# Stochastic Inversion Method for Concrete Dams on the Basis of Bayesian Back Analysis Theory

Authors: Chongshi Gu; Xin Cao; Bo Xu
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1]. 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 [2]. 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 [7] 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 [10] 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) [11].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 [12](1)δ=δH+δT+δϑ,where δH is the hydraulic component, δT is the thermal component, and δϑ is the time-effect component.Hydraulic componentδH 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:(2)δH=FH,E,where F is the export function of the constitutive model, H is the hydraulic load, and E is an m-dimensional undetermined mechanical parameter vector defined as(3)E=E1,E2,…,EmT.Undetermined mechanical parametersEii=1,2,…,m are regarded as random variables with certain probability distributions represented by the probability distribution function fi and spatial variations represented by correlation distance θi. These parameters are expressed as(4)Ei=gfi,θi,i=1,2,…,m,where g is the probability distribution function of the undetermined mechanical parameters.In practical engineering, the set of displacement monitoring pointsδH∗ is limited and is a subset of the entire δH, δH∗∈δH. The displacement function is transformed into(5)δHj∗=FH,Ei,i=1,2,…,m,j=1,2,…,n,where n is the number of displacement monitoring points.Structural response variablesδHj∗j=1,2,…,n 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](6)JE=δH∗−δHcalTcov−1εδH∗δH∗−δHcal+E−μETcov−1EE−μE,where δH∗ is the measured displacement matrix, δHcal is the corresponding calculated displacement vector, cov⋅ is the covariance matrix, εδH∗ is the measuring error matrix, E is the undetermined parameter matrix, and μE is the prior information matrix of E.Equation (6) is the objective function of Bayesian back analysis. The main objective of stochastic inversion is to obtain the minimum JE by generating different matrices E. Variables δH∗, εδH∗, and μE 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 E and calculate δHcal 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 (Ec) and the deformation moduli of the dam foundation (Er) and reservoir basin Eb, which are usually different from the original design values [7]. 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 [15]. 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](7)Kδ=RH,where K is the overall stiffness matrix, δ is the node displacement matrix, and RH is the node load matrix. K assumes the following form [7]:(8)K=∑ej∈Ω1CejTKejCej+∑ej∈Ω2CejTKejCej+∑ej∈Ω3CejTKejCej,where Ω1, Ω2, and Ω3 are the regions of the dam body, dam foundation, and reservoir basin, respectively, and Cej and Kej are the stiffness transformation matrix and the stiffness matrix of element ej, respectively.In stochastic inversion, the overall stiffness matrixK 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 matrixK is influenced by the fluctuations of the random variables and can be divided into two parts, namely, stiffness matrix K0 at the mean value of the random variable and fluctuation value ΔK; thus, K=K0+ΔK. According to the Neumann series expansion [18], the inverse matrix K−1 assumes the following form:(9)K−1=K0+ΔK−1=K0I+K0−1ΔK−1=I+K0−1ΔK−1K0−1=I+P−1K0−1=I−P+P2+⋯+−1m−1Pm−1+⋯K0−1,where(10)P=K0−1ΔK.Equation (7) can be transformed into(11)δ=K−1RH.If(12)δ0=K0−1RH,combining equations (9), (11), and (12), the following formula can be obtained:(13)δ=δ0−Pδ0+P2δ0+⋯+−1m−1Pm−1δ0.The number of terms in equation (13) is determined according to the accuracy requirement.If(14)δi=−1i−1Pi−1δ0,then the following recursive formula can be obtained:(15)δi=−Pδi−1=−K0−1ΔKδi−1,i=1,2,3,….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:(16)Eδ=1Ns∑i=1Nsδi,Varδ=1Ns∑i=1Nsδi−Eδ2,Mkδ=1Ns∑i=1Nsδi−Eδk,where Ns is the sample number; Mk is the kth-order moment, k≥3; and δi is the displacement response in the ith 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 matrixK in equation (7) is only determined by the random distribution of the mechanical parameters E. 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 fieldsZx, which can be expressed as(17)Zx=∫−∞∞eix⋅ωWdω,where Wdω is an interval white noise process with zero mean and Sωdω variance. Sω is the spectral density function. In practice, the 3D integral for discrete random fields Zxj,xk,xp defined on a domain of K1×K2×K3 is expressed as(18)Zxj,xk,xp=limK1,K2,K3⟶∞∑l=−K1K1∑m=−K2K2∑n=−K3K3Almncosxjωl+xkωm+xpωn+Blmnsinxjωl+xkωm+xpωn,where Almn and Blmn are Fourier coefficients.The correlation of a random variable [X = Xx,y,z] in 3D space between points [x,y,z] and x+Δx,y+Δy,z+Δz that are defined in the same domain and are, respectively, Δx, Δy, and Δz apart in the three principal axis direction is as follows:(19)ρXx,y,z,Xx+Δx,y+Δy,z+Δz=CXx,y,z,Xx+Δx,y+Δy,z+ΔzσX2=EXx,y,z−μXXx+Δx,y+Δy,z+Δz−μXσX2,where μX is the mean value of X and σX2 is the variance of X.According to the FFT algorithm, the one-sided spectral density function of a single exponential correlation coefficient is defined as(20)Gω1,ω2,ω3=σ2θ1θ2θ3π21+θ1ω1/221+θ2ω2/221+θ3ω3/22.UseL=K1−l, M=K2−m, and N=K3−n to denote the symmetric points in fields K1×K2×K3. The variances of the Fourier coefficients in three dimensions are as follows:(21)EAlmn2=116δlmnAΔωGlmnd+GlmNd+GlMnd+GLmnd+GlMNd+GLmNd+GLMnd+GLMNd,EBlmn2=116δlmnBΔωGlmnd+GlmNd+GlMnd+GLmnd+GlMNd+GLmNd+GLMnd+GLMNd,where Glmnd is the spectral density function and d is the number component of ω=ω1,ω2,ω3. Δω and Glmnd have the form: (22)Δω=∏i=13Δωi,Glmnd=Gωl,ωm,ωn2d,where Δωi is an interval of length π/Ki and δlmnA and δlmnB are given by(23)δlmnA=2,ifl=0or12K1,m=0or12K2,n=0or12K3,1,otherwise,δlmnA=0,ifl=0or12K1,m=0or12K2,n=0or12K3,1,otherwise.


## 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](7)Kδ=RH,where K is the overall stiffness matrix, δ is the node displacement matrix, and RH is the node load matrix. K assumes the following form [7]:(8)K=∑ej∈Ω1CejTKejCej+∑ej∈Ω2CejTKejCej+∑ej∈Ω3CejTKejCej,where Ω1, Ω2, and Ω3 are the regions of the dam body, dam foundation, and reservoir basin, respectively, and Cej and Kej are the stiffness transformation matrix and the stiffness matrix of element ej, respectively.In stochastic inversion, the overall stiffness matrixK 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 matrixK is influenced by the fluctuations of the random variables and can be divided into two parts, namely, stiffness matrix K0 at the mean value of the random variable and fluctuation value ΔK; thus, K=K0+ΔK. According to the Neumann series expansion [18], the inverse matrix K−1 assumes the following form:(9)K−1=K0+ΔK−1=K0I+K0−1ΔK−1=I+K0−1ΔK−1K0−1=I+P−1K0−1=I−P+P2+⋯+−1m−1Pm−1+⋯K0−1,where(10)P=K0−1ΔK.Equation (7) can be transformed into(11)δ=K−1RH.If(12)δ0=K0−1RH,combining equations (9), (11), and (12), the following formula can be obtained:(13)δ=δ0−Pδ0+P2δ0+⋯+−1m−1Pm−1δ0.The number of terms in equation (13) is determined according to the accuracy requirement.If(14)δi=−1i−1Pi−1δ0,then the following recursive formula can be obtained:(15)δi=−Pδi−1=−K0−1ΔKδi−1,i=1,2,3,….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:(16)Eδ=1Ns∑i=1Nsδi,Varδ=1Ns∑i=1Nsδi−Eδ2,Mkδ=1Ns∑i=1Nsδi−Eδk,where Ns is the sample number; Mk is the kth-order moment, k≥3; and δi is the displacement response in the ith 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 matrixK in equation (7) is only determined by the random distribution of the mechanical parameters E. 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 fieldsZx, which can be expressed as(17)Zx=∫−∞∞eix⋅ωWdω,where Wdω is an interval white noise process with zero mean and Sωdω variance. Sω is the spectral density function. In practice, the 3D integral for discrete random fields Zxj,xk,xp defined on a domain of K1×K2×K3 is expressed as(18)Zxj,xk,xp=limK1,K2,K3⟶∞∑l=−K1K1∑m=−K2K2∑n=−K3K3Almncosxjωl+xkωm+xpωn+Blmnsinxjωl+xkωm+xpωn,where Almn and Blmn are Fourier coefficients.The correlation of a random variable [X = Xx,y,z] in 3D space between points [x,y,z] and x+Δx,y+Δy,z+Δz that are defined in the same domain and are, respectively, Δx, Δy, and Δz apart in the three principal axis direction is as follows:(19)ρXx,y,z,Xx+Δx,y+Δy,z+Δz=CXx,y,z,Xx+Δx,y+Δy,z+ΔzσX2=EXx,y,z−μXXx+Δx,y+Δy,z+Δz−μXσX2,where μX is the mean value of X and σX2 is the variance of X.According to the FFT algorithm, the one-sided spectral density function of a single exponential correlation coefficient is defined as(20)Gω1,ω2,ω3=σ2θ1θ2θ3π21+θ1ω1/221+θ2ω2/221+θ3ω3/22.UseL=K1−l, M=K2−m, and N=K3−n to denote the symmetric points in fields K1×K2×K3. The variances of the Fourier coefficients in three dimensions are as follows:(21)EAlmn2=116δlmnAΔωGlmnd+GlmNd+GlMnd+GLmnd+GlMNd+GLmNd+GLMnd+GLMNd,EBlmn2=116δlmnBΔωGlmnd+GlmNd+GlMnd+GLmnd+GlMNd+GLmNd+GLMnd+GLMNd,where Glmnd is the spectral density function and d is the number component of ω=ω1,ω2,ω3. Δω and Glmnd have the form: (22)Δω=∏i=13Δωi,Glmnd=Gωl,ωm,ωn2d,where Δωi is an interval of length π/Ki and δlmnA and δlmnB are given by(23)δlmnA=2,ifl=0or12K1,m=0or12K2,n=0or12K3,1,otherwise,δlmnA=0,ifl=0or12K1,m=0or12K2,n=0or12K3,1,otherwise.


## 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 parametersE, 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 [21]. 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 [22]. 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 as(24)bj=f∑i=1nωijxi−ζj,i=1,2,…,n;j=1,2,…,s,where bj is the output of the hidden layer, f is the transfer function, ωij is the weight from the input layer to the hidden layer, ζj is the threshold value of the hidden layer, n is the input node number, and s is the node number of the hidden layer.Figure 1
The architecture of back-propagation neural network.The output of the output layeryk is as follows:(25)yk=f∑j=1sωjkbj−ζk′,j=1,2,…,s;k=1,2,…,m,where ωjk is the weight from the hidden layer to the output layer, ζk′ is the threshold value of the output layer, and m is the output node number.The following sigmoid function is adopted as the transfer function:(26)fx=11+e−x.The error function of the actual network system outpute is as follows:(27)e=∑k=1mdk−yk2,where dk is the desired output.The back-propagation algorithm is used in training the ANN. Figure1 illustrates the architecture of BPNN.The main steps of BPNN are as follows:Step 1. Initialize the weights through ωlq=Random⋅, where lq represents ij or jk.Step 2. Input P learning samples for calculation. Set the current learning sample as the pth sample.Step 3. Calculate the output bj and yk j=1,2,…,s;k=1,2,…,m of each layer.Step 4. Obtain the back-propagation errors of each layer through(28)ejkp=dkp−ykpykp1−ykp,k=1,2,…,m,(29)eijp=∑k=1mejkpωjkbjp1−bjp,j=1,2,…,s.Record the value ofxip and bjp.Step 5. Record the number p of the trained learning sample. If the inequation p<P is satisfied, go to Step 2. If p=P, go to Step 6.Step 6. Revise the weights and thresholds of each layer.Step 7. Calculate xip and bjp according to the revised weights. If the inequation dkp−ykp<ε is satisfied for every p and k 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 valueδHj∗j=1,2,…,n is treated as an approximation under a certain measure, which is the objective function JE given by equation (6).Ranges are assigned the statistical parameters of undetermined mechanical properties as follows:(30)μEimin<μEi<μEimax,i=1,2,…,N,σEimin<σEi<σEimax,i=1,2,…,N,θEimin<θEi<θEimax,i=1,2,…,N,where μEimin and μEimax are the minimum and maximum values of μEi, which is the mean value of Ei, respectively; σEimin and σEimax are the minimum and maximum values of σEi, which is the variance of Ei; and θEimin and θEimax are the minimum and maximum values of θEi, which is the correlation distance of Ei.The undetermined mechanical parameterEi can be obtained by searching the minimum JE 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 JE 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 Eii=1,2,…,N 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 μE and correlation covE of the deformation modulus Eii=1,2,…,N are obtained.Step 3. Deformation modulus E0ii=1,2,…,N is initialized with the randomly given probability random parameters (i.e., μE0i, σE0i, and θE0i) and with the constraint E0i∈Eimin,Eimax.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 K0 is generated according to the established finite element model and the random fields of deformation modulus. Deformation δ0 is calculated by the finite element method.Step 6. An implementation of the random fields Ekii=1,2,…,N;k=0,1,2,… is generated by utilizing the FFT algorithm for stochastic simulation based on the given probability random parameters.Step 7. Fluctuation value ΔK and recursive coefficient matrix P 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 JkE.Step 9. Random values are assigned to the probability random parameters μEki, σEki, and θEk. 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 JE.The inversion process is shown with flowchart in Figure2.Figure 2
Stochastic inversion algorithm for concrete dams.


## 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 [21]. 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 [22]. 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 as(24)bj=f∑i=1nωijxi−ζj,i=1,2,…,n;j=1,2,…,s,where bj is the output of the hidden layer, f is the transfer function, ωij is the weight from the input layer to the hidden layer, ζj is the threshold value of the hidden layer, n is the input node number, and s is the node number of the hidden layer.Figure 1
The architecture of back-propagation neural network.The output of the output layeryk is as follows:(25)yk=f∑j=1sωjkbj−ζk′,j=1,2,…,s;k=1,2,…,m,where ωjk is the weight from the hidden layer to the output layer, ζk′ is the threshold value of the output layer, and m is the output node number.The following sigmoid function is adopted as the transfer function:(26)fx=11+e−x.The error function of the actual network system outpute is as follows:(27)e=∑k=1mdk−yk2,where dk is the desired output.The back-propagation algorithm is used in training the ANN. Figure1 illustrates the architecture of BPNN.The main steps of BPNN are as follows:Step 1. Initialize the weights through ωlq=Random⋅, where lq represents ij or jk.Step 2. Input P learning samples for calculation. Set the current learning sample as the pth sample.Step 3. Calculate the output bj and yk j=1,2,…,s;k=1,2,…,m of each layer.Step 4. Obtain the back-propagation errors of each layer through(28)ejkp=dkp−ykpykp1−ykp,k=1,2,…,m,(29)eijp=∑k=1mejkpωjkbjp1−bjp,j=1,2,…,s.Record the value ofxip and bjp.Step 5. Record the number p of the trained learning sample. If the inequation p<P is satisfied, go to Step 2. If p=P, go to Step 6.Step 6. Revise the weights and thresholds of each layer.Step 7. Calculate xip and bjp according to the revised weights. If the inequation dkp−ykp<ε is satisfied for every p and k 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 valueδHj∗j=1,2,…,n is treated as an approximation under a certain measure, which is the objective function JE given by equation (6).Ranges are assigned the statistical parameters of undetermined mechanical properties as follows:(30)μEimin<μEi<μEimax,i=1,2,…,N,σEimin<σEi<σEimax,i=1,2,…,N,θEimin<θEi<θEimax,i=1,2,…,N,where μEimin and μEimax are the minimum and maximum values of μEi, which is the mean value of Ei, respectively; σEimin and σEimax are the minimum and maximum values of σEi, which is the variance of Ei; and θEimin and θEimax are the minimum and maximum values of θEi, which is the correlation distance of Ei.The undetermined mechanical parameterEi can be obtained by searching the minimum JE 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 JE 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 Eii=1,2,…,N 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 μE and correlation covE of the deformation modulus Eii=1,2,…,N are obtained.Step 3. Deformation modulus E0ii=1,2,…,N is initialized with the randomly given probability random parameters (i.e., μE0i, σE0i, and θE0i) and with the constraint E0i∈Eimin,Eimax.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 K0 is generated according to the established finite element model and the random fields of deformation modulus. Deformation δ0 is calculated by the finite element method.Step 6. An implementation of the random fields Ekii=1,2,…,N;k=0,1,2,… is generated by utilizing the FFT algorithm for stochastic simulation based on the given probability random parameters.Step 7. Fluctuation value ΔK and recursive coefficient matrix P 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 JkE.Step 9. Random values are assigned to the probability random parameters μEki, σEki, and θEk. 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 JE.The inversion process is shown with flowchart in Figure2.Figure 2
Stochastic inversion algorithm for concrete dams.


## 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 Figure3. 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.Figure 3
Material zoning of the dam body.Table 1
The dam body material zoning information.
Zone
Concrete strength grade
Concrete grading
Dam body zone A
C9025
Four-graded
Dam body zone B
C18020
Three-graded
Dam body zone C
C9030
Three-gradedOn 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. Figure4 shows the finite element mesh.Figure 4
The finite element mesh for the #7 left-bank water-retaining dam section.


### 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 Figure5 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:(31)δ=δH+δT+δϑ=a0+∑i=13aiHi+∑i=12b1isin2πit365+b2icos2πit365+c1ϑ−ϑ0+c2lnϑ−lnϑ0,where δH, δT, and δϑ are, respectively, the hydraulic component, the temperature component, and the time-effect component of the monitored displacement, H is the upstream water head, t is the cumulative days from the initial day to the present day, ϑ0 is the initial monitoring time which takes 1/100, ϑ is the current monitoring time which takes t/100, and a0, a1, a2, a3, b1i, b1i, b1i, b1i, c1, and c2 are model coefficients.Figure 5
Measured displacements of the #7 left-bank water-retaining dam section.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 Figure5. The decomposition results are shown in Figure 6.Figure 6
Displacement component result of the monitoring point PL1-1.


### 5.2. Random Properties of the Undetermined Parameters and Displacement

Mechanical parameter vectorE consists of five elements, namely, EcA, EcB, EcC, Er, and Eb. 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.Table 2
Initial modulus and ranges of different zones.
Zone
Initial modulus (GPa)
Range of modulus mean value (GPa)
Range of modulus variable coefficient
Range of correlation distance (m)
Min
Max
Min
Max
Min
Max
Zone A
27.2
10.0
40.0
0.02
0.3
0.2
10
Zone B
24.1
10.0
40.0
0.02
0.3
0.2
10
Zone C
35.4
10.0
40.0
0.01
0.2
0.2
10
Foundation
26.0
15.0
35.0
0.05
0.3
30
80
Reservoir basin
20.0
10.0
30.0
0.05
0.3
30
80The 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. Figure7 exhibits the 2D map of the GRF on the cross section of the dam foundation.Figure 7
2D map of the Gaussian random fields: (a)θ = 30 m, (b) θ = 50 m, and (c) θ = 80 m.
(a)
(b)
(c)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 Figure5. 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.Table 3
Properties of the hydraulic displacement component at the normal water level.
Monitoring points
Mean value (mm)
Variance (mm)
Variable coefficient
IP1
3.75
0.052
0.014
PL1-1
5.29
0.098
0.019
PL1-2
7.58
0.15
0.020
PL1-3
10.32
0.19
0.018


### 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 Figure6 based on the assignment of the random modulus generated in Section 5.2.Each mechanical parameterEi was independent of others, and Ei 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 Table2 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.Table 4
Inversion results of different calculation zones and comparison with the conventional method.
Calculation zone
Inversed distribution parameters
Inversion results of conventional method
μ
E
i (GPa)
σ
E
i (GPa)
θ
E
i (m)
E
i (GPa)
Zone A
28.5
0.20
0.51
29.2
Zone B
25.7
0.35
0.73
25.5
Zone C
38.9
2.5
0.75
37.3
Foundation
27.2
3.1
53
28.2
Reservoir basin
23.5
2.4
35
24.1The predicted displacements based on the inversed parameters are listed in Table5. 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.Table 5
Comparison between the predicted displacement and the measured data.
Monitoring points
E
δ
i
Var
δ
i
E
δ
i
cal
Var
δ
i
cal
IP1
3.75
0.052
3.72
0.072
PL1-1
5.29
0.098
5.36
0.15
PL1-2
7.58
0.15
7.75
0.20
PL1-3
10.32
0.19
10.48
0.27Figure 8
Distribution of the calculated displacement of the selected dam body section.Figure 9
Distribution of the measured hydraulic displacement of the selected dam body section.Figure 10
Comparison between the measured displacement and the predicted displacement.Figure 11
Comparison between the measured variance and the predicted variance.Figure12 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.Figure 12
Comparison between the initial modulus and the stochastic inversion results.Figure 13
Standard deviation of deformation modulus of different zones.Figure 14
Correlation distance of deformation modulus of different zones.


## 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 Figure5 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:(31)δ=δH+δT+δϑ=a0+∑i=13aiHi+∑i=12b1isin2πit365+b2icos2πit365+c1ϑ−ϑ0+c2lnϑ−lnϑ0,where δH, δT, and δϑ are, respectively, the hydraulic component, the temperature component, and the time-effect component of the monitored displacement, H is the upstream water head, t is the cumulative days from the initial day to the present day, ϑ0 is the initial monitoring time which takes 1/100, ϑ is the current monitoring time which takes t/100, and a0, a1, a2, a3, b1i, b1i, b1i, b1i, c1, and c2 are model coefficients.Figure 5
Measured displacements of the #7 left-bank water-retaining dam section.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 Figure5. The decomposition results are shown in Figure 6.Figure 6
Displacement component result of the monitoring point PL1-1.


## 5.2. Random Properties of the Undetermined Parameters and Displacement

Mechanical parameter vectorE consists of five elements, namely, EcA, EcB, EcC, Er, and Eb. 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.Table 2
Initial modulus and ranges of different zones.
Zone
Initial modulus (GPa)
Range of modulus mean value (GPa)
Range of modulus variable coefficient
Range of correlation distance (m)
Min
Max
Min
Max
Min
Max
Zone A
27.2
10.0
40.0
0.02
0.3
0.2
10
Zone B
24.1
10.0
40.0
0.02
0.3
0.2
10
Zone C
35.4
10.0
40.0
0.01
0.2
0.2
10
Foundation
26.0
15.0
35.0
0.05
0.3
30
80
Reservoir basin
20.0
10.0
30.0
0.05
0.3
30
80The 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. Figure7 exhibits the 2D map of the GRF on the cross section of the dam foundation.Figure 7
2D map of the Gaussian random fields: (a)θ = 30 m, (b) θ = 50 m, and (c) θ = 80 m.
(a)
(b)
(c)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 Figure5. 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.Table 3
Properties of the hydraulic displacement component at the normal water level.
Monitoring points
Mean value (mm)
Variance (mm)
Variable coefficient
IP1
3.75
0.052
0.014
PL1-1
5.29
0.098
0.019
PL1-2
7.58
0.15
0.020
PL1-3
10.32
0.19
0.018


## 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 Figure6 based on the assignment of the random modulus generated in Section 5.2.Each mechanical parameterEi was independent of others, and Ei 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 Table2 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.Table 4
Inversion results of different calculation zones and comparison with the conventional method.
Calculation zone
Inversed distribution parameters
Inversion results of conventional method
μ
E
i (GPa)
σ
E
i (GPa)
θ
E
i (m)
E
i (GPa)
Zone A
28.5
0.20
0.51
29.2
Zone B
25.7
0.35
0.73
25.5
Zone C
38.9
2.5
0.75
37.3
Foundation
27.2
3.1
53
28.2
Reservoir basin
23.5
2.4
35
24.1The predicted displacements based on the inversed parameters are listed in Table5. 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.Table 5
Comparison between the predicted displacement and the measured data.
Monitoring points
E
δ
i
Var
δ
i
E
δ
i
cal
Var
δ
i
cal
IP1
3.75
0.052
3.72
0.072
PL1-1
5.29
0.098
5.36
0.15
PL1-2
7.58
0.15
7.75
0.20
PL1-3
10.32
0.19
10.48
0.27Figure 8
Distribution of the calculated displacement of the selected dam body section.Figure 9
Distribution of the measured hydraulic displacement of the selected dam body section.Figure 10
Comparison between the measured displacement and the predicted displacement.Figure 11
Comparison between the measured variance and the predicted variance.Figure12 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.Figure 12
Comparison between the initial modulus and the stochastic inversion results.Figure 13
Standard deviation of deformation modulus of different zones.Figure 14
Correlation distance of deformation modulus of different zones.


## 6. Conclusions

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.



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