Realtime monitoring of the actual elastic modulus is essential and necessary to ensure the safe operation of arch dams. The zoning elastic modulus of a high arch dam is inversed by using deformation safety monitoring data in the operation period, based on the particle swarm optimization with gravitation search algorithm for support vector machine (PSOGSASVM) method. Firstly, the measured data of multipoints with a pendulum are separated to construct the initial sample training set; then, an optimal inversion model is established to reflect the complex nonlinear relationship between the mechanical parameters of the high arch dam and the deformation of measured points; finally, the PSOGSASVM method is used to train and dynamically update the training set so as to realize the optimization solution of the inversion model. The proposed inversion method is successfully applied to a high arch dam in China to verify its feasibility and validity. The results show that the actual elastic modulus of the dam body is much larger than the initial elastic modulus, which is beneficial to structural stability.
China has built a large number of 300meterhigh concrete arch dams in the upper reaches of the Yangtze River and the Yellow River, such as JinpingI of 305 m, Xiaowan of 294.5 m, Baihetan of 289 m, and Xiluodu of 285.5 m. These high dams and reservoirs are worldclass projects, which can switch water disasters to water conservancy and provide the source power for national economic development. The topographic and geological conditions of the dam area are very complex. Determining the accurate physical and mechanical parameters of materials is the key to the structural deformation, stress, and stability of dams [
Dam safety monitoring is a prototype test by 1 : 1. By using prototype observation data to carry out inversion analysis, the parameters most consistent with the actual working state of the project can be determined. According to the equation of finite element numerical simulation of concrete dams, the horizontal displacement of the dam crest is inversely proportional to the change of the dam body elastic modulus. Many scholars have carried out very meaningful research [
For the complex inversion analysis in practical engineering, the parameter inversion problem is often transformed into the optimization problem of the objective function. This process requires an efficient iterative method with strong nonlinear mapping capability and only a few samples. In recent years, such optimization methods have widely been developed [
This paper mainly focuses on the optimal inversion analysis of zoning elastic modulus of JinpingI high arch dam in China. A reasonable statistical model is established to obtain the displacement components caused by water pressure. On this basis, the actual average elastic modulus of JinpingI high arch dam is obtained by inversion. Finally, the real working strength of the arch dam is fed back through the simulation of the whole construction period. In the inversion analysis of the elastic modulus, considering the zoning characters, an effective PSOGSASVM method which can improve the speed and accuracy is established. This algorithm is simple, easy to implement, and fast to calculate when dealing with the extremum optimization problem. There is no requirement for the mathematical form of the objective function, and the gradient of the objective function, which is especially suitable for nonlinear and multiextremum optimization problems.
The inversion results show that the vertical deformation of JinpingI high arch dam is relatively small, and the elastic modulus is increasing from now on, which is beneficial to structural stability. In other words, measured displacement of this high arch dam is much smaller than expected, which should arouse our attention. This phenomenon has not yet been unanimously explained by experts. What we need to do now is to strengthen safety monitoring and ensure the stable operation of JinpingI high arch dam.
In the initial stage of impoundment, the high arch dam has not experienced other sudden loads. The dam concrete is in the range of elastic deformation. The average elastic modulus and Poisson’s ratio of the dam body and foundation are, respectively,
Then equation (
Considering the short storage time, the influence of temperature load and aging factor can be neglected. The increment of external load is mainly caused by the change of reservoir water level. Therefore, the water pressure component
Considering the zoning concrete characteristics of high arch dams, the increment of structural displacement under the uplift of reservoir water level is proportional to the elastic modulus combination of each zoning dam body and dam foundation.
The measured vertical displacement of the high arch dam is the superimposed effect of deformation of dam concrete, dam foundation, and reservoir bedrock. In structural numerical simulation, the change of horizontal displacement of the arch dam is obviously affected by three zoning mechanical parameters. The inversion method of the dam body and dam foundation parameters, especially the joint inversion method by zoning multipoints, is studied in combination with zoning characters and pendulums of #13 dam section of a high arch dam.
The high arch dam body is divided into several concrete zones. Taking JinpingI high arch dam as an example (Figure
Layout of concrete zoning and pendulum of an arch dam.
It can be concluded that the equations for calculating the elastic modulus inversion of high arch dams in different zones are as follows:
The first inversion step of the high arch dam is to determine rock mechanics parameters of dam foundation by the inversion of vertical monitoring data. The inverted pendulums of a high arch dam foundation monitor the relative displacement from the deep suspension point to the buoy near the dam foundation. The inverted vertical suspension depth can reach about 100 m. When the inverted vertical suspension point is deep enough, it can be regarded as the absolute displacement of the dam foundation. Using displacement monitoring data of the inverted pendulum, the deformation modulus of dam foundation at the location of the inverted pendulum can be calculated by using equation (
The second inversion step of the high arch dam is to determine the mechanical parameters of dam zoning concrete by inversion of dam body monitoring data. Taking point A at dam crest as an example, the deformation under water pressure is composed of three parts: dam body deformation caused by water pressure
Three parts of the displacement component under water pressure.
Therefore, equation (
The measured displacement at point A caused by the turning angle of the dam foundation is the superposition of the upstream overhang of the arch dam caused by the action of reservoir water on the reservoir plate and the upstream tilt displacement of the arch dam caused by the action of reservoir water on the dam body. In order to simplify the calculation, the deformation from dam heel to dam toe caused by these two actions is assumed to be uniform. Generally speaking, this assumption can meet the requirements of engineering accuracy. Then the displacement component
There are many pendulums inside the high arch dam. The measured deformations of each pendulum can be expressed as
Since the arch dam can selfadjust the stress state, it is necessary to further extract the pendulums which are sensitive to the elastic modulus of zoning dam concrete and to extract the relative displacement in the finite element model to improve the accuracy of the inversion results when using equation (
Sensitivity analysis of the pendulum system is determined by the finite element numerical simulation method. Firstly, the characteristic water levels
The inversion of the elastic modulus of high arch dams can be summarized as the following fitting optimization problems:
The support vector machine (SVM) [
According to the optimization theory and Wolfe duality technique [
The idea of solving the abovementioned nonlinear problems is to map the lowdimensional input samples into the highdimensional space by introducing the kernel function and transform the nonlinear support vector regression into a linear problem so as to construct the optimal hyperplane solution. According to functional theory, the inner product function
The key to obtain the optimal solution of equation (
In order to consider the memory of particles, the idea of particle evolutionary computation in particle swarm optimization (PSO) is introduced to improve the iteration speed. Then, the velocity update equation is transformed into
In the optimized inversion of the multipoint zoning concrete method, based on the construction of the initial sample training set, the set is updated dynamically and continuously trained to establish the optimal inversion model which can reflect the complex nonlinear relationship between the elastic modulus of the zoning concrete and the displacement of the measured points. The elastic modulus of zoning concrete of the high arch dam can be inverted by inputting the measured displacement of the separated measured points. Its main implementation process is shown in Figure
PSOGSASVM inversion process of zoning high arch dams.
Establish the safety monitoring model between measured displacement and environmental factors of the pendulum system, separate the water pressure component, and separate the displacement caused by the deformation of reservoir plate and the shear of dam foundation further.
Considering engineering materials and structural zone of the dam body concrete, dam foundation rock, replacement body, cracks, faults, and weak structural surfaces, establish a threedimensional finite element model of the arch dam.
Determine the elastic modulus parameter groups of
Separate the displacement values calculated by FEM corresponding to each pendulum, carry out sensitivity analysis, determine the contribution of each pendulum to the inversion, and then select the main contributing factors to construct the training sample set and test sample set of SVM regression modeling.
Load the radial basis function parameter
Load the parameters to be inverted
The JinpingI high arch dam is located in southwest China, which is mainly used for power generation and flood control of the Yalong River. Its layout chart is shown in Figure
Layout chart of the JinpingI high arch dam.
In the dam monitoring system, the pendulum can directly reflect the dam deformation with high accuracy. The measured data show that the radial deformation of the middle part of the dam is obviously affected by water pressure and is less restricted by the mountains on both sides of the river. Therefore, 24 measured points in sections #5, #9, #11, #13, #16, and #19 are taken as calculation data in the process of inversion. According to the installation drawings of the pendulum system, the corresponding notes of the finite element model are processed by accumulating and transforming the
Pendulum system of some main dam sections.
Figure
Radial displacement increment nephogram (to the downstream is positive).
Tangential displacement increment nephogram (to the left bank is positive).
From the radial displacement and tangential displacement increment nephograms of the pendulum system, it can be seen that the overall zoning rule of radial displacement is obvious. In addition, the increment value of radial displacement is large, which can be taken as the main basis for inversion. Figure
Fitting components of radial displacement of PL133.
Based on engineering design and geological data, a threedimensional finite element model of the high arch dam is established to study dam working state. The finite element is built according to the twodimensional engineering drawings, and the simulation calculation is conducted with the finite element software. Figures
Threedimensional finite element model of the JinpingI arch dam (upstream viewpoint).
Threedimensional finite element model of the JinpingI arch dam (downstream viewpoint).
Finite element model of zoning concrete of the dam body.
Selection of the coordinate system in the model:
The geological conditions of the dam foundation and the mountain body on both sides of the river are complex, and the rock mass parameters of each part are different, which have little influence on the calculation value of the dam deformation. Therefore, the simplified calculation regards the dam foundation and the mountain body on both sides of the river as a whole, that is, the dam foundation and the mountain body on both sides of the river are regarded as zone D, with a comprehensive elastic modulus parameter
For the inversion range of elastic modulus of A, B, C, and D zones, consulting the research papers of relevant scholars, the elastic modulus of the JinpingI high arch dam body and dam foundation are usually larger than the initial elastic modulus. In addition, considering that A, B, and C zones concrete in dam construction is
We select the statistical model of the stepwise regression algorithm to preprocess the inversion data and select
The water pressure component of pendulum radial displacement increment from June 1, 2017, to December 31, 2018, is extracted on the basis of finite element simulation inversion.
The separation results of pendulum radial displacement increment of some main measuring points are as follows by using statistical model.
From Table
Separation results of pendulum radial displacement increment.
Points 






PL52  20.54  16.26  16.26  3.40  0.19 
PL53  12.35  9.82  9.82  1.49  0.13 
PL54  4.79  3.06  3.06  0.78  0.04 
PL93  23.65  21.49  21.49  1.86  0.01 
PL94  13.81  12.85  12.85  0.67  −0.02 
PL95  3.43  3.20  3.20  0.14  −0.02 
PL112  37.90  33.46  33.46  4.24  −0.01 
PL113  30.47  27.69  27.69  2.15  −0.10 
PL114  21.52  19.68  19.68  1.04  −0.15 
PL115  10.06  9.02  9.02  0.30  −0.10 
IP111  2.65  2.24  2.24  0.02  −0.03 
PL133  30.38  27.89  27.89  1.94  0.09 
PL134  22.33  20.93  20.93  0.83  −0.01 
PL135  11.49  10.73  10.73  0.23  0.06 
IP132  3.94  3.04  3.04  0.05  0.13 
PL162  26.94  22.69  22.69  2.59  0.13 
PL163  22.88  19.69  19.69  0.91  0.02 
PL164  17.52  14.90  14.90  0.28  −0.04 
PL165  8.96  7.83  7.83  0.29  0.01 
IP161  2.83  2.15  2.15  0.08  0.03 
PL192  15.43  11.23  11.23  2.66  −0.27 
PL193  11.18  8.59  8.59  1.22  0.06 
PL194  7.70  6.12  6.12  0.43  −0.02 
PL195  3.43  2.23  2.23  0.20  0.03 
Separation results of pendulum radial displacement increment.
Based on assumption that the dam body is homogenous and dam foundation is simplified, which means
The incremental value
When the value of
Standard deviations of the combined results of



30 GPa  33 GPa  36 GPa  39 GPa  42 GPa 

48 GPa  


24 GPa  30.59  22.97  16.67  11.43  7.22  4.49  4.46 

27.89  20.34  14.11  9.03  5.29 

5.91 
30 GPa  25.73  18.24  12.11  7.27  4.38  5.03  7.60 
33 GPa  23.96  16.53  10.53  6.05  4.39  6.28  9.16 
36 GPa  22.49  15.13  9.29  5.32  4.98  7.52  10.54 
39 GPa  21.24  13.97  8.32  5.01  5.77  8.66  11.75 
From Table
Threedimensional graph of standard deviation of combined results of
Next, considering the zoning of the dam body, the assumed elastic modulus range of A, B, C, and D zones are as shown in Table
Assumed elastic modulus range of A, B, C, and D zones (GPa).



 

Homogeneous value  45.0  45.0  45.0  45.0 
Range of actual value  44.0–46.0  43.5–45.5  43.0–45.0  26.0–28.0 
The PSOGSASVM algorithm described above is used to construct training samples set and testing samples set. The results are as follows.
From Figure
Standard deviation of combined results of
Figure
Water pressure component
Then, we take the inversion results into the calculation of the pendulum system. The comparison nephograms between the calculated radial displacement and tangential displacement and those of the measured are as follows.
From Figures
Comparison nephogram between the calculated radial displacement and that of the measured.
Comparison nephogram between the calculated tangential displacement and that of the measured.
An optimal analysis method has been proposed to invert the zoning elastic modulus of the JinpingI high arch dam. The water pressure component of radial displacement of main measured points is obtained with the statistic model as the calibration. A new PSOGSASVM optimization method is carried out to solve the inversion problem, which is of great significance to improve the efficiency and accuracy. The results are reasonable and reliable.
Considering with the similar studies of other scholars, the results of
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest.
This work was supported by the National Key R&D Program of China (Grant no. 2016YFC0401908), National Natural Science Foundation of China (Grant nos. 51739003, 51579085, 51779086, 51579086, 51379068, 51579083, and 51609074), project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (YS11001), special project funded by the National Key Laboratory (20165042112), and Key R&D Program of Guangxi (AB17195074).