be used to estimate model coefficients with multicollinearity considered. To date, HST models have been used to analyze the monitoring data of many dams, and various methods have been proposed to obtain the specific expressions and coefficients of practical models [8][9][10][11][12][13][14][15][16]. Traditional displacement monitoring model research studies focused on model calculation [17], but ignored key factor selection. However, many cracks exist in a lot of concrete dams which have been operating for years [18]. A mouth opening degree of a large surface crack varies regularly with the external loads, and the opening degree variations directly affect the concrete dam displacement [19,20]. Crack-induced displacement is incorporated into the time effect component of those classical models, so the accuracy and interpretability of those models are weakened somewhat. It is necessary to analyze a large surface crack's effect on concrete dam displacement, and a crack-considered monitoring model is yet to be built to improve interpretability. e original theoretical displacement monitoring model of concrete dams conservatively includes as many explaining variables as possible to minimize the model deviation caused by the lack of important explaining variables. However, the actually applied model only needs those explaining variables which have a closer relationship with the explained response to improve its interpretability and predictive precision.
Stepwise regression combined with AIC or BIC criterion has been generally used to select the optimal model. However, it has some inevitable drawbacks. Breiman [21] pointed out that the stepwise regression method is unstable when applied to select the actual model. And Fan et al. [22] indicated that a random error exists in the calculation process of stepwise regression and the theoretical nature of the method is difficult to study. In order to solve the above problems, Breiman [23] puts forward the nonnegative garrote method based on penalized least squares to get the actual interpretable model. Combining the nonnegative garrote method and bridge regression proposed by Frank [24], Tibshirani [25] proposed the Lasso (least absolute shrinkage and selection operator) method to select the explaining variables. Efron et al. [26] put forward the least angle regression algorithm to promote the practical application of the Lasso method. e Lasso method is being carried out in many fields [27][28][29][30][31], including statistics, engineering, biology, and information science.
e Lasso method excessively shrinks the coefficients, which have larger absolute values sometimes, resulting in a larger model error in some cases. In view of this, Fan and Liu [22] proposed the SCAD method, and Zou [32] presented the adaptive Lasso method to solve the deficient coefficient shrinkage problems in some situations. Zou and Hastie [33] introduced the quadratic penalty of coefficients into the Lasso method and put forward the another improved Lasso method called the elastic net method. e elastic net method can select significant explaining variables and estimate the corresponding coefficients simultaneously. Moreover, the method moderately shrinks the coefficients. e elastic net method with better stability is more suitable to solve the displacement monitoring model. is article is organized as follows. In Section 2, large surface crack's effect on the concrete dam displacement is analyzed theoretically and then the crack factor is introduced to develop a theoretical crack-considered monitoring model of concrete dam displacement. In Section 3, the elastic net method with better stability is used to solve the crackconsidered displacement monitoring model. In Section 4, the proposed model is applied to analyze the radial displacement of a gravity arch dam, and the analysis results demonstrate better interpretability and higher predictive precision. Concluding remarks complete the paper in Section 5.

Theoretical Model Development
Traditional concrete dam displacement monitoring models divide the displacement response into the hydrostatic pressure component, temperature component, and time effect component, which can be expressed as follows: where δ represents the concrete dam displacement response, δ H denotes the displacement caused by hydrostatic pressure, δ T indicates the displacement caused by temperature, δ θ refers to the displacement caused by the time effect, andε means the random error term. Based on the dam engineering theory and engineering mechanics, the mathematical expressions for each component of the concrete dam horizontal displacement response in equation (1) have been deduced [6].

Hydrostatic-Pressure Component.
e hydrostaticpressure-induced horizontal displacement of any point in a concrete dam consists of three parts [34] which can be expressed as follows: (2) e first part δ 1H comes from the dam body deformation caused by the reservoir water pressure. e second part δ 2H is derived from the foundation deformation induced by reservoir water pressure. e third part δ 3H is from foundation surface rotation caused by reservoir water gravity.
For the gravity dam, the cross section can be simplified as a triangular wedge with a vertical upstream surface and the dam can be simplified as a cantilever beam fixed on the foundation, as shown in Figure 1. e following explanation can be obtained through the mechanical relationship between the water level and dam deformation. Reservoir water pressure on the beam has a linear correlation with water depth.
e water pressure to the dam body causes the horizontal displacement δ 1H : 2 Mathematical Problems in Engineering where H is the upstream water level, h is the height of the dam, m is the downstream slope, d is the distance between the observation point and the dam crest, E c and G c are the elastic modulus and shear modulus of dam concrete, respectively, and c 0 is the water density. e effect of reservoir water pressure is transferred by the cantilever beam to the foundation. e water pressure to the bottom of a dam causes the horizontal displacement δ 2H : where E r and μ r are elastic modulus and Poisson's ratio of the dam foundation, respectively. e topography and geological conditions of the reservoir area are complex, so a strict derivation for the mathematical expressions of δ 3H is difficult. Here, vertical reservoir water pressure is regarded as acting uniformly on an infinite elastic body surface with the assumption that the reservoir bottom near the dam is horizontal and has a certain width. e horizontal displacement δ 3H caused by the rotation of dam foundation is, approximately, as follows: where α is the rotation angle of the dam foundation surface at the dam heel. From the above analysis, it can be found that δ 1H of the gravity dam is derived to have a linear relation with H, H 2 , and H 3 . And δ 2H is linearly related with H 2 and H 3 . For the arch dam, reservoir water pressure on the upstream surface is borne by both the horizontal arch and the vertical cantilever beam which are simplified in its structural analysis, so the water pressure distribution on the beam is nonlinear. Water pressure on the beam for the arch dam has a quadratic relation with water depth H. us, δ 1H of the arch dam has a linear relation with H 2 , H 3 , and H 4 , and δ 2H is linearly related with H 3 and H 4 . In addition, δ 3H is approximately considered to be proportional to H. erefore, the mathematical expression for the hydrostatic-pressure-induced horizontal displacement of any point in a concrete dam can be expressed as follows: where a i stands for the coefficients corresponding to the hydrostatic-pressure component and m 1 takes 3 and 4, respectively, for the gravity dam and the arch dam.

Temperature Component.
e temperature component of concrete dam displacement is mainly caused by the temperature variation of dam concrete and bedrock, so the measurements of the thermometers buried in the dam body and foundation are selected as temperature factors. However, the number of thermometers embedded in many dams is small and some thermometers have been failed after years of operation. Besides, the dam temperature field tends to be quasi-steady over time, so the temperature of any point in the dam is approximately considered to be periodic [35].
us, the temperature component can be simply represented by some periodic functions as follows: where t denotes the accumulated days from the initial measured day to the observation day, b 1i and b 2i represent the corresponding regression coefficients, and m 2 refers to the cycle parameter which takes 1 for a year cycle and 2 for half a year cycle, and so on.

Time Effect Component.
e time effect component of concrete dam displacement comes from many aspects, including creep and plastic deformation of dam concrete and bedrock, irreversible deformation caused by long-term adverse loads, and self-grown volume deformation [14]. In general, time effect displacement changed dramatically during initial water impoundment and then gradually steadily over time. For the concrete dam operating for years, by analyzing the developing trend of displacement observations and the remaining value δ − δ H − δ T , the expression of δ θ can be determined reasonably as follows: where c 1 and c 2 indicate the corresponding regression coefficients and θ � t/100 and t has the same meaning with that in equation (7).

Crack Component.
e mouth opening degree of some large surface cracks, such as horizontal surface cracks and longitudinal ones, varies regularly with the external loads, including hydrostatic pressure and temperature. e opening degree variations of these cracks obviously affect the concrete dam displacement. Herein, we take a horizontal surface crack as an example to analyze the crack's effect on the dam displacement.
Assuming the concrete dam body is rigid, without considering water pressure, Figure 2 shows the effect of large surface crack on the dam horizontal displacement. e rotation angles α and β are deemed approximately equal (sinα � sinβ); thus equations (9) and (10) can be approximately given: where l A and l K indicate the distances shown in Figure 2, ΔK represents the opening degree variation of the crack, and δ AJ denotes the horizontal displacement of point A. For a specific point A, l A /l K is the deemed constant; then, the crack-induced displacement can be approximately expressed as follows: where K i represents the opening degree of the ith large surface crack, d i stands for the regression coefficients corresponding to the crack component, and m 3 refers to the number of large surface cracks.
where a 0 denotes the constant term, εindicates the random error term, and the other symbols have the same meanings with those in the above equations.

Elastic Net. Given a linear regression with standardized predictors x ij and centered response values
is defined by the following equation: For all t > 0, the solution for α is y. Assuming without loss of generality that y � 0 and hence omitting α, the lasso solves the L 1 -penalized regression problem of finding β � e lasso is similar to ridge regression with constraint j β 2 j ≤ t. However, ridge regression only shrinks the coefficients, whereas the lasso estimate does both variable selection and shrinkage in the above L 1 -penalized regression. e lasso and ridge regression can share a more general penalty of the form ( j β q j ) 1/q ≤ t, where q takes 1 for the lasso and 2 for ridge regression. q takes the smaller value for the lasso (i.e., closer to subset selection which appears as q tends to zero). It is a convex problem which is attractive for computational purposes.
e Lasso method is actually to optimize a function with loss item N i�1 (y i − j x ij β j ) 2 and penalty item λ p j�1 |β j |. e improved Lasso methods were given by changing the penalty function item. Among these improved lasso methods, the elastic net estimate β is defined as follows: where λ 1 and λ 2 represent the penalty parameters and λ 1 , λ 2 ≥ 0. λ 1 takes 0 for the ridge regression estimate. e Lasso estimate is given when λ 2 takes zero. Asymptotics for Lasso-type estimators has been analyzed by Knight and Fu [36]. However, the Lasso method shrinks all the coefficients to the same degree resulting in a large model error, and the Lasso method does not have oracle properties. In comparison, improved lasso methods including the elastic net method commonly have oracle properties, that is, the optimal model obtained by elastic net has the following properties.
Letting A � i|β i ≠ 0 , p 0 represents the number of elements in the set A and p 0 < p. Assuming β(η) is the estimator of model coefficients obtained by the method η, the method η has oracle properties indicating that β(η) asymptotically meets the following two conditions: where Σ stands for the covariance of real model parameters. e oracle properties are also referred to as super-efficiency in the parameter estimation.

Practical Elastic Net Crack-Considered Monitoring Model of Concrete Dam Displacement.
Combining with the elastic net method with better stability, moderate coefficients' shrinkage, and oracle properties, the practically applied elastic net crack-considered monitoring model of concrete dam displacement is obtained as follows: where (α, β) represents the elastic net estimator of the model coefficients, N denotes the number of observations, and nindicates the number of explaining variables. Determination of penalty parameters λ 1 and λ 2 is critical in the model solution. In general, n-fo1d cross va1idation [37] is applied to determine penalty parameters. e following equations briefly introduce the n-fo1d cross va1idation.
Suppose T represents the complete dataset, T − T n represents the training set, and T n represents the testing set, where n � l,2,. . .. For each λ 1 , λ 2 , and n, the estimate (α, β) (n) (λ 1 , λ 2 ) of (α, β) can be obtained from the training set T − T n . en, the standard CV(λ 1 , λ 2 ) of n-fo1d cross va1idation can be given as follows: e estimator λ 1 , λ 2 of the penalty parameters can be obtained as follows: (18) Figure 3 shows how to solve the elastic net crack-considered monitoring model of concrete dam displacement.

Displacement Model Development.
e crest horizontal displacement of a concrete dam is analyzed by the elastic net crack-considered monitoring model. is dam is a concrete gravity arch dam, as shown in Figure 4, located in the east of China. Its maximum height is 76.3 m, and it has 28 sections. e dam was constructed in three phases. e concrete lift in Phase II was poured in such a hurry that its shrinkage distortion was restrained by the concrete in Phase I, resulting in a horizontal crack in the longitudinal direction at the interface. e crack extends from dam section No. 5 to No. 28 and is up to 5 m deep into the dam body. A large vertical crack in the longitudinal direction also exists in the dam crest. e dam crest radial displacement monitoring data in dam section No. 26 from May 8, 2013, to July 25, 2019, are analyzed. e monitoring data provided in the supplementary material "Monitoring data of dam section No. 26 pdf" were obtained from the pendulums buried in the dam, as shown in Figure 5 and Figure 6. e elevation in Figure 6 is in meters.
According to the elastic net crack-considered model, with the actual project situation considered, the radial displacement monitoring model of the dam crest is obtained as follows: Obtain the elastic net estimate with the ultimate λ 1 , λ 2 (α , β) = arg min Output the practically-applied elastic net crack-considered monitoring model of concrete dam displacement   a 2 , a 3 , a 4 , b 11 , b 12 , b 21 , b 22 , c 1 , c 2   where K 1 represents the opening degree of the vertical crack described above, K 2 indicates the opening degree of the horizontal crack described above, and the other symbols refer to the meanings of those variables in the aforementioned equations. e recorded water level curve of the reservoir is shown in Figure 7. Figure 8 displays the opening degree curves of the two cracks mentioned above.

Results and Discussion
e radial displacement monitoring data from May 8, 2013, to December 31, 2018, were used to establish the monitoring model. Tables 1 and 2 show the results obtained by different models. In Table 2, R stands for the multiple correlation coefficient. Larger the multiple correlation coefficient, better the fitted effect. In Table 2, S means standard deviation. Smaller standard deviation represents better fitted effect. R and S are defined as follows: (20) Figures 9-12 give comparisons between measured displacement curves and fitted ones obtained by four different models. Displacement components extracted by the crackconsidered elastic net model are given in Figure 13. (1) e multiple correlation coefficient R (0.961) of the crack-unconsidered stepwise model is smaller than that (0.975) of the crack-considered stepwise model. e standard deviation S (0.218) of the crack-unconsidered stepwise model is larger than that (0.199) of the crack-considered stepwise model. For models obtained by the elastic net method, R (0.964) of the uncrack-considered model is smaller than that (0.978) of the crack-considered model and S (0.215) of the uncrack-considered model is larger than that (0.193) of the crack-considered model. ese results show that the crack-considered models have a better fitting effect than the crack-unconsidered ones.
(2) e values of the coefficient d 1 are positive for both the crack-considered stepwise model and the crackconsidered elastic net model, which means that the dam crest displacement is positively correlated with the vertical crack opening degree, that is, the opening of the vertical crack contributes to the downstream displacement of the dam crest. Meanwhile, the negative values of d 2 obtained by the two crackconsidered models indicate that the opening of the horizontal crack contributes to the upstream displacement of the dam crest. It can also be seen from Figure 13 that the crack components account for considerable proportions in the dam crest displacement. erefore, it is necessary to extract the crack component separately to explain the variation behavior of the dam displacement, that is, the crackconsidered model has better interpretability.  1 and c 2 ), that is, the time effect is regarded not to contribute to the displacement. However, it is confirmed that the time effect component cannot be a zero value by analyzing the measured series and referring to general engineering experience. e time effect component obtained by the elastic net models is not a zero value. e results imply that the elastic net models have better interpretability than the stepwise regression models.
In order to evaluate the forecast effect of the monitoring model, the radial displacement monitoring data from January 1, 2019, to July 25, 2019, were used as predictive samples. Figures 16 and 17 give comparisons between measured displacement residual curves and predicted ones obtained by different models. Figure 18 displays a comparison between the measured displacement curve and the predicted one obtained by the crack-considered elastic net model. Table 2 shows the predictive effect (R pre and S pre ) of different models. R pre and S pre are multiple correlation coefficient and standard deviation, respectively, representing the predicted goodness. rough comparison from the table and figures, the following analysis result can be obtained.

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Mathematical Problems in Engineering (2) For the crack-considered models, R pre (0.975) of the stepwise regression model is approximately equal with that (0.978) of the elastic net model, while S pre (0.174) of the elastic net model is smaller than that (0.201) of the stepwise regression model. e results indicate that the elastic net models have a better predictive effect than the stepwise regression models.

Conclusions
is study was devoted to establish and solve the crackconsidered elastic net monitoring model of the concrete dam displacement. e main conclusions are as follows: (1) Concrete dam safety operation and management need analysis and forecast on the dam displacement monitoring data. By inputting displacement monitoring data into theoretical monitoring models, the final actually applied monitoring and forecasting models can be obtained to interpret and predict the work behavior of concrete dams. (2) Crack-induced displacement in classical displacement monitoring models is incorporated into the time effect component so that the accuracy and interpretability of those models are weakened somewhat. In this paper, the crack factor was introduced into the model to be an independent crack component. rough the theoretical analysis of large surface crack's effect on the concrete dam displacement, a mathematical expression of the crackinduced displacement component was derived, and the theoretical crack-considered monitoring model of the concrete dam displacement was built to improve the interpretability of the model. (3) e theoretical monitoring model of the concrete dam displacement conservatively includes as many explaining variables as possible to minimize the model deviation. However, the actually applied model only needs those explaining variables which have a closer relationship with the explained response to improve its interpretability and predictive precision. Conventional stepwise regression methods used in variable selection have instability in some cases. e elastic net method with better stability and moderate coefficient shrinkage was used to develop the practically applied crack-considered elastic net monitoring model of concrete dam displacement. e proposed model gives more reasonable factors and parameter estimation. (4) study on the crack-considered elastic net monitoring model of concrete dam displacement has been conducted tentatively in the paper, and a project case exhibits better interpretability and higher prediction accuracy of the proposed model. Furthermore, the thought and method of establishing the crack-

Data Availability
e data used to support the findings of this study are from a large water conservancy project and not suitable to upload to the network. e data are included within the Supplementary Information files.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.