# Integrating the Finite Element Method with a Data-Driven Approach for Dam Displacement Prediction

Authors: Chenfei Shao; Chongshi Gu; Zhenzhu Meng; Yating Hu
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

Both numerical simulations and data-driven methods have been applied in dam’s displacement modeling. For monitored displacement data-driven methods, the physical mechanism and structural correlations were rarely discussed. In order to take the spatial and temporal correlations among all monitoring points into account, we took the first step toward integrating the finite element method into a data-driven model. As the data-driven method, we selected the random coefficient model, which can make each explanatory variable coefficient of all monitoring points following one or several normal distributions. In this way, explanatory variables are constrained. Another contribution of the proposed model is that the actual elastic modulus at each monitoring point can be back-calculated. Moreover, with a Lagrange polynomial interpolation, we can obtain the distribution field of elastic modulus, rather than gaining one value for the whole dam in previous studies. The proposed model was validated by a case study of the concrete arch dam in Jinping-I hydropower station. It has a better prediction precision than the random coefficient model without the finite element method.


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## Body

## 1. Introduction

For dams that have been running for years, their actual physical and mechanical properties often have considerable differences from the initial or design values, due to the effect of external environment. Hence, it is crucial to monitor and evaluate the dam’s structural behaviors and its actual material properties. Using monitoring devices installed inside the dam, displacement, rotation, seepage, and stress can be measured, among which displacement is the most important [1]. Dam’s displacement is not only affected by the external conditions (e.g., water load, temperature, and aging), but also depends on its material properties such as elastic modulus. Elastic modulus, the ratio of applied external stress to the elastic deformation of material, is usually difficult to measure; hence, it is often obtained by back-analysis based on monitored displacement data.For the dam’s displacement prediction, two approaches have been commonly used: one is numerical simulation using finite element method [2, 3]; another is data-driven method based on monitored displacement data [4, 5]. The principle of numerical simulations is to solve the constitutive equations; emphasis is given on physical mechanism governing the displacement [6, 7]. Yet the numerical simulation often has considerable inaccuracies induced by unavoidable simplifications and idealizations of the dam’s structural and material properties. Moreover, the simulation results fully depend on the initial setting of parameters, without any calibration with the actual displacement. Many researches have improved the accuracy of finite element method, especially when complex situations are involved [8–10]. However, there is still a gap in integrating the numerical simulation with monitored data in real world and avoiding the inaccuracies due to structural simplifications.For data-driven methods, with the development of soft computing techniques, the modeling algorithms improved from the simple linear regression methods [11–13] to the advanced machine learning methods such as support vector machines (SVM) [14, 15] and artificial neural network (ANN) [16]. In recent years, many synthetic models that integrate several machine learning methods also have been carried out [17, 18]. Unlike the finite element methods that simulate the dam’s behavior based on the design structural and material properties, data-driven methods cope with monitored data in real world. However, compared with the fact that finite element method can simulate the displacement of arbitrary point on the dam, the statistical model can merely model the displacement at the monitoring point. In addition, the data recorded at each monitoring point were modeled independently; the spatial and temporal correlations among the monitoring points were rarely discussed, which may weaken the generalization ability and sometimes result in overfitting problems [19, 20].In order to take considerations of spatial correlations among monitoring points in data-driven models, recent researches integrated clustering models with statistical models; they first clustered the monitoring points into several groups based on one or several indicators and then constrained the coefficients of explanatory variables in statistical model based on the clustering results. Shao et al. clustered the monitoring points based on their spatial characteristics, namely, distance from the monitoring point to the dam foundation [21]. Hu et al. provided a clustering model that considers the temporal-spatial characteristics [22]. However, their constraint ability is still not satisfactory, as they provided constraint for each group rather than for each monitoring point. The primary objective of the present study is to consider spatial correlations and provide constraints for each monitoring point rather than for each group.With this objective in mind, we took advantage of finite element method that can simulate the deformation at any arbitrary point on the dam to consider the structural correlations of each monitoring point. Specifically, we determined that the water pressure component and temperature component of the whole region of dam with the finite element model, the ratio of design and actual elastic modulus, and the ratio of design and actual coefficient of linear expansion are only variables in the proposed model. Then we constructed variables relating to time in aging component to consider the temporal correlations. The random coefficient model assumes the monitoring points as a panel and the coefficients of all explanatory variables in the model following a Gaussian distribution, which constrains the coefficients at a certain extent. The multicollinearity problem is solved by constraining the explanatory variables and the model becomes more stable and has a better prediction ability. Another contribution of the proposed model is that the actual elastic modulus can be back-calculated for each monitoring point. Rather than gaining one elastic modulus for the whole dam in previous studies, with a Lagrange polynomial interpolation, we obtained a distribution field of elastic modulus from the value at each monitoring point.This article is organized as follows. Section2 presents the development of the synthetic model that combines the finite element method with the random coefficient model. The engineering project and dataset are exhibited in Section 3. The modeling results are presented in Section 4. Section 5 discusses the prediction precision of the proposed model and the distribution field of elastic modulus. Concluding remarks complete the paper in Section 6.


## 2. Methods

### 2.1. Statistical Model

In the statistical model, the concrete dam’s displacementδ is commonly divided into three components: the water pressure component δH, the temperature component δT, and the aging component δθ [23]. The dam’s displacement is expressed as follows:(1)δ=δH+δθ+δT.


#### 2.1.1. Water Pressure Component

The water pressure componentδH includes displacement caused by itself δ1H, dam foundation δ2H, and rotation of the dam bedrock δ3H, which are made up of polynomials of H,H2,H3, H4 [1]. δH can be simplified as(2)δH=∑i=1naiHi,where H is the upstream water level, ai are the coefficients of water pressure component which are related to the dam height h, m is the downstream slope angle, and d is the distance from the monitoring point to the dam foundation, which are constants for indicated monitoring points. In addition, ai also depends on dam’s material properties including elastic modulus of the dam body Ec, Poisson's ratio of the dam body μc, elastic modulus of the foundation Er, and Poisson’s ratio of the foundation μr. Poisson’s ratios μc and μr have negligible effect on the dam’s displacement. Therefore, the elastic modulus of the dam body and elastic modulus of the foundation Ec and Er are the merely random coefficients in the water pressure component.


#### 2.1.2. Temperature Component

The temperature componentδT is mainly composed of the displacement produced by the temperature variation of dam body concrete and dam foundation rock. When there is lack of temperature data, multiperiod harmonic is used to describe the temperature component:(3)δT=∑i=1m3b1isin2πit365+b2icos2πit365,where i indicates the period; for the annual cycle i = 1, and for half a cycle i = 2; b1i and b2i represent coefficients; t is the cumulative number of days from the initial value to monitoring value; and m3 is usually taken as 1 or 2.


#### 2.1.3. Aging Component

Under long-term external loads, the dam produces inelastic displacement related to time; a fixed form of trend function is used to describe the aging component:(4)δθ=c1θ+lnθ,where θ=t/100.In summary, according to the characteristics of concrete dam, the statistical model of concrete dam can be depicted as(5)δ=a0+∑i=1naiHi+∑i=1m3b1isin2πit365+b2icos2πit365+c1θ+lnθ,where a0 denotes the constant term; other symbols mean the same as above.


### 2.2. Combination of the Finite Element Method and Statistical Model

In the proposed model, we can obtain simulated water pressure component and temperature component by the finite element method with design material parameters. The actual elastic modulus of the dam bodyEc and actual elastic modulus of the dam foundation Er can be expressed as Er=λEc; then, δH becomes(6)δH=1Ec∑i=1nai0Hi,where ai0 are the coefficients determined by h, m, d, and λ. As expressed in equation (7), with a given design elastic modulus of the dam body Ec0 and upstream water level H, we can obtain a numerical solution of water pressure component δH0 using the finite element method. Then, we can obtain a series of coefficients ai1 by fitting the upstream water level H and numerical solution δH0:(7)δH0=1Ec0∑i=1nai1Hi.We then applyai1 instead of ai0 to equation (6); the displacement component δH can be expressed as(8)δH=Ec0Ec1Ec0∑i=1nai1Hi=Ec0EcδH0=ξδH0,where ξ is the ratio from the design elastic modulus of the dam body Ec0 to the actual elastic modulus of the dam body Ec.Similar toδH, the coefficients of explanatory variables in the temperature component δT can also be determined using the finite element method. The displacement depending on temperature load in one dimension can be described by dL=αLdT, where α is the coefficient of linear expansion, L is the length, and T is temperature. The coefficient of linear expansion αc can be considered as the only random coefficient. The temperature component δT can be expressed as(9)δT=αc∑i=1nbi0Ti,where bi0 represents the coefficients of temperature component, αc is the actual coefficient of linear expansion of the dam body, and T is the environmental temperature. With a given design coefficient of linear expansion αc0 and temperature T, we can obtain a numerical solution of temperature component δT0 using finite element method. By fitting the temperature T and the numerical solution δT0, the coefficients bi1 of T can be obtained as(10)δT0=∑i=1nbi1Ti.We then applybi1 instead of bi0 in equation (9), and δH then can be expressed as(11)δT=αcαc0αc01∑i=1nbi1Ti=αcαc0δT0=ψδT0,where ψ is the ratio from the actual coefficient of linear expansion αc to design coefficient of linear expansion αc0.The aging componentδθ characterizes irreversible displacement caused by factors such as creep and fatigue of concrete. Due to the difficulties of expressing δθ theoretically, here, we provide a formula, which includes time and monitoring point’s coordinates as variables, to describe its tendency:(12)δθ=∑l=0L∑m=0Md1lmθ+d2lmlnθxlzm,where x is the horizontal coordinate and z is the vertical coordinate of the monitoring point, θ=t/100 , t is the cumulative number of days from the initial value to monitoring value, and d1lm and d2lm are unknown coefficients. Then, the radial displacement can be exhibited as(13)δi=δiH+δiT+δiθ,=ξiδH0+ψiδT0+∑l=0L∑m=0Md1lmθ+d2lmlnθxilzim+u,where i is the index of measuring point, u is the random error term, δH0 are series of water pressure component calculated under groups of upstream water level H with finite element method, δT0 are series of temperature component calculated under groups of temperature T with finite element method, and coefficients ξi, ψi, d1lm, d2lm, and u are unknown coefficients in the model of all monitoring points which can be calculated with random coefficient model.In the model, we can obtain the actual material parametersEc and ψ when ξ and ψ are calculated. In addition, ξiδH0 and ψiδT0 represent the displacement produced by water pressure and fluctuation of temperature; ∑l=0L∑m=0Md1lmθ+d2lmlnθxilzim represents the displacement produced owing to aging behavior of material such as creep and deterioration of material strength and so forth.


### 2.3. Solving with the Random Coefficient Model

With random coefficient model, pending coefficients in equation (9) of all monitoring points can be calculated simultaneously obeying an asymptotic normal distribution. The monitoring displacement data are two-dimensional panel data containing time series and cross-section panel. Data on one panel represent the displacement data at an indicated time. The regression coefficients of one panel can be expressed by the following formula:(14)yit=∑k=1Kβkixkit+uit=∑k=1Kβk+γkixkit+u,where yit represents the two-dimensional dam displacement data; xkit represents the two-dimensional data of explanatory variables; t is the time index; i is the index of section; k is the index of the explanatory variables; the coefficient βki is independent with time, which can be decomposed into βk and γki; β=β1,…,βK′ is the common mean coefficient vector; γ=γ1i,…,γKi′ is the degree of the individual deviated from the common mean; u is the random error term.The objective is to obtainβki from the two-dimensional data series yit and xkit. According to the above analysis, we consider that the random coefficients βki obey an asymptotic normal distribution. Here, βi=β+γi is considered as a random coefficient with a common mean value β; Swamy gave the following assumptions [24]. In the equations below, the symbol at the upper right corner of the letter represents the transposition operation:(15)Eγiγj′=Δ,i=j,0,i≠j.Eγi=0,Exitγj′=0,Euiuj′=σi2IT,i=j,0,i≠j.By integrating the observation data, the matrix equation can be obtained as follows:(16)y=Xβ+X˜γ+u,where(17)yNT×1=y1′,…,yN′,XNT×1=X1X2⋮XN,X˜NT×NK=X1⋯0⋮⋱⋮0⋯XN,u=u1′,…,uN′',γ=γ1′,…,γN′',where N is the number of panels, T is the amount of data in the panel, the compound error term X˜γ+u is a diagonal matrix, and the i-th diagonal block is ψi=XiΔXi′+σi2IT. Under Swamy’s assumption, the estimator of β using the ordinary least squares is biased. When X′X/NT converges to a nonzero constant matrix, we can get a consistent noneffective estimation. The generalized least squares estimation β^GLS is the optimal linear unbiased estimator of β:(18)β^GLS=∑i=1NXi′ψi−1Xi−1∑i=1NXi′ψi−1yi=∑i=1NWiβ^i,Wi=∑i=1NΔ+σi2Xi′Xi−1−1−1Δ+σi2Xi′Xi−1−1,β^=Xi′Xi−1Xi′yi.The variance of the estimator can be written as(19)Varβ^GLS=∑i=1NXi′ψi−1Xi−1=∑i=1NΔ+σi2Xi′Xi−1−1−1.Usingβ^i=Xi′Xi−1Xi′yi and u^i=yi−Xiβ^i, we can obtain the unbiased estimator of σi2 and Δ.(20)σi2=u^i′uiT−K=1T−Kyi′I−XiXi′Xi−1Xi′yi,Δ^=1N−1∑i=1Nβ^i−N−1∑i=1Nβ^iβ^i−N−1∑i=1Nβ^i′−1N∑i=1Nσ^i2Xi′Xi−1.However,Δ^ obtained from estimator might be negative. In this case, we chose(21)Δ^=1N−1∑i=1Nβ^i−N−1∑i=1Nβ^iβ^i−N−1∑i=1Nβ^i′.Thus, even though the estimator is not unbiased, it is invariable nonnegative. Moreover, whenT⟶∞, the estimate is consistent. Therefore, β^GLS obeys an asymptotic normal distribution and is a valid estimator of β.Figure1 presents the flowchart of the proposed synthetic model which integrates the finite element method into a random coefficient model based on the statistical model. The cross-section displacement data are analyzed as a panel in the random coefficient model, and the independent random coefficients ξi and ψi obey normal distributions. In addition, the proposed model can derivate the actual elastic modulus and actual coefficient of linear expansion at each monitoring point, which are dominant factors representing dam’s properties.Figure 1
Flowchart of the proposed model which integrates the finite element method into a random coefficient model based on the statistical model.


## 2.1. Statistical Model

In the statistical model, the concrete dam’s displacementδ is commonly divided into three components: the water pressure component δH, the temperature component δT, and the aging component δθ [23]. The dam’s displacement is expressed as follows:(1)δ=δH+δθ+δT.


### 2.1.1. Water Pressure Component

The water pressure componentδH includes displacement caused by itself δ1H, dam foundation δ2H, and rotation of the dam bedrock δ3H, which are made up of polynomials of H,H2,H3, H4 [1]. δH can be simplified as(2)δH=∑i=1naiHi,where H is the upstream water level, ai are the coefficients of water pressure component which are related to the dam height h, m is the downstream slope angle, and d is the distance from the monitoring point to the dam foundation, which are constants for indicated monitoring points. In addition, ai also depends on dam’s material properties including elastic modulus of the dam body Ec, Poisson's ratio of the dam body μc, elastic modulus of the foundation Er, and Poisson’s ratio of the foundation μr. Poisson’s ratios μc and μr have negligible effect on the dam’s displacement. Therefore, the elastic modulus of the dam body and elastic modulus of the foundation Ec and Er are the merely random coefficients in the water pressure component.


### 2.1.2. Temperature Component

The temperature componentδT is mainly composed of the displacement produced by the temperature variation of dam body concrete and dam foundation rock. When there is lack of temperature data, multiperiod harmonic is used to describe the temperature component:(3)δT=∑i=1m3b1isin2πit365+b2icos2πit365,where i indicates the period; for the annual cycle i = 1, and for half a cycle i = 2; b1i and b2i represent coefficients; t is the cumulative number of days from the initial value to monitoring value; and m3 is usually taken as 1 or 2.


### 2.1.3. Aging Component

Under long-term external loads, the dam produces inelastic displacement related to time; a fixed form of trend function is used to describe the aging component:(4)δθ=c1θ+lnθ,where θ=t/100.In summary, according to the characteristics of concrete dam, the statistical model of concrete dam can be depicted as(5)δ=a0+∑i=1naiHi+∑i=1m3b1isin2πit365+b2icos2πit365+c1θ+lnθ,where a0 denotes the constant term; other symbols mean the same as above.


## 2.1.1. Water Pressure Component

The water pressure componentδH includes displacement caused by itself δ1H, dam foundation δ2H, and rotation of the dam bedrock δ3H, which are made up of polynomials of H,H2,H3, H4 [1]. δH can be simplified as(2)δH=∑i=1naiHi,where H is the upstream water level, ai are the coefficients of water pressure component which are related to the dam height h, m is the downstream slope angle, and d is the distance from the monitoring point to the dam foundation, which are constants for indicated monitoring points. In addition, ai also depends on dam’s material properties including elastic modulus of the dam body Ec, Poisson's ratio of the dam body μc, elastic modulus of the foundation Er, and Poisson’s ratio of the foundation μr. Poisson’s ratios μc and μr have negligible effect on the dam’s displacement. Therefore, the elastic modulus of the dam body and elastic modulus of the foundation Ec and Er are the merely random coefficients in the water pressure component.


## 2.1.2. Temperature Component

The temperature componentδT is mainly composed of the displacement produced by the temperature variation of dam body concrete and dam foundation rock. When there is lack of temperature data, multiperiod harmonic is used to describe the temperature component:(3)δT=∑i=1m3b1isin2πit365+b2icos2πit365,where i indicates the period; for the annual cycle i = 1, and for half a cycle i = 2; b1i and b2i represent coefficients; t is the cumulative number of days from the initial value to monitoring value; and m3 is usually taken as 1 or 2.


## 2.1.3. Aging Component

Under long-term external loads, the dam produces inelastic displacement related to time; a fixed form of trend function is used to describe the aging component:(4)δθ=c1θ+lnθ,where θ=t/100.In summary, according to the characteristics of concrete dam, the statistical model of concrete dam can be depicted as(5)δ=a0+∑i=1naiHi+∑i=1m3b1isin2πit365+b2icos2πit365+c1θ+lnθ,where a0 denotes the constant term; other symbols mean the same as above.


## 2.2. Combination of the Finite Element Method and Statistical Model

In the proposed model, we can obtain simulated water pressure component and temperature component by the finite element method with design material parameters. The actual elastic modulus of the dam bodyEc and actual elastic modulus of the dam foundation Er can be expressed as Er=λEc; then, δH becomes(6)δH=1Ec∑i=1nai0Hi,where ai0 are the coefficients determined by h, m, d, and λ. As expressed in equation (7), with a given design elastic modulus of the dam body Ec0 and upstream water level H, we can obtain a numerical solution of water pressure component δH0 using the finite element method. Then, we can obtain a series of coefficients ai1 by fitting the upstream water level H and numerical solution δH0:(7)δH0=1Ec0∑i=1nai1Hi.We then applyai1 instead of ai0 to equation (6); the displacement component δH can be expressed as(8)δH=Ec0Ec1Ec0∑i=1nai1Hi=Ec0EcδH0=ξδH0,where ξ is the ratio from the design elastic modulus of the dam body Ec0 to the actual elastic modulus of the dam body Ec.Similar toδH, the coefficients of explanatory variables in the temperature component δT can also be determined using the finite element method. The displacement depending on temperature load in one dimension can be described by dL=αLdT, where α is the coefficient of linear expansion, L is the length, and T is temperature. The coefficient of linear expansion αc can be considered as the only random coefficient. The temperature component δT can be expressed as(9)δT=αc∑i=1nbi0Ti,where bi0 represents the coefficients of temperature component, αc is the actual coefficient of linear expansion of the dam body, and T is the environmental temperature. With a given design coefficient of linear expansion αc0 and temperature T, we can obtain a numerical solution of temperature component δT0 using finite element method. By fitting the temperature T and the numerical solution δT0, the coefficients bi1 of T can be obtained as(10)δT0=∑i=1nbi1Ti.We then applybi1 instead of bi0 in equation (9), and δH then can be expressed as(11)δT=αcαc0αc01∑i=1nbi1Ti=αcαc0δT0=ψδT0,where ψ is the ratio from the actual coefficient of linear expansion αc to design coefficient of linear expansion αc0.The aging componentδθ characterizes irreversible displacement caused by factors such as creep and fatigue of concrete. Due to the difficulties of expressing δθ theoretically, here, we provide a formula, which includes time and monitoring point’s coordinates as variables, to describe its tendency:(12)δθ=∑l=0L∑m=0Md1lmθ+d2lmlnθxlzm,where x is the horizontal coordinate and z is the vertical coordinate of the monitoring point, θ=t/100 , t is the cumulative number of days from the initial value to monitoring value, and d1lm and d2lm are unknown coefficients. Then, the radial displacement can be exhibited as(13)δi=δiH+δiT+δiθ,=ξiδH0+ψiδT0+∑l=0L∑m=0Md1lmθ+d2lmlnθxilzim+u,where i is the index of measuring point, u is the random error term, δH0 are series of water pressure component calculated under groups of upstream water level H with finite element method, δT0 are series of temperature component calculated under groups of temperature T with finite element method, and coefficients ξi, ψi, d1lm, d2lm, and u are unknown coefficients in the model of all monitoring points which can be calculated with random coefficient model.In the model, we can obtain the actual material parametersEc and ψ when ξ and ψ are calculated. In addition, ξiδH0 and ψiδT0 represent the displacement produced by water pressure and fluctuation of temperature; ∑l=0L∑m=0Md1lmθ+d2lmlnθxilzim represents the displacement produced owing to aging behavior of material such as creep and deterioration of material strength and so forth.


## 2.3. Solving with the Random Coefficient Model

With random coefficient model, pending coefficients in equation (9) of all monitoring points can be calculated simultaneously obeying an asymptotic normal distribution. The monitoring displacement data are two-dimensional panel data containing time series and cross-section panel. Data on one panel represent the displacement data at an indicated time. The regression coefficients of one panel can be expressed by the following formula:(14)yit=∑k=1Kβkixkit+uit=∑k=1Kβk+γkixkit+u,where yit represents the two-dimensional dam displacement data; xkit represents the two-dimensional data of explanatory variables; t is the time index; i is the index of section; k is the index of the explanatory variables; the coefficient βki is independent with time, which can be decomposed into βk and γki; β=β1,…,βK′ is the common mean coefficient vector; γ=γ1i,…,γKi′ is the degree of the individual deviated from the common mean; u is the random error term.The objective is to obtainβki from the two-dimensional data series yit and xkit. According to the above analysis, we consider that the random coefficients βki obey an asymptotic normal distribution. Here, βi=β+γi is considered as a random coefficient with a common mean value β; Swamy gave the following assumptions [24]. In the equations below, the symbol at the upper right corner of the letter represents the transposition operation:(15)Eγiγj′=Δ,i=j,0,i≠j.Eγi=0,Exitγj′=0,Euiuj′=σi2IT,i=j,0,i≠j.By integrating the observation data, the matrix equation can be obtained as follows:(16)y=Xβ+X˜γ+u,where(17)yNT×1=y1′,…,yN′,XNT×1=X1X2⋮XN,X˜NT×NK=X1⋯0⋮⋱⋮0⋯XN,u=u1′,…,uN′',γ=γ1′,…,γN′',where N is the number of panels, T is the amount of data in the panel, the compound error term X˜γ+u is a diagonal matrix, and the i-th diagonal block is ψi=XiΔXi′+σi2IT. Under Swamy’s assumption, the estimator of β using the ordinary least squares is biased. When X′X/NT converges to a nonzero constant matrix, we can get a consistent noneffective estimation. The generalized least squares estimation β^GLS is the optimal linear unbiased estimator of β:(18)β^GLS=∑i=1NXi′ψi−1Xi−1∑i=1NXi′ψi−1yi=∑i=1NWiβ^i,Wi=∑i=1NΔ+σi2Xi′Xi−1−1−1Δ+σi2Xi′Xi−1−1,β^=Xi′Xi−1Xi′yi.The variance of the estimator can be written as(19)Varβ^GLS=∑i=1NXi′ψi−1Xi−1=∑i=1NΔ+σi2Xi′Xi−1−1−1.Usingβ^i=Xi′Xi−1Xi′yi and u^i=yi−Xiβ^i, we can obtain the unbiased estimator of σi2 and Δ.(20)σi2=u^i′uiT−K=1T−Kyi′I−XiXi′Xi−1Xi′yi,Δ^=1N−1∑i=1Nβ^i−N−1∑i=1Nβ^iβ^i−N−1∑i=1Nβ^i′−1N∑i=1Nσ^i2Xi′Xi−1.However,Δ^ obtained from estimator might be negative. In this case, we chose(21)Δ^=1N−1∑i=1Nβ^i−N−1∑i=1Nβ^iβ^i−N−1∑i=1Nβ^i′.Thus, even though the estimator is not unbiased, it is invariable nonnegative. Moreover, whenT⟶∞, the estimate is consistent. Therefore, β^GLS obeys an asymptotic normal distribution and is a valid estimator of β.Figure1 presents the flowchart of the proposed synthetic model which integrates the finite element method into a random coefficient model based on the statistical model. The cross-section displacement data are analyzed as a panel in the random coefficient model, and the independent random coefficients ξi and ψi obey normal distributions. In addition, the proposed model can derivate the actual elastic modulus and actual coefficient of linear expansion at each monitoring point, which are dominant factors representing dam’s properties.Figure 1
Flowchart of the proposed model which integrates the finite element method into a random coefficient model based on the statistical model.


## 3. Case Study

We selected the concrete dam at Jinping-I hydropower station as an example, which is the tallest arch dam in the world at present with a height of 305 m (the elevation of the dam crest is 1885 m and the elevation of dam foundation is 1580 m). It is a double-curved arch dam with the crest width of 16 m and the reservoir volume of 474 × 106 m3. The dam is located in the southwest of China at the Yalong River (see Figure 2). The construction of the dam’s body was completed on June 2013.Figure 2
Photograph of Jinping-I hydropower station.The displacement of an arch dam consists of three parts: vertical, tangential, and radial displacement, among which radial displacement is the most important indicator to evaluate the health of a concrete arch dam. Hence, we took the radial displacement as an example in this study. In order to monitor the radial displacement, the plumb system, consisting of 40 direct plumb meters and 30 reversed plumb meters, was installed in the selected dam. The structures of plumb meters are illustrated in Figure3. For the direct plumb meter, the anchor is fitted on the dam crest with a block and, for the reversed plumb meter, the anchor is buried in the bedrock of dam foundation. The model of plumb meters used in this project is the CCD 50 manufactured by Xi’an Huateng Co., Ltd., with a measurement range of 0–0.1 m, a resolution of 0.1×10−4 m, and a measurement accuracy of 0.1×10−3 m.Figure 3
Plumb meters used in the selected concrete dam: (a) direct plumb and (b) reversed plumb [25].
(a)
(b)In this study, we selected a dataset of 24 monitoring points that were recorded during the period from June 16, 2013, to August 25, 2015. There are 26 sections from left bank to right bank and 6 vertical lines (5#, 9#, 11#, 13#, 16#, and 19#) in the dam. The 24 monitoring points were located on the 6 indicated vertical lines (see Figure4).Figure 4
Distribution of dam sections and the monitoring points from the upstream side view.Since the dataset had been selected, we first eliminated the misdata and obtained 274 validated data samples for each monitoring point. As shown in Figure5, the temperature evolution has a significant regular periodicity relative to time. The upstream water level is manually controlled depending on the water resources allocation of the whole river basin. The water level in June 2013 was fairly low, as it was just newly constructed and putted into operation. Since then, the water level was gradually increased into a normal level.Figure 5
Time evolution of the (a) upstream level and (b) temperature.
(a)
(b)Figure6 shows the time evolution of the monitored radial displacement for all the monitoring points. We classified the monitoring points by their spatial distribution (i.e., the vertical lines where they are located). When coupled with the graphic information in Figures 5(a) and 6, it is notable that the general trend of the monitored radial displacement is approximately in line with the upstream water level. Another noticeable trend is the similarity in radial displacement for monitoring points at one vertical line and for adjacent monitoring points, which means that the displacement has a strong structural correlation.Figure 6
Time variation of the monitored radial displacement for monitoring points located at (a) vertical line 5#, (b) vertical line 9#, (c) vertical line 11#, (d) vertical line 13#, (e) vertical line 16#, and (f) vertical line 19#.
(a)
(b)
(c)
(d)
(e)
(f)


## 4. Results

### 4.1. Coefficient Determination Using the Finite Element Model

In order to quantify the spatial distribution of displacement of all monitoring points, we first established a three-dimensional finite element model based on the actual geological characteristics of the dam (see Figure7). The three-dimensional model includes three sections: dam body, dam foundation, and surrounded mountains; and the dam body was discretized into 38537 elements and 31941 nodes. The boundary of surrounding mountains was set to 2–3 times higher than the dam body in all the directions (x, y, and z). The model was constrained in the normal direction for all the lateral boundaries and was fixed in all directions for the bottom boundary; the crest of the dam and the dam body were free to move. The water load was merely applied to the upstream surface of dam and the bottom of reservoir; the elastic modulus of dam body and dam foundation were initially set to 30 GPa and 25 GPa, respectively.Figure 7
Finite element model of the Jinping-I arch dam: (a) upstream side view and (b) downstream side view.
(a)
(b)


#### 4.1.1. Coefficients for Water Pressure Component

To shed light on the water pressure component acting on the dam displacement, we merely apply water load in this case. As the upstream water level of Jinping-I hydropower station varied from 1700 m to 1880 m during the selected period, here we simulated the dam displacement under six water levels within this range, that is, 1700 m, 1740 m, 1780 m, 1820 m, 1860 m, and 1880 m.Figure8 exhibits the radial displacement simulated by the finite element method under three water levels (1700 m, 1780 m, and 1880 m). We can see that the displacements in all these subfigures are approximately horizontal symmetry, with the displacement at the midline being more significant than in border areas. In addition, with the increasing of upstream water level, the vertical coordinate of the center of minimum displacement region moves from the bottom to the dam crest.Figure 8
Radial displacement simulated by finite element method under varying upstream water levels: (a) 1700 m; (b) 1780 m; and (c) 1880 m.
(a)
(b)
(c)After the radial displacement field on the cross section had been simulated, we can obtain the displacement at each monitoring point under the six selected upstream water levels. Simulated radial displacement of all the monitoring points versus upstream water level is illustrated in Figure9. For all monitoring points, the radial displacement increased sharply with the increase of the upstream water level.Figure 9
The radial displacement versus upstream water level simulated by finite element method for monitoring points located at vertical line: (a) 5#; (b) 9#; (c) 11#; (d) 13#; (e) 16#; and (f) 19#.
(a)
(b)
(c)
(d)
(e)
(f)We then calculated the coefficientsai1 (introduced in equation (8) in Section 2) in the water pressure components by ordinary least square (OLS) regression based on simulated radial displacement. As an arch dam, i=1,2,3,4 The coefficients a11, a21, a31, and a41 for each monitoring point are listed in Table 1. The coefficients ai1 will be used in the statistical model solved by random coefficient model in Section 4.2.Table 1
Coefficientsai1 fitted by the simulation results of the finite element model.
Measuring points
a
4
1
a
3
1
a
2
1
a
1
1
PL5-2
3.95E − 09
−2.05E − 05
3.54E − 02
−2.04E + 01
PL5-3
6.96E − 10
−3.35E − 06
5.35E − 03
−2.83E + 00
PL5-4
−1.95E − 10
1.15E − 06
−2.22E − 03
1.41E + 00
PL9-3
9.25E − 10
−4.13E − 06
6.01E − 03
−2.84E + 00
PL9-4
−6.41E − 10
3.85E − 06
−7.50E − 03
4.79E + 00
PL9-5
−2.51E − 10
1.40E − 06
−2.57E − 03
1.56E + 00
PL11-2
5.15E − 09
−2.62E − 05
4.46E − 02
−2.53E + 01
PL11-3
7.45E − 10
−2.97E − 06
3.65E − 03
−1.28E + 00
PL11-4
−1.01E − 09
6.00E − 06
−1.16E − 02
7.35E + 00
PL11-5
−5.47E − 10
3.08E − 06
−5.68E − 03
3.44E + 00
IP11-1
−8.74E − 11
4.78E − 07
−8.53E − 04
4.99E − 01
PL13-3
5.72E − 10
−2.06E − 06
2.05E − 03
−3.54E − 01
PL13-4
−1.12E − 09
6.63E − 06
−1.27E − 02
8.03E + 00
PL13-5
−6.33E − 10
3.56E − 06
−6.56E − 03
3.97E + 00
IP13-2
−1.21E − 10
6.67E − 07
−1.19E − 03
6.99E − 01
PL16-2
3.61E − 09
−1.85E − 05
3.15E − 02
−1.79E + 01
PL16-3
1.97E − 10
−3.88E − 07
−3.84E − 04
8.05E − 01
PL16-4
−1.03E − 09
5.93E − 06
−1.12E − 02
6.92E + 00
PL16-5
−4.84E − 10
2.69E − 06
−4.90E − 03
2.95E + 00
IP16-1
−6.23E − 11
3.39E − 07
−6.03E − 04
3.51E − 01
PL19-2
2.37E − 09
−1.23E − 05
2.13E − 02
−1.23E + 01
PL19-3
1.06E − 10
−2.92E − 07
6.88E − 05
2.05E − 01
PL19-4
−5.24E − 10
2.94E − 06
−5.43E − 03
3.33E + 00
PL19-5
−1.32E − 10
7.17E − 07
−1.28E − 03
7.60E − 01


#### 4.1.2. Coefficients for Temperature Component

Similar to the calculation of the coefficients in water pressure component, as the temperature of Jinping region varies from 4°C to 24°C in the real world, we selected six temperatures within this range: 4°C, 8°C, 12°C, 16°C, 20°C, and 24°C. We simulated the radial displacement under these temperatures using the finite element method and then obtained the relation between simulated radial displacement and indicated temperatures. As shown in Figure10, in contrast with the water load impact, the radial displacement declines steadily with the rise in temperature.Figure 10
Variation of the radial displacement with external temperature simulated by finite element method for monitoring points located at vertical line: (a) 5#; (b) 9#; (c) 11#; (d) 13#; (e) 16#; and (f) 19#.
(a)
(b)
(c)
(d)
(e)
(f)Fitted with the simulation results, we obtained the coefficientsbi1 (i = 1, (2) in equation (11) (see section 2.1) for each monitoring point. It should be noted that the displacement curves in Figure 10 never pass the origin of coordinate axes, which implies the necessity of adjusting equation (11) by adding a constant term. Table 2 shows the fitting results of the coefficients bi1 and the constant term.Table 2
Coefficientsbi1 and the constant term fitted by simulation results of the finite element model.
Measuring points
b
2
1
b
1
1
Constant term
PL5-2
−0.00691
−9.24E − 01
19.3949
PL5-3
−0.00715
−3.83E − 01
10.8835
PL5-4
−0.00333
−1.43E − 01
4.50277
PL9-3
−0.01623
−9.22E − 01
25.5644
PL9-4
−0.0121
−4.57E − 01
15.3685
PL9-5
−0.0044
−1.73E − 01
5.70124
PL11-2
−0.0196
−1.91E + 00
43.6112
PL11-3
−0.0233
−1.35E + 00
37.0455
PL11-4
−0.02127
−9.91E − 01
30.0274
PL11-5
−0.01347
−5.38E − 01
17.5743
IP11-1
−0.00356
−1.66E − 01
5.02897
PL13-3
−0.02454
−1.41E + 00
38.9179
PL13-4
−0.0231
−1.09E + 00
32.817
PL13-5
−0.01599
−6.55E − 01
21.119
IP13-2
−0.0052
−2.25E − 01
7.0605
PL16-2
−0.01288
−1.28E + 00
29.1189
PL16-3
−0.0174
−9.13E − 01
26.1968
PL16-4
−0.01677
−7.10E − 01
22.5246
PL16-5
−0.01088
−4.17E − 01
13.919
IP16-1
−0.00255
−1.08E − 01
3.42312
PL19-2
−0.00712
−6.07E − 01
14.4608
PL19-3
−0.00903
−3.30E − 01
11.2959
PL19-4
−0.0075
−2.27E − 01
8.62348
PL19-5
−0.00276
−1.01E − 01
3.44706


### 4.2. Fitting Using the Random Coefficient Model

After the coefficients in the water pressure component and temperature component had been obtained, we fitted the monitoring data based on equation (13) using the random coefficient model.The coefficients of the aging componentsd1lm and d2lm for each monitoring point were fitted by the random coefficient model (see Table 3). Meanwhile, we can obtain the ratio between design and actual elastic modulus ξ and the ratio between actual and design coefficient of linear expansion ψ; then we can derive the actual elastic modulus Ec and actual coefficient of linear expansion αc (see Table 4). The elastic modulus reflects the ability of being deformed elastically when a stress is applied on an object. The coefficient of linear expansion represents the capacity of the length changes when material is heated or cooled; namely, the length changes by an amount proportional to the original length and the change in temperature. The elastic modulus may serve to evaluate the dam’s running status from the view of material.Table 3
The coefficientsd1lm and d2lm in the aging component.
Measuring points
d
1
l
m
d
2
l
m
PL5-2
3.87
−0.17
PL5-3
2.73
5.62
PL5-4
4.84
4.67
PL9-3
−5.95
12.19
PL9-4
1.85
5.25
PL9-5
0.51
4.83
PL11-2
−2.00
−5.57
PL11-3
0.28
6.84
PL11-4
0.30
14.40
PL11-5
7.23
3.15
IP11-1
4.68
0.74
PL13-3
1.53
3.17
PL13-4
2.94
10.39
PL13-5
3.18
9.43
IP13-2
0.69
6.40
PL16-2
3.44
−0.50
PL16-3
−1.04
13.92
PL16-4
−0.43
20.68
PL16-5
3.75
12.28
IP16-1
2.92
7.33
PL19-2
−1.49
3.37
PL19-3
0.62
7.76
PL19-4
6.19
5.55
PL19-5
4.18
5.65Table 4
The actual elastic modulusEc and actual coefficient of linear expansion αc at each monitoring point.
Monitoring points
E
c (GPa)
α
c (m)
IP11-1
23.64
1.61E − 06
IP13-2
20.10
1.08E − 06
IP16-1
32.51
3.02E − 06
PL19-5
21.00
3.41E − 06
PL19-4
20.70
1.53E − 06
PL19-3
20.10
2.12E − 06
PL19-2
19.50
2.80E − 06
PL16-5
21.41
1.39E − 06
PL16-4
16.60
1.79E − 06
PL16-3
17.13
2.37E − 06
PL16-2
16.23
3.04E − 06
PL13-5
18.30
1.03E − 06
PL13-4
18.11
1.50E − 06
PL13-3
18.32
2.03E − 06
PL11-5
19.25
1.12E − 06
PL11-4
16.27
1.64E − 06
PL11-3
16.87
2.19E − 06
PL11-2
17.18
2.63E − 06
PL9-5
17.89
1.54E − 06
PL9-4
17.86
1.87E − 06
PL9-3
14.89
2.27E − 06
PL5-4
30.67
3.46E − 07
PL5-3
23.00
1.97E − 06
PL5-2
21.79
2.91E − 06It is interesting from Table4 that Ec is dependent on the spatial location of monitoring points, where the minimum is 14.89 GPa at the monitoring point PL9-3, and the maximum is 32.51 GPa at IP16-1. We will provide a detailed discussion in section 5.2.With the random coefficient model, the radial displacement of all the monitoring points can be modeled simultaneously. The datasets were divided into two groups; 80% of the whole data were selected as fitting data, and the other 20% were selected to validate the model (validation data). We developed the prediction model with the fitting data. Then, we used the validation data to evaluate the performance of the model. Taking the monitoring points located at vertical line 5# as an example, the fitting and predicting results obtained from the synthetic model are shown in Figure11. Both the fitting results (left side of the line) and validating results (right side of the line) fit well with the monitored data.Figure 11
Modeling results for monitoring points located at vertical line 5#: (a) PL5-4, (b) PL5-3, and (c) PL5-2.
(a)
(b)
(c)In order to validate the synthetic model, we calculated the correlation coefficientR and the residual standard deviation s for the data series of all the monitoring points. The expressions of R and s are presented as follows:(22)R=∑i=1ny^i−y¯i2∑i=1nyi−y¯i2,s=∑i=1ny^i−y¯i2n−1.Usually, a model can be validated once the correlation coefficientR is above 0.9. As exhibited in Table 5, R for all the monitored points are higher than 0.987, which means that the fitting results and the monitoring data have a quite good agreement. The residual standard deviation s is located in the range of 0.186 to 1.550. The data lead us to the conclusion that the model performed well in fitting radial displacement of concrete dam.Table 5
Correlation coefficientR and residual standard deviation s of the synthetic model for each monitoring point.
Measuring points
R (/)
S (mm)
IP11-1
0.999
0.256
IP13-2
0.994
0.360
IP16-1
0.991
0.287
PL19-5
0.997
0.528
PL19-4
0.993
0.186
PL19-3
0.999
0.234
PL19-2
0.998
0.445
PL16-5
0.992
0.893
PL16-4
0.993
0.873
PL16-3
0.998
0.514
PL16-2
0.987
0.270
PL13-5
0.985
0.292
PL13-4
0.998
0.281
PL13-3
0.988
0.231
PL11-5
0.987
0.488
PL11-4
0.993
0.223
PL11-3
0.999
1.550
PL11-2
0.994
0.581
PL9-5
0.998
0.365
PL9-4
0.995
0.363
PL9-3
0.990
0.283
PL5-4
0.992
1.223
PL5-3
0.987
0.473
PL5-2
0.997
1.410


## 4.1. Coefficient Determination Using the Finite Element Model

In order to quantify the spatial distribution of displacement of all monitoring points, we first established a three-dimensional finite element model based on the actual geological characteristics of the dam (see Figure7). The three-dimensional model includes three sections: dam body, dam foundation, and surrounded mountains; and the dam body was discretized into 38537 elements and 31941 nodes. The boundary of surrounding mountains was set to 2–3 times higher than the dam body in all the directions (x, y, and z). The model was constrained in the normal direction for all the lateral boundaries and was fixed in all directions for the bottom boundary; the crest of the dam and the dam body were free to move. The water load was merely applied to the upstream surface of dam and the bottom of reservoir; the elastic modulus of dam body and dam foundation were initially set to 30 GPa and 25 GPa, respectively.Figure 7
Finite element model of the Jinping-I arch dam: (a) upstream side view and (b) downstream side view.
(a)
(b)


### 4.1.1. Coefficients for Water Pressure Component

To shed light on the water pressure component acting on the dam displacement, we merely apply water load in this case. As the upstream water level of Jinping-I hydropower station varied from 1700 m to 1880 m during the selected period, here we simulated the dam displacement under six water levels within this range, that is, 1700 m, 1740 m, 1780 m, 1820 m, 1860 m, and 1880 m.Figure8 exhibits the radial displacement simulated by the finite element method under three water levels (1700 m, 1780 m, and 1880 m). We can see that the displacements in all these subfigures are approximately horizontal symmetry, with the displacement at the midline being more significant than in border areas. In addition, with the increasing of upstream water level, the vertical coordinate of the center of minimum displacement region moves from the bottom to the dam crest.Figure 8
Radial displacement simulated by finite element method under varying upstream water levels: (a) 1700 m; (b) 1780 m; and (c) 1880 m.
(a)
(b)
(c)After the radial displacement field on the cross section had been simulated, we can obtain the displacement at each monitoring point under the six selected upstream water levels. Simulated radial displacement of all the monitoring points versus upstream water level is illustrated in Figure9. For all monitoring points, the radial displacement increased sharply with the increase of the upstream water level.Figure 9
The radial displacement versus upstream water level simulated by finite element method for monitoring points located at vertical line: (a) 5#; (b) 9#; (c) 11#; (d) 13#; (e) 16#; and (f) 19#.
(a)
(b)
(c)
(d)
(e)
(f)We then calculated the coefficientsai1 (introduced in equation (8) in Section 2) in the water pressure components by ordinary least square (OLS) regression based on simulated radial displacement. As an arch dam, i=1,2,3,4 The coefficients a11, a21, a31, and a41 for each monitoring point are listed in Table 1. The coefficients ai1 will be used in the statistical model solved by random coefficient model in Section 4.2.Table 1
Coefficientsai1 fitted by the simulation results of the finite element model.
Measuring points
a
4
1
a
3
1
a
2
1
a
1
1
PL5-2
3.95E − 09
−2.05E − 05
3.54E − 02
−2.04E + 01
PL5-3
6.96E − 10
−3.35E − 06
5.35E − 03
−2.83E + 00
PL5-4
−1.95E − 10
1.15E − 06
−2.22E − 03
1.41E + 00
PL9-3
9.25E − 10
−4.13E − 06
6.01E − 03
−2.84E + 00
PL9-4
−6.41E − 10
3.85E − 06
−7.50E − 03
4.79E + 00
PL9-5
−2.51E − 10
1.40E − 06
−2.57E − 03
1.56E + 00
PL11-2
5.15E − 09
−2.62E − 05
4.46E − 02
−2.53E + 01
PL11-3
7.45E − 10
−2.97E − 06
3.65E − 03
−1.28E + 00
PL11-4
−1.01E − 09
6.00E − 06
−1.16E − 02
7.35E + 00
PL11-5
−5.47E − 10
3.08E − 06
−5.68E − 03
3.44E + 00
IP11-1
−8.74E − 11
4.78E − 07
−8.53E − 04
4.99E − 01
PL13-3
5.72E − 10
−2.06E − 06
2.05E − 03
−3.54E − 01
PL13-4
−1.12E − 09
6.63E − 06
−1.27E − 02
8.03E + 00
PL13-5
−6.33E − 10
3.56E − 06
−6.56E − 03
3.97E + 00
IP13-2
−1.21E − 10
6.67E − 07
−1.19E − 03
6.99E − 01
PL16-2
3.61E − 09
−1.85E − 05
3.15E − 02
−1.79E + 01
PL16-3
1.97E − 10
−3.88E − 07
−3.84E − 04
8.05E − 01
PL16-4
−1.03E − 09
5.93E − 06
−1.12E − 02
6.92E + 00
PL16-5
−4.84E − 10
2.69E − 06
−4.90E − 03
2.95E + 00
IP16-1
−6.23E − 11
3.39E − 07
−6.03E − 04
3.51E − 01
PL19-2
2.37E − 09
−1.23E − 05
2.13E − 02
−1.23E + 01
PL19-3
1.06E − 10
−2.92E − 07
6.88E − 05
2.05E − 01
PL19-4
−5.24E − 10
2.94E − 06
−5.43E − 03
3.33E + 00
PL19-5
−1.32E − 10
7.17E − 07
−1.28E − 03
7.60E − 01


### 4.1.2. Coefficients for Temperature Component

Similar to the calculation of the coefficients in water pressure component, as the temperature of Jinping region varies from 4°C to 24°C in the real world, we selected six temperatures within this range: 4°C, 8°C, 12°C, 16°C, 20°C, and 24°C. We simulated the radial displacement under these temperatures using the finite element method and then obtained the relation between simulated radial displacement and indicated temperatures. As shown in Figure10, in contrast with the water load impact, the radial displacement declines steadily with the rise in temperature.Figure 10
Variation of the radial displacement with external temperature simulated by finite element method for monitoring points located at vertical line: (a) 5#; (b) 9#; (c) 11#; (d) 13#; (e) 16#; and (f) 19#.
(a)
(b)
(c)
(d)
(e)
(f)Fitted with the simulation results, we obtained the coefficientsbi1 (i = 1, (2) in equation (11) (see section 2.1) for each monitoring point. It should be noted that the displacement curves in Figure 10 never pass the origin of coordinate axes, which implies the necessity of adjusting equation (11) by adding a constant term. Table 2 shows the fitting results of the coefficients bi1 and the constant term.Table 2
Coefficientsbi1 and the constant term fitted by simulation results of the finite element model.
Measuring points
b
2
1
b
1
1
Constant term
PL5-2
−0.00691
−9.24E − 01
19.3949
PL5-3
−0.00715
−3.83E − 01
10.8835
PL5-4
−0.00333
−1.43E − 01
4.50277
PL9-3
−0.01623
−9.22E − 01
25.5644
PL9-4
−0.0121
−4.57E − 01
15.3685
PL9-5
−0.0044
−1.73E − 01
5.70124
PL11-2
−0.0196
−1.91E + 00
43.6112
PL11-3
−0.0233
−1.35E + 00
37.0455
PL11-4
−0.02127
−9.91E − 01
30.0274
PL11-5
−0.01347
−5.38E − 01
17.5743
IP11-1
−0.00356
−1.66E − 01
5.02897
PL13-3
−0.02454
−1.41E + 00
38.9179
PL13-4
−0.0231
−1.09E + 00
32.817
PL13-5
−0.01599
−6.55E − 01
21.119
IP13-2
−0.0052
−2.25E − 01
7.0605
PL16-2
−0.01288
−1.28E + 00
29.1189
PL16-3
−0.0174
−9.13E − 01
26.1968
PL16-4
−0.01677
−7.10E − 01
22.5246
PL16-5
−0.01088
−4.17E − 01
13.919
IP16-1
−0.00255
−1.08E − 01
3.42312
PL19-2
−0.00712
−6.07E − 01
14.4608
PL19-3
−0.00903
−3.30E − 01
11.2959
PL19-4
−0.0075
−2.27E − 01
8.62348
PL19-5
−0.00276
−1.01E − 01
3.44706


## 4.1.1. Coefficients for Water Pressure Component

To shed light on the water pressure component acting on the dam displacement, we merely apply water load in this case. As the upstream water level of Jinping-I hydropower station varied from 1700 m to 1880 m during the selected period, here we simulated the dam displacement under six water levels within this range, that is, 1700 m, 1740 m, 1780 m, 1820 m, 1860 m, and 1880 m.Figure8 exhibits the radial displacement simulated by the finite element method under three water levels (1700 m, 1780 m, and 1880 m). We can see that the displacements in all these subfigures are approximately horizontal symmetry, with the displacement at the midline being more significant than in border areas. In addition, with the increasing of upstream water level, the vertical coordinate of the center of minimum displacement region moves from the bottom to the dam crest.Figure 8
Radial displacement simulated by finite element method under varying upstream water levels: (a) 1700 m; (b) 1780 m; and (c) 1880 m.
(a)
(b)
(c)After the radial displacement field on the cross section had been simulated, we can obtain the displacement at each monitoring point under the six selected upstream water levels. Simulated radial displacement of all the monitoring points versus upstream water level is illustrated in Figure9. For all monitoring points, the radial displacement increased sharply with the increase of the upstream water level.Figure 9
The radial displacement versus upstream water level simulated by finite element method for monitoring points located at vertical line: (a) 5#; (b) 9#; (c) 11#; (d) 13#; (e) 16#; and (f) 19#.
(a)
(b)
(c)
(d)
(e)
(f)We then calculated the coefficientsai1 (introduced in equation (8) in Section 2) in the water pressure components by ordinary least square (OLS) regression based on simulated radial displacement. As an arch dam, i=1,2,3,4 The coefficients a11, a21, a31, and a41 for each monitoring point are listed in Table 1. The coefficients ai1 will be used in the statistical model solved by random coefficient model in Section 4.2.Table 1
Coefficientsai1 fitted by the simulation results of the finite element model.
Measuring points
a
4
1
a
3
1
a
2
1
a
1
1
PL5-2
3.95E − 09
−2.05E − 05
3.54E − 02
−2.04E + 01
PL5-3
6.96E − 10
−3.35E − 06
5.35E − 03
−2.83E + 00
PL5-4
−1.95E − 10
1.15E − 06
−2.22E − 03
1.41E + 00
PL9-3
9.25E − 10
−4.13E − 06
6.01E − 03
−2.84E + 00
PL9-4
−6.41E − 10
3.85E − 06
−7.50E − 03
4.79E + 00
PL9-5
−2.51E − 10
1.40E − 06
−2.57E − 03
1.56E + 00
PL11-2
5.15E − 09
−2.62E − 05
4.46E − 02
−2.53E + 01
PL11-3
7.45E − 10
−2.97E − 06
3.65E − 03
−1.28E + 00
PL11-4
−1.01E − 09
6.00E − 06
−1.16E − 02
7.35E + 00
PL11-5
−5.47E − 10
3.08E − 06
−5.68E − 03
3.44E + 00
IP11-1
−8.74E − 11
4.78E − 07
−8.53E − 04
4.99E − 01
PL13-3
5.72E − 10
−2.06E − 06
2.05E − 03
−3.54E − 01
PL13-4
−1.12E − 09
6.63E − 06
−1.27E − 02
8.03E + 00
PL13-5
−6.33E − 10
3.56E − 06
−6.56E − 03
3.97E + 00
IP13-2
−1.21E − 10
6.67E − 07
−1.19E − 03
6.99E − 01
PL16-2
3.61E − 09
−1.85E − 05
3.15E − 02
−1.79E + 01
PL16-3
1.97E − 10
−3.88E − 07
−3.84E − 04
8.05E − 01
PL16-4
−1.03E − 09
5.93E − 06
−1.12E − 02
6.92E + 00
PL16-5
−4.84E − 10
2.69E − 06
−4.90E − 03
2.95E + 00
IP16-1
−6.23E − 11
3.39E − 07
−6.03E − 04
3.51E − 01
PL19-2
2.37E − 09
−1.23E − 05
2.13E − 02
−1.23E + 01
PL19-3
1.06E − 10
−2.92E − 07
6.88E − 05
2.05E − 01
PL19-4
−5.24E − 10
2.94E − 06
−5.43E − 03
3.33E + 00
PL19-5
−1.32E − 10
7.17E − 07
−1.28E − 03
7.60E − 01


## 4.1.2. Coefficients for Temperature Component

Similar to the calculation of the coefficients in water pressure component, as the temperature of Jinping region varies from 4°C to 24°C in the real world, we selected six temperatures within this range: 4°C, 8°C, 12°C, 16°C, 20°C, and 24°C. We simulated the radial displacement under these temperatures using the finite element method and then obtained the relation between simulated radial displacement and indicated temperatures. As shown in Figure10, in contrast with the water load impact, the radial displacement declines steadily with the rise in temperature.Figure 10
Variation of the radial displacement with external temperature simulated by finite element method for monitoring points located at vertical line: (a) 5#; (b) 9#; (c) 11#; (d) 13#; (e) 16#; and (f) 19#.
(a)
(b)
(c)
(d)
(e)
(f)Fitted with the simulation results, we obtained the coefficientsbi1 (i = 1, (2) in equation (11) (see section 2.1) for each monitoring point. It should be noted that the displacement curves in Figure 10 never pass the origin of coordinate axes, which implies the necessity of adjusting equation (11) by adding a constant term. Table 2 shows the fitting results of the coefficients bi1 and the constant term.Table 2
Coefficientsbi1 and the constant term fitted by simulation results of the finite element model.
Measuring points
b
2
1
b
1
1
Constant term
PL5-2
−0.00691
−9.24E − 01
19.3949
PL5-3
−0.00715
−3.83E − 01
10.8835
PL5-4
−0.00333
−1.43E − 01
4.50277
PL9-3
−0.01623
−9.22E − 01
25.5644
PL9-4
−0.0121
−4.57E − 01
15.3685
PL9-5
−0.0044
−1.73E − 01
5.70124
PL11-2
−0.0196
−1.91E + 00
43.6112
PL11-3
−0.0233
−1.35E + 00
37.0455
PL11-4
−0.02127
−9.91E − 01
30.0274
PL11-5
−0.01347
−5.38E − 01
17.5743
IP11-1
−0.00356
−1.66E − 01
5.02897
PL13-3
−0.02454
−1.41E + 00
38.9179
PL13-4
−0.0231
−1.09E + 00
32.817
PL13-5
−0.01599
−6.55E − 01
21.119
IP13-2
−0.0052
−2.25E − 01
7.0605
PL16-2
−0.01288
−1.28E + 00
29.1189
PL16-3
−0.0174
−9.13E − 01
26.1968
PL16-4
−0.01677
−7.10E − 01
22.5246
PL16-5
−0.01088
−4.17E − 01
13.919
IP16-1
−0.00255
−1.08E − 01
3.42312
PL19-2
−0.00712
−6.07E − 01
14.4608
PL19-3
−0.00903
−3.30E − 01
11.2959
PL19-4
−0.0075
−2.27E − 01
8.62348
PL19-5
−0.00276
−1.01E − 01
3.44706


## 4.2. Fitting Using the Random Coefficient Model

After the coefficients in the water pressure component and temperature component had been obtained, we fitted the monitoring data based on equation (13) using the random coefficient model.The coefficients of the aging componentsd1lm and d2lm for each monitoring point were fitted by the random coefficient model (see Table 3). Meanwhile, we can obtain the ratio between design and actual elastic modulus ξ and the ratio between actual and design coefficient of linear expansion ψ; then we can derive the actual elastic modulus Ec and actual coefficient of linear expansion αc (see Table 4). The elastic modulus reflects the ability of being deformed elastically when a stress is applied on an object. The coefficient of linear expansion represents the capacity of the length changes when material is heated or cooled; namely, the length changes by an amount proportional to the original length and the change in temperature. The elastic modulus may serve to evaluate the dam’s running status from the view of material.Table 3
The coefficientsd1lm and d2lm in the aging component.
Measuring points
d
1
l
m
d
2
l
m
PL5-2
3.87
−0.17
PL5-3
2.73
5.62
PL5-4
4.84
4.67
PL9-3
−5.95
12.19
PL9-4
1.85
5.25
PL9-5
0.51
4.83
PL11-2
−2.00
−5.57
PL11-3
0.28
6.84
PL11-4
0.30
14.40
PL11-5
7.23
3.15
IP11-1
4.68
0.74
PL13-3
1.53
3.17
PL13-4
2.94
10.39
PL13-5
3.18
9.43
IP13-2
0.69
6.40
PL16-2
3.44
−0.50
PL16-3
−1.04
13.92
PL16-4
−0.43
20.68
PL16-5
3.75
12.28
IP16-1
2.92
7.33
PL19-2
−1.49
3.37
PL19-3
0.62
7.76
PL19-4
6.19
5.55
PL19-5
4.18
5.65Table 4
The actual elastic modulusEc and actual coefficient of linear expansion αc at each monitoring point.
Monitoring points
E
c (GPa)
α
c (m)
IP11-1
23.64
1.61E − 06
IP13-2
20.10
1.08E − 06
IP16-1
32.51
3.02E − 06
PL19-5
21.00
3.41E − 06
PL19-4
20.70
1.53E − 06
PL19-3
20.10
2.12E − 06
PL19-2
19.50
2.80E − 06
PL16-5
21.41
1.39E − 06
PL16-4
16.60
1.79E − 06
PL16-3
17.13
2.37E − 06
PL16-2
16.23
3.04E − 06
PL13-5
18.30
1.03E − 06
PL13-4
18.11
1.50E − 06
PL13-3
18.32
2.03E − 06
PL11-5
19.25
1.12E − 06
PL11-4
16.27
1.64E − 06
PL11-3
16.87
2.19E − 06
PL11-2
17.18
2.63E − 06
PL9-5
17.89
1.54E − 06
PL9-4
17.86
1.87E − 06
PL9-3
14.89
2.27E − 06
PL5-4
30.67
3.46E − 07
PL5-3
23.00
1.97E − 06
PL5-2
21.79
2.91E − 06It is interesting from Table4 that Ec is dependent on the spatial location of monitoring points, where the minimum is 14.89 GPa at the monitoring point PL9-3, and the maximum is 32.51 GPa at IP16-1. We will provide a detailed discussion in section 5.2.With the random coefficient model, the radial displacement of all the monitoring points can be modeled simultaneously. The datasets were divided into two groups; 80% of the whole data were selected as fitting data, and the other 20% were selected to validate the model (validation data). We developed the prediction model with the fitting data. Then, we used the validation data to evaluate the performance of the model. Taking the monitoring points located at vertical line 5# as an example, the fitting and predicting results obtained from the synthetic model are shown in Figure11. Both the fitting results (left side of the line) and validating results (right side of the line) fit well with the monitored data.Figure 11
Modeling results for monitoring points located at vertical line 5#: (a) PL5-4, (b) PL5-3, and (c) PL5-2.
(a)
(b)
(c)In order to validate the synthetic model, we calculated the correlation coefficientR and the residual standard deviation s for the data series of all the monitoring points. The expressions of R and s are presented as follows:(22)R=∑i=1ny^i−y¯i2∑i=1nyi−y¯i2,s=∑i=1ny^i−y¯i2n−1.Usually, a model can be validated once the correlation coefficientR is above 0.9. As exhibited in Table 5, R for all the monitored points are higher than 0.987, which means that the fitting results and the monitoring data have a quite good agreement. The residual standard deviation s is located in the range of 0.186 to 1.550. The data lead us to the conclusion that the model performed well in fitting radial displacement of concrete dam.Table 5
Correlation coefficientR and residual standard deviation s of the synthetic model for each monitoring point.
Measuring points
R (/)
S (mm)
IP11-1
0.999
0.256
IP13-2
0.994
0.360
IP16-1
0.991
0.287
PL19-5
0.997
0.528
PL19-4
0.993
0.186
PL19-3
0.999
0.234
PL19-2
0.998
0.445
PL16-5
0.992
0.893
PL16-4
0.993
0.873
PL16-3
0.998
0.514
PL16-2
0.987
0.270
PL13-5
0.985
0.292
PL13-4
0.998
0.281
PL13-3
0.988
0.231
PL11-5
0.987
0.488
PL11-4
0.993
0.223
PL11-3
0.999
1.550
PL11-2
0.994
0.581
PL9-5
0.998
0.365
PL9-4
0.995
0.363
PL9-3
0.990
0.283
PL5-4
0.992
1.223
PL5-3
0.987
0.473
PL5-2
0.997
1.410


## 5. Discussion

### 5.1. Improvement of Prediction Precision

Here, we selected correlation coefficientsR as the indicator to evaluate the prediction accuracy of proposed model and two previous models. One is the most commonly used statistical model based on ordinary least squares (OLS) regression. Another model was a random coefficient model improved by a clustering model named Gaussian mixture model (Clustering-RCM), because recent researches used clustering models to classify the monitoring points into several classes for the sake of considering the structural correlations among monitoring points. As introduced in Section 1, with only two explanatory variables, finite element method can simulate the water pressure component and temperature component of all monitoring points. In addition, synthetic model provides constraints for each monitoring point, whereas the clustering model merely provides constraint for each class; the number of explanatory variables in the proposed model is reduced more. Hence, the results of coefficients estimates are more stable and the prediction ability is better in the proposed model.As shown in Figure12, the correlation coefficients R of three models based on validation dataset of 24 monitoring points are compared; the proposed model had the highest R in 14 monitoring points and Clustering-RCM model had the highest R in 7 monitoring points. For IP16-1, PL19-3, PL5-2, and PL16-2, the difference of R between the proposed model and Clustering-RCM model is smaller than 0.001. In summary, the proposed model has a better prediction precision than the Clustering-RCM model, and the Clustering-RCM model performs better than OLS regression.Figure 12
Comparing the correlation coefficient R of the FEM-RCM model with the clustering-RCM model and statistical model using OLS regression.


### 5.2. Back-Analysis of Elastic Modulus

From the actual elastic modulusEc at each monitoring point presented in Table 4, we can derive the field distribution of Ec for the whole dam section using a Lagrange polynomial interpolation (see Figures 13(a) and 13(b)). It is striking that the actual elastic modulus at one point is strongly dependent on its spatial location on the dam section. In particular, the actual elastic modulus is small in the central area and large in the border area.Figure 13
The maps of (a) field distribution of actual elastic modulusEc, (b) actual elastic modulus Ec, (c) annual radial displacement in 2014, and (d) zoning designation of concrete material [26].
(a)
(b)
(c)
(d)Elastic modulus is a variable material property during the running status owing to the external environments and time. It is usually obtained by back-calculation based on monitored displacement and corresponding upstream water level, which is roughly written asEc=fH/δH, where fH is the expression of upstream water level. One reason for the regularity of elastic modulus distribution is that the displacement δH near foundation is limited owing to the strong restraint of dam foundation. Taking the annual radial displacement in 2014 as an example, the displacement in the central area is also higher than in border area (see Figure 13(c)). Hence, the elastic modulus of concrete used in the region of surrounding the dam foundation is higher than the dam center.In Figure13(d), the design elastic modulus of three divided region areas A, B, and C is 24 GPa, 23.5 GPa, and 23 GPa, respectively. First, it is noted that the actual elastic modulus for most areas on the dam has a deviation from the design value. Moreover, the actual elastic modulus in central areas is commonly smaller than the design value, whereas the values near the foundation are larger than design value. It is inevitable that each material may have different disintegration rate under the external load. Mainly, the distribution of residual between the actual and design elastic modulus roughly agrees with the spatial location and deformation under external load.


### 5.3. Limitations

The proposed method can be merely applied in projects with known topography data and design data which include material parameters of dam, dam foundation and mountains, and physical dimension. Another issue that should be noticed is that the correctness of data at all monitoring points should be guaranteed. With random coefficient model, all data series are analyzed simultaneously, so the measurement errors in one monitoring point may affect the accuracy of other monitoring points.


## 5.1. Improvement of Prediction Precision

Here, we selected correlation coefficientsR as the indicator to evaluate the prediction accuracy of proposed model and two previous models. One is the most commonly used statistical model based on ordinary least squares (OLS) regression. Another model was a random coefficient model improved by a clustering model named Gaussian mixture model (Clustering-RCM), because recent researches used clustering models to classify the monitoring points into several classes for the sake of considering the structural correlations among monitoring points. As introduced in Section 1, with only two explanatory variables, finite element method can simulate the water pressure component and temperature component of all monitoring points. In addition, synthetic model provides constraints for each monitoring point, whereas the clustering model merely provides constraint for each class; the number of explanatory variables in the proposed model is reduced more. Hence, the results of coefficients estimates are more stable and the prediction ability is better in the proposed model.As shown in Figure12, the correlation coefficients R of three models based on validation dataset of 24 monitoring points are compared; the proposed model had the highest R in 14 monitoring points and Clustering-RCM model had the highest R in 7 monitoring points. For IP16-1, PL19-3, PL5-2, and PL16-2, the difference of R between the proposed model and Clustering-RCM model is smaller than 0.001. In summary, the proposed model has a better prediction precision than the Clustering-RCM model, and the Clustering-RCM model performs better than OLS regression.Figure 12
Comparing the correlation coefficient R of the FEM-RCM model with the clustering-RCM model and statistical model using OLS regression.


## 5.2. Back-Analysis of Elastic Modulus

From the actual elastic modulusEc at each monitoring point presented in Table 4, we can derive the field distribution of Ec for the whole dam section using a Lagrange polynomial interpolation (see Figures 13(a) and 13(b)). It is striking that the actual elastic modulus at one point is strongly dependent on its spatial location on the dam section. In particular, the actual elastic modulus is small in the central area and large in the border area.Figure 13
The maps of (a) field distribution of actual elastic modulusEc, (b) actual elastic modulus Ec, (c) annual radial displacement in 2014, and (d) zoning designation of concrete material [26].
(a)
(b)
(c)
(d)Elastic modulus is a variable material property during the running status owing to the external environments and time. It is usually obtained by back-calculation based on monitored displacement and corresponding upstream water level, which is roughly written asEc=fH/δH, where fH is the expression of upstream water level. One reason for the regularity of elastic modulus distribution is that the displacement δH near foundation is limited owing to the strong restraint of dam foundation. Taking the annual radial displacement in 2014 as an example, the displacement in the central area is also higher than in border area (see Figure 13(c)). Hence, the elastic modulus of concrete used in the region of surrounding the dam foundation is higher than the dam center.In Figure13(d), the design elastic modulus of three divided region areas A, B, and C is 24 GPa, 23.5 GPa, and 23 GPa, respectively. First, it is noted that the actual elastic modulus for most areas on the dam has a deviation from the design value. Moreover, the actual elastic modulus in central areas is commonly smaller than the design value, whereas the values near the foundation are larger than design value. It is inevitable that each material may have different disintegration rate under the external load. Mainly, the distribution of residual between the actual and design elastic modulus roughly agrees with the spatial location and deformation under external load.


## 5.3. Limitations

The proposed method can be merely applied in projects with known topography data and design data which include material parameters of dam, dam foundation and mountains, and physical dimension. Another issue that should be noticed is that the correctness of data at all monitoring points should be guaranteed. With random coefficient model, all data series are analyzed simultaneously, so the measurement errors in one monitoring point may affect the accuracy of other monitoring points.


## 6. Conclusions

In this article, we presented a synthetic model that integrates the finite element method into a random coefficient model. The primary objective is to take the structural correlations among each monitoring point into account in a statistical model.With this objective in mind, we first used the finite element method to simulate the water pressure component and temperature component of displacement at each monitoring point, so as to constrain the spatial correlations among monitoring points. Then, with random coefficient method, explanatory variable coefficients of all monitoring points are calculated, obeying an asymptotic normal distribution. Finally, we selected the concrete arch dam in Jinping-I hydropower station as an example to validate the proposed model. Remarkable results can be concluded in two aspects.First, with the proposed model, the modeling displacement data fitted well with the monitored data. We find that the proposed model performed better than a combined model of random coefficient model and Gaussian mixture clustering model, owing to the fact that finite element model can provide more rigorous constraints than clustering methods for each monitoring point.Second, using a Lagrange interpolation, we can obtain the distribution field of the actual elastic modulus based on the actual elastic modulus at each monitoring point. The field distribution of the actual elastic modulus has deviation from the design value due to a coeffect of external load. In addition, with the help of finite element method, the distribution field of the displacement on the basis of monitored data can also be obtained.



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*Source: 4961963-2020-04-22.xml*