Stochastic Inversion Method for Concrete Dams on the Basis of Bayesian Back Analysis Theory
Inverse analysis is necessary for concrete dams in normal operation to overcome the discrepancy between the true mechanical parameters and test results. In view of the uncertain characteristics of concrete dams, a stochastic inverse model is proposed in this study to solve the undetermined mechanical parameters with sequential and spatial randomness using measured displacement data and Bayesian back analysis theory. An inversion method for the mechanical parameters of concrete dams is proposed. Fast Fourier transform algorithm is introduced to generate random fields for SFEM analysis. The case study shows that the proposed inversion method can reflect the random characteristics of concrete dams, the mechanical parameters obtained are reasonable, and the inverse model is feasible.
The determination of mechanical parameters is a key issue that affects the accuracy and reliability of positive analysis in the safety assessment of large-scale structures, such as concrete dams . Mechanical parameters are usually obtained from laboratory or in situ tests. However, the results of laboratory tests are influenced by sample disturbance, and in situ tests cannot be conducted extensively due to workload limitations . Thus, the parameters obtained from these tests present large discrepancies with the true values and cannot reflect the real operational situation of projects, especially aging concrete dams [3, 4]. Therefore, the mechanical parameters of concrete dams should be determined through inverse analysis based on prototype observation data [5, 6]. Gu and Wu  proposed a deterministic model and a mixed model to obtain the mechanical parameters of concrete dams.
Uncertainties exist in the dam body, dam foundation, and reservoir basin because of the complexity of dam foundation and the heterogeneity of concrete. The corresponding main physical and mechanical parameters usually vary with time and space in a random manner [8, 9]. Therefore, a stochastic inversion method is more suitable for obtaining parameters that can reflect these uncertain characteristics. With its strong capability for nonlinear dynamic processing and mapping highly nonlinear systems, an artificial neural network (ANN) gives more accurate identification results than the conventional methods based on linear models. Aiming at improving the computational efficiency and accuracy of conventional inversion algorithm, the inversion algorithm for random parameters of the dam body, dam foundation, and reservoir basin is put forward using ANN.
In this study, the undetermined physical and mechanical parameters are viewed as random variables with certain probability distributions and spatial variations. Each inversion result is regarded as an implementation of these variables. The objective function is established by utilizing Bayesian back analysis theory. A fast Fourier transform (FFT) algorithm  is used to generate Gaussian random fields for the undetermined parameters. The mean value, variance, and correlation distance of each parameter are obtained using a back-propagation neural network and a stochastic finite element method (SFEM) .
A numerical simulation is performed on a certain concrete dam in the commercial finite element software ABAQUS. The mechanical parameters are assigned by utilizing a UMAT subroutine. The inversion algorithm is implemented using the commercial mathematical computing software MATLAB. Results show that the proposed methods can realize stochastic inversion.
2. Basic Principles of the Stochastic Inversion of the Physical and Mechanical Parameters of Concrete Dams
In the stochastic inversion of physical and mechanical parameters, the structural response and the external load are regarded as random variables. The deformation of a concrete dam consists of a hydraulic deformation component, a temperature deformation component, and time-effect deformation component and is defined as where is the hydraulic component, is the thermal component, and is the time-effect component.
Hydraulic component is considered in the inversion of mechanical parameters. After deducing the thermal component and time-effect component, the hydraulic component can be expressed as follows:where is the export function of the constitutive model, is the hydraulic load, and is an m-dimensional undetermined mechanical parameter vector defined as
Undetermined mechanical parameters are regarded as random variables with certain probability distributions represented by the probability distribution function and spatial variations represented by correlation distance . These parameters are expressed aswhere is the probability distribution function of the undetermined mechanical parameters.
In practical engineering, the set of displacement monitoring points is limited and is a subset of the entire , . The displacement function is transformed intowhere is the number of displacement monitoring points.
Structural response variables are viewed as random variables in the stochastic inverse analysis. According to Bayesian inversion theory, the error estimation function of back analysis is expressed as [13, 14]where is the measured displacement matrix, is the corresponding calculated displacement vector, is the covariance matrix, is the measuring error matrix, is the undetermined parameter matrix, and is the prior information matrix of .
Equation (6) is the objective function of Bayesian back analysis. The main objective of stochastic inversion is to obtain the minimum by generating different matrices . Variables , , and were obtained from monitoring and design information. Therefore, the key to solving the problem is to determine the appropriate method for generating the random variable vector and calculate through the finite element method.
3. Inverse Analysis Method
Inverse analysis for concrete dams is widely used in determining the elastic modulus of the dam body () and the deformation moduli of the dam foundation () and reservoir basin , which are usually different from the original design values . Therefore, numerical analysis is necessary for the inverse analysis of concrete dams. The current inversion methods are deterministic, and they aim at obtaining the comprehensive mechanical parameters by mean averaging. However, the physical and mechanical parameters of the dam body, foundation, and reservoir basin present randomness and spatial variation characteristics owing to the complexity of geological conditions at the dam site and the heterogeneity of dam body concrete . Numerical simulation involves the simulation of random fields, such as the 3D space random distributions of the mechanical parameters of rock and concrete. Therefore, SFEM is introduced into the stochastic inversion.
3.1. Analysis Model for Stochastic Inversion
The finite element governing equation of the concrete dam system within the linear elastic range is expressed as [16, 17]where is the overall stiffness matrix, is the node displacement matrix, and is the node load matrix. assumes the following form :where , , and are the regions of the dam body, dam foundation, and reservoir basin, respectively, and and are the stiffness transformation matrix and the stiffness matrix of element , respectively.
In stochastic inversion, the overall stiffness matrix in each inversion calculation is regarded as an implementation of the entire stochastic simulation. Monte Carlo simulation is adopted in the stochastic simulation. To increase calculation efficiency, the Neumann expansion equation is introduced to reduce the calculation times of stiffness matrix inversion.
Stiffness matrix is influenced by the fluctuations of the random variables and can be divided into two parts, namely, stiffness matrix at the mean value of the random variable and fluctuation value ; thus, . According to the Neumann series expansion , the inverse matrix assumes the following form:where
Equation (7) can be transformed into
The number of terms in equation (13) is determined according to the accuracy requirement.
Ifthen the following recursive formula can be obtained:
The Neumann series expansion improves the calculation efficiency of the stochastic simulation by recursion. In the presented method, the stiffness matrix inversion in each Monte Carlo simulation sampling is performed only once. Other simulation samples can be obtained by simply multiplying the fluctuation values [11, 19].
The moments of the displacement responses are obtained by the following equations through the analysis of the concrete dam structure by SFEM:where is the sample number; is the th-order moment, ; and is the displacement response in the th stochastic simulation.
3.2. Random Fields of Mechanical Parameters
Three-graded concrete and four-graded concrete are widely used in the construction of concrete dams. However, the dam foundation is a natural rock mass that consists of microcracks, weak layers, and faults. The heterogeneity of concrete and the complexity of dam foundation result in the uncertain characteristics of the mechanical parameters of the dam body, dam foundation, and reservoir basin. Gaussian random fields (GRF) [10, 20] are introduced to describe these mechanical parameters with random characteristics.
The geometric structure and finite elements of the concrete dam calculation model are kept constant during the process of stochastic simulation. Therefore, stiffness matrix in equation (7) is only determined by the random distribution of the mechanical parameters . Weibull distribution, gamma distribution, and lognormal distribution are common in geotechnical engineering. Thus, the JC method is utilized in this study to transform nonnormal distributions into normal ones. Spatial discretization and spectral decomposition are two common methods for random field generation. An FFT algorithm is introduced to generate the Gaussian random fields of the mechanical parameters of the concrete dam system.
The FFT algorithm is based on the spectral representation of the homogeneous mean-square continuous random fields , which can be expressed aswhere is an interval white noise process with zero mean and variance. is the spectral density function. In practice, the 3D integral for discrete random fields defined on a domain of is expressed aswhere and are Fourier coefficients.
The correlation of a random variable [ = ] in 3D space between points  and that are defined in the same domain and are, respectively, , , and apart in the three principal axis direction is as follows:where is the mean value of and is the variance of .
According to the FFT algorithm, the one-sided spectral density function of a single exponential correlation coefficient is defined as
Use , , and to denote the symmetric points in fields . The variances of the Fourier coefficients in three dimensions are as follows:where is the spectral density function and is the number component of . and have the form: where is an interval of length and and are given by
4. Inversion Method Based on a BP Neural Network
The inversion of the mechanical parameters of the dam body and foundation can be summarized as an optimization problem. Since the deformation is an implicit equation of mechanical parameters , the back-propagation algorithm of the artificial neural network is introduced into the inversion of the mechanical parameters of the dam body, foundation, and reservoir basin of the concrete dam.
4.1. Basic Theory of BP Neural Network
Artificial neural networks (ANNs) are widely used artificial intelligent algorithms in many fields owing to their excellent capability of modeling nonlinear systems and superior data fitting performance . The feed-forward ANN in Figure 1 consists of an input layer, a hidden layer, and an output layer and can simulate nonlinear mapping from an M-dimensional space to an N-dimensional space . The ANN can perform nonlinear mapping with different precision levels by choosing an appropriate number of nodes in a hidden layer. The back-propagation (BP) algorithm is a mature and widely applied algorithm among ANN algorithms. The hidden layer can be written aswhere is the output of the hidden layer, is the transfer function, is the weight from the input layer to the hidden layer, is the threshold value of the hidden layer, is the input node number, and is the node number of the hidden layer.
The output of the output layer is as follows:where is the weight from the hidden layer to the output layer, is the threshold value of the output layer, and is the output node number.
The following sigmoid function is adopted as the transfer function:
The error function of the actual network system output is as follows:where is the desired output.
The back-propagation algorithm is used in training the ANN. Figure 1 illustrates the architecture of BPNN.
The main steps of BPNN are as follows: Step 1. Initialize the weights through , where represents or . Step 2. Input learning samples for calculation. Set the current learning sample as the sample. Step 3. Calculate the output and of each layer. Step 4. Obtain the back-propagation errors of each layer through Record the value of and . Step 5. Record the number of the trained learning sample. If the inequation is satisfied, go to Step 2. If , go to Step 6. Step 6. Revise the weights and thresholds of each layer. Step 7. Calculate and according to the revised weights. If the inequation is satisfied for every and or the max training time is reached, terminate the training procedure. Otherwise, go to Step 2.
4.2. Deformation Modulus Inversion Method
The inversion of parameters is a process, wherein the basic physical and mechanical parameters are regarded as input variables and the response value is treated as an approximation under a certain measure, which is the objective function given by equation (6).
Ranges are assigned the statistical parameters of undetermined mechanical properties as follows:where and are the minimum and maximum values of , which is the mean value of , respectively; and are the minimum and maximum values of , which is the variance of ; and and are the minimum and maximum values of , which is the correlation distance of .
The undetermined mechanical parameter can be obtained by searching the minimum under the end condition given in equation (28). In the inversion algorithm that uses a BP neural network, the statistical parameters are regarded as the nodes in the input layer, and objective function is considered the output. Subsequently, the map from the former to the latter can be obtained with a small set of training samples. The main steps of the inversion algorithm are as follows: Step 1. Prior distribution information is obtained, namely, the probability distribution parameters and spatial distribution information of the deformation modulus on the basis of the design information and laboratory or in situ tests. Step 2. The nonnormal distributions obtained in Step 1 are transformed into normal ones through the JC method. Prior information and correlation of the deformation modulus are obtained. Step 3. Deformation modulus is initialized with the randomly given probability random parameters (i.e., , and ) and with the constraint . Step 4. A 3D finite element model for the concrete dam body, dam foundation, and reservoir basin is established. Step 5. The initial stiffness matrix is generated according to the established finite element model and the random fields of deformation modulus. Deformation is calculated by the finite element method. Step 6. An implementation of the random fields is generated by utilizing the FFT algorithm for stochastic simulation based on the given probability random parameters. Step 7. Fluctuation value and recursive coefficient matrix are calculated. The updated deformation is obtained by utilizing the Neumann expansion equation (14). Step 8: Steps 6–7 are repeated and substituted the calculation results into the error estimation function (6) to obtain . Step 9. Random values are assigned to the probability random parameters , , and . Steps 5–8 are repeated and a sufficient number of training samples are generated for the BP neural network. The distribution parameters of the mechanical properties can be obtained by minimizing the objective function .
The inversion process is shown with flowchart in Figure 2.
5. Case Study
A concrete gravity dam in southwest China is considered the research object. The crest elevation is 384 m, the minimum foundation surface is 222 m, the maximum dam height is 162 m, and the dam crest length is 909.26 m. The dam body is divided into 43 sections. The normal water level is 380 m, and the dead water level is 370 m. The reservoir storage capacity under normal water level 380 m is 5.16 × 109 m3, and the effective storage capacity is 9.03 × 108 m3.
A 3D finite element model of the #7 left-bank water-retaining dam section was established for the back-and-forth analysis of the concrete dam. The model included the dam body, dam foundation, and reservoir basin. The near-dam-area model consisted of the dam body, dam foundation, and reservoir basin. To monitor the dam displacements, a reversed pendulum IP1 was planted at 222 m elevation and three pendulums, namely, PL1-1, PL1-2, and PL1-3, were planted at 322.0, 350.0, and 384.0 m elevations of the dam section, respectively.
The dam section is divided into three zones, namely, A, B, and C, as shown in Figure 3. The concrete of different zones has different concrete grades and strength grades. Table 1 shows the dam body material zoning information of the dam section.
On the basis of the material distributions and the locations of the pendulums, the finite element model was divided into five calculation areas, including the dam body areas A, B, and C, dam foundation, and reservoir basin. The established finite element model was composed of 1590 elements and 2392 nodes, among which 333 elements and 536 nodes were for the dam body. Figure 4 shows the finite element mesh.
5.1. Decomposition of the Hydraulic Displacement Component
The time series of the horizontal displacement of the #7 left-bank water-retaining dam section and that of the water level are given in Figure 5 according to the actual pendulum monitoring information. In accordance with dam safety monitoring theory, the statistical models of dam body deformation were established based on the horizontal displacement time series between October 19, 2012, and April 22, 2015, which were monitored by the pendulums and the reversed pendulums. The expression of the established models are as follows:where , , and are, respectively, the hydraulic component, the temperature component, and the time-effect component of the monitored displacement, is the upstream water head, is the cumulative days from the initial day to the present day, is the initial monitoring time which takes 1/100, is the current monitoring time which takes , and , , , , , , , , , and are model coefficients.
The hydraulic displacements were obtained by subtracting the temperature component and time-effect component from the horizontal displacements. The measured displacements of different monitoring points are shown in Figure 5. The decomposition results are shown in Figure 6.
5.2. Random Properties of the Undetermined Parameters and Displacement
Mechanical parameter vector consists of five elements, namely, , , , , and . The nonnormal distributions were transformed into normal ones by the JC method. By combining the design information, laboratory material test results, and engineering experience with similar projects, the ranges of the statistical parameters of the mechanical properties were obtained and are given in Table 2.
The input mechanical properties of the finite element model for each stochastic implementation were described by Gaussian random fields, which were generated by the FFT method. Figure 7 exhibits the 2D map of the GRF on the cross section of the dam foundation.
The selected hydraulic structure experienced five impounding processes since the initial impounding in October 2012 and achieved normal water level five times, as shown in Figure 5. The distribution of each measured horizontal displacement was obtained by the stochastic analysis of the measured data at the normal water level. The equivalent normal distribution parameters were obtained by the JC method. The hydraulic displacement components were obtained by subtracting the temperature displacement and the time-effect displacement from the measured data, as shown in Table 3.
5.3. Numerical Simulation with Stochastic FEM
The sample number of each stochastic FEM calculation is 100. The displacement responses and the corresponding modulus distribution are shown in Figure 6 based on the assignment of the random modulus generated in Section 5.2.
Each mechanical parameter was independent of others, and was independent of the displacement errors between the measured data and calculated displacements.
5.4. Inversion Results
Inversion was performed on all five calculation zones of the selected dam section. The zoning deformation modulus is obtained utilizing the initial deformation modulus ranges in Table 2 as optimization constraints and deformation modulus inversion method based on BPNN. The BPNN is trained 1000 times. The inversion results of different calculation zones and comparison with the conventional method are listed in Table 4. The data in Table 4 show that the differences between the inversion results of the stochastic inversion method proposed in this paper and the conventional method are all below 5% which reflect the rationality of the inversion results.
The predicted displacements based on the inversed parameters are listed in Table 5. Figures 8 and 9 show the distribution of the calculated and measured hydraulic displacement of the selected dam body section. Figures 10 and 11 show the comparison between the predicted mean value and variance of hydraulic displacement and the measured data. The comparison of the predicted displacements with the measured data showed that the former are approximate to the latter, verifying the excellence performance of the proposed method.
Figure 12 shows that the true deformation modulus of the dam body, foundation, and reservoir basin is higher than the initial value. This means that the integral stiffness is higher than the design stiffness, which is conducive to the safety of the dam and the foundation. Figures 13 and 14 show the random parameters of deformation modulus of different zones. It can be seen from Figures 13 and 14 that the maximum values of standard deviation and correlation distance both appear in the foundation zone. The standard deviation and correlation distance of the reservoir basin are relatively large. And zone C has a much larger standard deviation than the other two dam body zones. This means that the deformation modulus of zone C, foundation, and reservoir basin plays a controlling role in the accuracy of displacement prediction.
This study focuses on the stochastic inversion of the elastic and deformation moduli of the dam foundation and the reservoir basin. On the basis of the finite element method, the error function of the stochastic inversion is established using Bayesian back analysis theory, and the implicit and nonlinear relationships between dam displacements and mechanical parameters are determined using the BP neural network. The FFT algorithm is used to generate the random fields of the undetermined parameters in stochastic sampling, and the Neumann expansion function is used to increase the calculation efficiency of the finite element analysis. Numerical simulation is performed on a certain concrete dam, and the stochastic parameters of the dam body mechanical parameters are obtained by the proposed inverse analysis method. The following conclusions are drawn:(1)The stochastic inversion method proposed in this paper introduces randomness of mechanical parameters and monitoring data into the inversion procedure which break the limit of conventional inversion method that can only deal with deterministic situation. The mechanical parameters of the dam body, dam foundation, and reservoir basin are viewed as random variables with certain probability distribution and spatial correlation. Thus, the inversion results of the stochastic inversion method put forward can reflect the uncertain characteristics of the mechanical parameters.(2)The actual zoning deformation modulus of the dam body, foundation, and reservoir basin is put forward based on the comprehensive utilization of SFEM and BP neural network. Furthermore, the objective function of the stochastic inversion is obtained on the basis of Bayesian theory. The results of the inversion case of a certain concrete dam show that the proposed stochastic inversion method is feasible.(3)The Neumann expansion function is used to increase the calculation efficiency of the stochastic finite element simulation. The FFT algorithm is used to generate random fields efficiently for each sampling of stochastic simulation. The BP neural network has been found to be a fast, accurate, and objective method for nonlinear mapping and increases the efficiency of the optimization by using a small set of training samples.
|, :||Fourier coefficients|
|:||Output of the hidden layer|
|:||Stiffness transformation matrix|
|, (unit: GPa for deformation modulus):||Undetermined mechanical parameters|
|:||Probability distribution function|
|:||One-sided spectral density function|
|(unit: m):||Hydraulic load|
|, , :||Stiffness matrix|
|:||Fluctuation value of stiffness matrix|
|:||The th-order moment|
|:||Node load matrix|
|:||Interval white noise process with zero mean|
|:||Input of the ANN|
|:||Output of the output layer|
|, (unit: mm):||Concrete dam deformation|
|, , , (unit: mm):||Hydraulic component|
|(unit: mm):||Calculated displacement vector|
|(unit: mm):||Thermal component|
|(unit: mm):||Time-effect component|
|:||Measuring error matrix|
|:||Threshold value of the hidden layer|
|:||Threshold value of the output layer|
|:||Prior information matrix of|
|, , , , (unit: m):||Correlation distance|
|, (unit: 1/100 d):||Monitoring time|
|:||Correlation of a random variable|
|:||Weight from the input layer to the hidden layer|
|:||Weight from the hidden layer to the output layer|
|, , :||Regions of the dam body.|
No additional unpublished data are available.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
This research was supported by the National Natural Science Foundation of China (grant nos. 51739003, 51479054, 51779086, 51579086, 51379068, 51579083, 51579085, and 51609074), National Key R&D Program of China (2016YFC0401601 and 2017YFC0804607), Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (YS11001), Jiangsu Natural Science Foundation (grant no. BK20160872), Special Project Funded of National Key Laboratory (20145027612 and 20165042112), Key R&D Program of Guangxi (AB17195074), and Central University Basic Research Project (2017B11114).
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