# Zoning Modulus Inversion Method for Concrete Dams Based on Chaos Genetic Optimization Algorithm

Authors: Hao Gu; Zhongru Wu; Xiaofei Huang; Jintao Song
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

For dams and rock foundations of ages, the actual mechanical parameters sometimes differed from the design and the experimental values. Therefore, it is necessary to carry out the inversion analysis on main physical and mechanical parameters of dams and rock foundations. However, only the integrated deformation modulus can be inversed by utilizing the conventional inversion method, and it does not meet the actual situation. Therefore, a new method is developed in this paper to inverse the actual initial zoning deformation modulus and to determine the inversion objective function for the actual zoning deformation modulus, based on the dam displacement measured data and finite element calculation results. Furthermore, based on the chaos genetic optimization algorithm, the inversion method for zoning deformation modulus of dam, dam foundation and, reservoir basin is proposed. Combined with the project case, the feasibility and validity of the proposed method are verified.


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## Body

## 1. Introduction

In design and construction of major projects, in the past people mostly focused on the planning, design, and construction of the project, and the safety assessment is usually utilized with the traditional positive analysis method [1], but the various parameters of the mathematical model for positive analysis are generally determined by experts experience or experiment. The values of physical and mechanical parameters of dams are the foundation of design and analysis of the operation behavior of dam body, dam foundation, and reservoir basin. But for dams of ages, the actual mechanical parameters differed from the design and experiment. It is often difficult to reflect the overall mechanical behaviors of dams by utilizing the values of mechanical parameters determined by design or experiment [2–4].Therefore, it is necessary to carry out the inversion analysis of the main physical and mechanical parameters of dam body, dam foundation, and reservoir basin [5, 6]. By utilizing the dam zoning displacement measured data, application of inverse theory and method for quantitative calculation is the basic idea of inversion analysis which can effectively solve the inversion problem for zoning deformation modulus of dam body, dam foundation, and reservoir basin [7]. An inversion analysis model and method are proposed by considering the influence factors of deformation, load, the dam zoning material characteristics, and topographical and geological conditions of dam body, dam foundation, and reservoir basin. Aiming at the shortage of conventional inversion method, the inversion method for zoning deformation modulus of dam body, dam foundation, and reservoir basin is put forward and with the help of chaos genetic optimization algorithm [8], the inversion process can be realized.


## 2. The Inversion Analysis Method

### 2.1. The Conventional Inversion Method

For the dam of ages, the elastic modulus of dam bodyEc and deformation moduli of dam foundation and reservoir basin Er, Eb differed with the original design values. If utilized the original design modulus and other mechanical parameters directly for the analysis of dam deformation behavior may result in a big difference with the actual situation. Therefore, the inversion method is needed through the structure analysis and calculation [9, 10] and, based on the measured data of dam displacement to determine the average elastic modulus of dam body, deformation modulus of dam foundation and reservoir basin. The conventional inversion analysis principle is as follows.


#### 2.1.1. Determination of Dam Displacement Field

Within the linear elastic range and under water pressureH, the balance equation [11] of dam system composed by dam, dam foundation, and reservoir basin is(1)Kδ=RH.In this equation, K, δ, and RH are the overall stiffness matrix, node displacement matrix, and node load matrix, respectively. One has(2)K=∑ej∈Ω1CejTKejCej+∑ej∈Ω2CejTKejCej+∑ej∈Ω3CejTKejCej.In the above equation, Ω1, Ω2, and Ω3, respectively, represent the calculation domains of dam, dam foundation, and reservoir rock foundation and Cej and Kej represent stiffness transformation matrix and the stiffness matrix of element ej.Analysis shows that Poisson ratiosμc, μr, and μb of dam, dam foundation, and reservoir basin could be considered as constant parameters due to their small changes. As for the dam system, the geometric properties basically remained unchanged in the loading process. So K in (2) can be expressed as follows [12]:(3)K=∑ej∈Ω1CejTK-ejCej+∑ej∈Ω2CejTK-ejCej+∑ej∈Ω3CejTK-ejCej.In (3), K-ej is mainly concerned with Poisson ratio of corresponding part and geometric characteristics.Node displacement array according to (1) can be expressed as(4)δ=K-1RH,where RH is a fixed value under water pressure H. The nodal displacement fH,x,y,z can be shown as the sum of displacement caused by deformation fcH,x,y,z, frH,x,y,z, and fbH,x,y,z of dam, dam foundation, and reservoir basin. That is,(5)fH,x,y,z=fcH,x,y,z+frH,x,y,z+fbH,x,y,z.In (5), H is the depth of the water; x, y, and z are the node coordinates.Reference [1] shows that node displacement caused by the dam, dam foundation, and reservoir basin is(6)fH,x,y,z=∑k=034∑l,m,n=03AklmnHkxlymzn+∑k=034∑l,m,n=03BklmnHkxlymzn+∑k=034∑l,m,n=03CklmnHkxlymzn.In (6), Aklmn, Bklmn, and Cklmn are fitting coefficients of node displacement  component factors of dam, dam foundation, and reservoir basin under water pressure H.Dam displacement field can be determined by (6).


#### 2.1.2. Inversion ofEc, Er, and Eb

Under the assumption of average deformation moduliEc0, Er0, and Eb0 of dam, dam foundation, and reservoir basin, dam displacement component field caused by deformation of dam, dam foundation, and reservoir basin can be obtained under different water pressure H as fc0H,x,y,z, fr0H,x,y,z, and fb0H,x,y,z in (5). Aklmn, Bklmn, and Cklmn in (6) can be obtained by utilizing optimization method. Due to the assumption of average deformation modulus of dam, dam foundation, and reservoir basin in structural calculation, each field component should be adjusted to get the real dam displacement field fH,x,y,z and the node displacement equation after adjustment is(7)fH,x,y,z=Xfc0H,x,y,z+Yfr0H,x,y,z+Zfb0H,x,y,z=X∑k=034∑l,m,n=03AklmnHkxlymzn+Y∑k=034∑l,m,n=03BklmnHkxlymzn+Z∑k=034∑l,m,n=03CklmnHkxlymzn.In (7), Aklmn, Bklmn, and Cklmn are fitting coefficients after the optimization of node displacement in structural calculation under different water pressure.Put (3) into (4) and obtain (7) after expansion; the component adjustment coefficients of X, Y, and Z are(8)X=Ec0Ec;Y=Er0EcEc0Er;Z=Eb0EcEc0Eb.Utilize various measured displacement data to obtain adjustment coefficients X, Y, and Z in (7) through the optimization analysis. Then average elastic modulus of dam body Ec and average deformation moduli of dam foundation and reservoir basin Er, Eb can be obtained by utilizing (8).


### 2.2. Zoning Modulus Inversion Method

Since conventional inversion methods can only inverse the integrated modulus of dam, dam foundation, and reservoir basin, the material composition differed. The dam foundation can be divided into different zones, as the dam body and reservoir basin. Therefore, the zoning deformation measured data is utilized to inverse dam zoning elastic modulus, dam foundation zoning, and reservoir basin zoning deformation modulus. The following analyzes the detailed zoning inversion analysis method.


#### 2.2.1. Method for Determining Initial Zoning Modulus

Dam elastic modulus is referred to deformation modulus for the convenience below. Assume that dam, dam foundation, and reservoir basin are divided inton zones, and the corresponding modulus is Ei(i=1,2,…,n). For the zone i, the deformation measured data of corresponding zone can most reflect the impact. Make use of deformation measured data to establish deformation safety monitoring model and the equation is similar to (5). Utilize (5) to separate the water pressure component fjHm from the value of deformation measuring point j in zone i under the action of water head Hm. On the basis of above, assume that the deformation modulus of zone i is Ei′ and the rest of moduli of the other zones utilize the design values. Then calculate water pressure component δjHm  (j=1,2,…,k, where k is the number of measuring points; m=1,2,…,M, where M is the number of water pressure components) of measuring point j in zone i under different water level by structure analysis method such as finite element. The equation of δjHm is(9)δjHm=FHm,Ei′.Assume(10)Q=∑j=1k∑m=1MfjHm-xδjHm2.Minimize Q, without constraint conditions; then(11)x=∑j=1k∑m=1MfjHmδjHm∑j=1k∑m=1MδjHm2.Assume that the initial value of the actual modulus is E0i in zone i and the equation of x according to its physical meaning is(12)x=Ei′E0i.Equation (12) can be used to determine the actual initial modulus in zone i as(13)E0i=Ei′x.


#### 2.2.2. Method for Determining Actual Zoning Deformation Modulus

E
0
i is the initial deformation modulus, which assumes that other zones utilize design values except zone i. It differed from the actual situation, so multipoint deformation measured data combined with the finite element structure analysis results is needed to inverse the actual deformation modulus [13–16]. The inversion method is as follows.Firstly, the zoning initial actual deformation modulusE0i(i=1,2,…,n) obtained above is utilized to calculate deformation value δHl of deformation measuring point under effect of different water head Hl with application of structural analysis method. For example, the calculated deformation value of measuring point p in zone i is δipHl as (14)δipHl=FHl,E0ii=1,2,…,n,where δipHl′ is the measured deformation value of measuring point p, which is separated by safety monitoring mathematical model from the measured data. Make(15)Q=∑i=1n∑pi=1npi∑l=1NδipHl′-xpiδipHl2=∑i=1n∑pi=1npi∑l=1NδipHl′-Ei′E0iδipHl2.In (15), n is the number of zones. npi is the number of measuring points in zone i. N is deformation monitoring frequency. The meaning of xpi is the same as that in (12).x
p
i is the final deformation modulus adjusting coefficient of zone i when Q in (15) reaches the minimum. Then the actual deformation modulus of zone i can be obtained by (13). Therefore, the following focuses on determining method of xpi, which is also the process of zoning modulus inversion of dam, dam foundation, and reservoir basin.


## 2.1. The Conventional Inversion Method

For the dam of ages, the elastic modulus of dam bodyEc and deformation moduli of dam foundation and reservoir basin Er, Eb differed with the original design values. If utilized the original design modulus and other mechanical parameters directly for the analysis of dam deformation behavior may result in a big difference with the actual situation. Therefore, the inversion method is needed through the structure analysis and calculation [9, 10] and, based on the measured data of dam displacement to determine the average elastic modulus of dam body, deformation modulus of dam foundation and reservoir basin. The conventional inversion analysis principle is as follows.


### 2.1.1. Determination of Dam Displacement Field

Within the linear elastic range and under water pressureH, the balance equation [11] of dam system composed by dam, dam foundation, and reservoir basin is(1)Kδ=RH.In this equation, K, δ, and RH are the overall stiffness matrix, node displacement matrix, and node load matrix, respectively. One has(2)K=∑ej∈Ω1CejTKejCej+∑ej∈Ω2CejTKejCej+∑ej∈Ω3CejTKejCej.In the above equation, Ω1, Ω2, and Ω3, respectively, represent the calculation domains of dam, dam foundation, and reservoir rock foundation and Cej and Kej represent stiffness transformation matrix and the stiffness matrix of element ej.Analysis shows that Poisson ratiosμc, μr, and μb of dam, dam foundation, and reservoir basin could be considered as constant parameters due to their small changes. As for the dam system, the geometric properties basically remained unchanged in the loading process. So K in (2) can be expressed as follows [12]:(3)K=∑ej∈Ω1CejTK-ejCej+∑ej∈Ω2CejTK-ejCej+∑ej∈Ω3CejTK-ejCej.In (3), K-ej is mainly concerned with Poisson ratio of corresponding part and geometric characteristics.Node displacement array according to (1) can be expressed as(4)δ=K-1RH,where RH is a fixed value under water pressure H. The nodal displacement fH,x,y,z can be shown as the sum of displacement caused by deformation fcH,x,y,z, frH,x,y,z, and fbH,x,y,z of dam, dam foundation, and reservoir basin. That is,(5)fH,x,y,z=fcH,x,y,z+frH,x,y,z+fbH,x,y,z.In (5), H is the depth of the water; x, y, and z are the node coordinates.Reference [1] shows that node displacement caused by the dam, dam foundation, and reservoir basin is(6)fH,x,y,z=∑k=034∑l,m,n=03AklmnHkxlymzn+∑k=034∑l,m,n=03BklmnHkxlymzn+∑k=034∑l,m,n=03CklmnHkxlymzn.In (6), Aklmn, Bklmn, and Cklmn are fitting coefficients of node displacement  component factors of dam, dam foundation, and reservoir basin under water pressure H.Dam displacement field can be determined by (6).


### 2.1.2. Inversion ofEc, Er, and Eb

Under the assumption of average deformation moduliEc0, Er0, and Eb0 of dam, dam foundation, and reservoir basin, dam displacement component field caused by deformation of dam, dam foundation, and reservoir basin can be obtained under different water pressure H as fc0H,x,y,z, fr0H,x,y,z, and fb0H,x,y,z in (5). Aklmn, Bklmn, and Cklmn in (6) can be obtained by utilizing optimization method. Due to the assumption of average deformation modulus of dam, dam foundation, and reservoir basin in structural calculation, each field component should be adjusted to get the real dam displacement field fH,x,y,z and the node displacement equation after adjustment is(7)fH,x,y,z=Xfc0H,x,y,z+Yfr0H,x,y,z+Zfb0H,x,y,z=X∑k=034∑l,m,n=03AklmnHkxlymzn+Y∑k=034∑l,m,n=03BklmnHkxlymzn+Z∑k=034∑l,m,n=03CklmnHkxlymzn.In (7), Aklmn, Bklmn, and Cklmn are fitting coefficients after the optimization of node displacement in structural calculation under different water pressure.Put (3) into (4) and obtain (7) after expansion; the component adjustment coefficients of X, Y, and Z are(8)X=Ec0Ec;Y=Er0EcEc0Er;Z=Eb0EcEc0Eb.Utilize various measured displacement data to obtain adjustment coefficients X, Y, and Z in (7) through the optimization analysis. Then average elastic modulus of dam body Ec and average deformation moduli of dam foundation and reservoir basin Er, Eb can be obtained by utilizing (8).


## 2.1.1. Determination of Dam Displacement Field

Within the linear elastic range and under water pressureH, the balance equation [11] of dam system composed by dam, dam foundation, and reservoir basin is(1)Kδ=RH.In this equation, K, δ, and RH are the overall stiffness matrix, node displacement matrix, and node load matrix, respectively. One has(2)K=∑ej∈Ω1CejTKejCej+∑ej∈Ω2CejTKejCej+∑ej∈Ω3CejTKejCej.In the above equation, Ω1, Ω2, and Ω3, respectively, represent the calculation domains of dam, dam foundation, and reservoir rock foundation and Cej and Kej represent stiffness transformation matrix and the stiffness matrix of element ej.Analysis shows that Poisson ratiosμc, μr, and μb of dam, dam foundation, and reservoir basin could be considered as constant parameters due to their small changes. As for the dam system, the geometric properties basically remained unchanged in the loading process. So K in (2) can be expressed as follows [12]:(3)K=∑ej∈Ω1CejTK-ejCej+∑ej∈Ω2CejTK-ejCej+∑ej∈Ω3CejTK-ejCej.In (3), K-ej is mainly concerned with Poisson ratio of corresponding part and geometric characteristics.Node displacement array according to (1) can be expressed as(4)δ=K-1RH,where RH is a fixed value under water pressure H. The nodal displacement fH,x,y,z can be shown as the sum of displacement caused by deformation fcH,x,y,z, frH,x,y,z, and fbH,x,y,z of dam, dam foundation, and reservoir basin. That is,(5)fH,x,y,z=fcH,x,y,z+frH,x,y,z+fbH,x,y,z.In (5), H is the depth of the water; x, y, and z are the node coordinates.Reference [1] shows that node displacement caused by the dam, dam foundation, and reservoir basin is(6)fH,x,y,z=∑k=034∑l,m,n=03AklmnHkxlymzn+∑k=034∑l,m,n=03BklmnHkxlymzn+∑k=034∑l,m,n=03CklmnHkxlymzn.In (6), Aklmn, Bklmn, and Cklmn are fitting coefficients of node displacement  component factors of dam, dam foundation, and reservoir basin under water pressure H.Dam displacement field can be determined by (6).


## 2.1.2. Inversion ofEc, Er, and Eb

Under the assumption of average deformation moduliEc0, Er0, and Eb0 of dam, dam foundation, and reservoir basin, dam displacement component field caused by deformation of dam, dam foundation, and reservoir basin can be obtained under different water pressure H as fc0H,x,y,z, fr0H,x,y,z, and fb0H,x,y,z in (5). Aklmn, Bklmn, and Cklmn in (6) can be obtained by utilizing optimization method. Due to the assumption of average deformation modulus of dam, dam foundation, and reservoir basin in structural calculation, each field component should be adjusted to get the real dam displacement field fH,x,y,z and the node displacement equation after adjustment is(7)fH,x,y,z=Xfc0H,x,y,z+Yfr0H,x,y,z+Zfb0H,x,y,z=X∑k=034∑l,m,n=03AklmnHkxlymzn+Y∑k=034∑l,m,n=03BklmnHkxlymzn+Z∑k=034∑l,m,n=03CklmnHkxlymzn.In (7), Aklmn, Bklmn, and Cklmn are fitting coefficients after the optimization of node displacement in structural calculation under different water pressure.Put (3) into (4) and obtain (7) after expansion; the component adjustment coefficients of X, Y, and Z are(8)X=Ec0Ec;Y=Er0EcEc0Er;Z=Eb0EcEc0Eb.Utilize various measured displacement data to obtain adjustment coefficients X, Y, and Z in (7) through the optimization analysis. Then average elastic modulus of dam body Ec and average deformation moduli of dam foundation and reservoir basin Er, Eb can be obtained by utilizing (8).


## 2.2. Zoning Modulus Inversion Method

Since conventional inversion methods can only inverse the integrated modulus of dam, dam foundation, and reservoir basin, the material composition differed. The dam foundation can be divided into different zones, as the dam body and reservoir basin. Therefore, the zoning deformation measured data is utilized to inverse dam zoning elastic modulus, dam foundation zoning, and reservoir basin zoning deformation modulus. The following analyzes the detailed zoning inversion analysis method.


### 2.2.1. Method for Determining Initial Zoning Modulus

Dam elastic modulus is referred to deformation modulus for the convenience below. Assume that dam, dam foundation, and reservoir basin are divided inton zones, and the corresponding modulus is Ei(i=1,2,…,n). For the zone i, the deformation measured data of corresponding zone can most reflect the impact. Make use of deformation measured data to establish deformation safety monitoring model and the equation is similar to (5). Utilize (5) to separate the water pressure component fjHm from the value of deformation measuring point j in zone i under the action of water head Hm. On the basis of above, assume that the deformation modulus of zone i is Ei′ and the rest of moduli of the other zones utilize the design values. Then calculate water pressure component δjHm  (j=1,2,…,k, where k is the number of measuring points; m=1,2,…,M, where M is the number of water pressure components) of measuring point j in zone i under different water level by structure analysis method such as finite element. The equation of δjHm is(9)δjHm=FHm,Ei′.Assume(10)Q=∑j=1k∑m=1MfjHm-xδjHm2.Minimize Q, without constraint conditions; then(11)x=∑j=1k∑m=1MfjHmδjHm∑j=1k∑m=1MδjHm2.Assume that the initial value of the actual modulus is E0i in zone i and the equation of x according to its physical meaning is(12)x=Ei′E0i.Equation (12) can be used to determine the actual initial modulus in zone i as(13)E0i=Ei′x.


### 2.2.2. Method for Determining Actual Zoning Deformation Modulus

E
0
i is the initial deformation modulus, which assumes that other zones utilize design values except zone i. It differed from the actual situation, so multipoint deformation measured data combined with the finite element structure analysis results is needed to inverse the actual deformation modulus [13–16]. The inversion method is as follows.Firstly, the zoning initial actual deformation modulusE0i(i=1,2,…,n) obtained above is utilized to calculate deformation value δHl of deformation measuring point under effect of different water head Hl with application of structural analysis method. For example, the calculated deformation value of measuring point p in zone i is δipHl as (14)δipHl=FHl,E0ii=1,2,…,n,where δipHl′ is the measured deformation value of measuring point p, which is separated by safety monitoring mathematical model from the measured data. Make(15)Q=∑i=1n∑pi=1npi∑l=1NδipHl′-xpiδipHl2=∑i=1n∑pi=1npi∑l=1NδipHl′-Ei′E0iδipHl2.In (15), n is the number of zones. npi is the number of measuring points in zone i. N is deformation monitoring frequency. The meaning of xpi is the same as that in (12).x
p
i is the final deformation modulus adjusting coefficient of zone i when Q in (15) reaches the minimum. Then the actual deformation modulus of zone i can be obtained by (13). Therefore, the following focuses on determining method of xpi, which is also the process of zoning modulus inversion of dam, dam foundation, and reservoir basin.


## 2.2.1. Method for Determining Initial Zoning Modulus

Dam elastic modulus is referred to deformation modulus for the convenience below. Assume that dam, dam foundation, and reservoir basin are divided inton zones, and the corresponding modulus is Ei(i=1,2,…,n). For the zone i, the deformation measured data of corresponding zone can most reflect the impact. Make use of deformation measured data to establish deformation safety monitoring model and the equation is similar to (5). Utilize (5) to separate the water pressure component fjHm from the value of deformation measuring point j in zone i under the action of water head Hm. On the basis of above, assume that the deformation modulus of zone i is Ei′ and the rest of moduli of the other zones utilize the design values. Then calculate water pressure component δjHm  (j=1,2,…,k, where k is the number of measuring points; m=1,2,…,M, where M is the number of water pressure components) of measuring point j in zone i under different water level by structure analysis method such as finite element. The equation of δjHm is(9)δjHm=FHm,Ei′.Assume(10)Q=∑j=1k∑m=1MfjHm-xδjHm2.Minimize Q, without constraint conditions; then(11)x=∑j=1k∑m=1MfjHmδjHm∑j=1k∑m=1MδjHm2.Assume that the initial value of the actual modulus is E0i in zone i and the equation of x according to its physical meaning is(12)x=Ei′E0i.Equation (12) can be used to determine the actual initial modulus in zone i as(13)E0i=Ei′x.


## 2.2.2. Method for Determining Actual Zoning Deformation Modulus

E
0
i is the initial deformation modulus, which assumes that other zones utilize design values except zone i. It differed from the actual situation, so multipoint deformation measured data combined with the finite element structure analysis results is needed to inverse the actual deformation modulus [13–16]. The inversion method is as follows.Firstly, the zoning initial actual deformation modulusE0i(i=1,2,…,n) obtained above is utilized to calculate deformation value δHl of deformation measuring point under effect of different water head Hl with application of structural analysis method. For example, the calculated deformation value of measuring point p in zone i is δipHl as (14)δipHl=FHl,E0ii=1,2,…,n,where δipHl′ is the measured deformation value of measuring point p, which is separated by safety monitoring mathematical model from the measured data. Make(15)Q=∑i=1n∑pi=1npi∑l=1NδipHl′-xpiδipHl2=∑i=1n∑pi=1npi∑l=1NδipHl′-Ei′E0iδipHl2.In (15), n is the number of zones. npi is the number of measuring points in zone i. N is deformation monitoring frequency. The meaning of xpi is the same as that in (12).x
p
i is the final deformation modulus adjusting coefficient of zone i when Q in (15) reaches the minimum. Then the actual deformation modulus of zone i can be obtained by (13). Therefore, the following focuses on determining method of xpi, which is also the process of zoning modulus inversion of dam, dam foundation, and reservoir basin.


## 3. Zoning Modulus Inversion of Dam, Dam Foundation, and Reservoir Basin

Q in (15) is the undetermined parameter. Under the condition of satisfying constraints, get the minimum Q. Then inversing zoning deformation modulus of dam, dam foundation, and reservoir basin is actually an optimization problem [17, 18]. Zoning deformation modulus Ei(i=1,2,…,n) is denoted as S vector. That is,(16)S=E1,E2,…,EnT.Assume(17)Q=SEi⟶mini=1,2,…,n.To meet the end condition(18)Eimin≤Ei≤Eimaxi=1,2,…,n.In (18), Eimin, Eimax are the minimum and maximum value of Ei, respectively.The inversion problem of zoning parameters in (16) cannot be directly solved to meet the constraints of (18), which needs positive analysis methods for repeated trial calculation, and finally get the solution of the optimal zoning parameters. This paper applies chaos genetic optimization algorithm [19, 20] to inverse analysis.In the process of zoning modulus inversion by chaos genetic optimization algorithm, the equation of random disturbance of zoning modulus is(19)Ek′=1-αE∗+αEk.In (19), E∗ is the optimal chaotic vector of zoning deformation modulus obtained by mapping the current optimal solution (E1∗,E2∗,…,En∗) to the interval of [0,1]. Ek is the chaotic vector of zoning deformation modulus after iteration for k times. Ek′ is the corresponding chaotic vector of zoning deformation modulus after adding random disturbance (E1,E2,…,En). α takes value between 0 and 1. When (E1,E2,…,En) are gradually close to the optimal solution, α takes the smaller value [21].Under normal circumstances,α can be determined by the following equation:(20)α=1-k-1km.In (20), m is the integer according to the optimization objective function and k is the number of iterations [8].Chaos genetic parameter disturbance method is clear after the random disturbance analysis of zoning deformation modulus above. The following focuses on inversion process and steps of zoning deformation modulus utilizing chaos genetic algorithm.(1)
Initialize zoning deformation modulus. Under the condition of constraint as (18) generates randomly a set of initial values of zoning modulus, E1,i0,E2,i0,…,Ek,i0∈[Eimin,Eimax], i=1,2,…,n. n is the number of parameters to be optimized. k is the number of initial values generated by deformation modulus of each zone.(2)
Produce the initial values in chaotic sequence of zoning deformation modulus. MapEj,i0 to the interval [0,1] via(21)Xj,i0=E1,i0-EiminEimax-Eimin,i=1,2,…,n;j=1,2,…,k.(3)
Produce zoning deformation modulus chaotic sequence of this generation. The sequenceXj,i(0,l)(l=0,1,…,m) is obtained utilizing the Logistic equation with the initial value Xj,i0. Among those, Xj,i0=Xj,i(0,0) and m is the number of chaotic iterations.(4)
Generate family members of zoning deformation modulus. After inverse mapping the following sequence is obtained:(22)Ej,i0,l=Eimin+Eimax-EiminXj,i0,l,i=1,2,…,n;j=1,2,…,k;l=1,2,…,m.(5)
Choose and calculateF(Ej,i(0,l)). Identify individuals in each zoning deformation modulus family, so that they have the opportunity to become the father generation.(6)
Coding: encode the selected father generation.(7)
Cross: cross according to crossover probability.(8)
Variation: produce variation according to the mutation probability.(9)
Make chaos genetic optimization operations to the individual of highest fitness.(10)
If the new individual fitness is higher than that of the original individual, then replace the original individual; otherwise the individual remains unchanged. Repeat steps (2)~(10), untilQ in (15) reaches the minimum. Then zoning deformation modulus inversion results are obtained. The process is shown in Figure 1.Figure 1
Flow diagram for chaos genetic algorithm to inverse deformation modulus.


## 4. Case Study

A concrete hyperbolic arch dam is located in Southwest China. The crest elevation is 1245.0 m, the minimum foundation surface is 950.5 m, and the maximum dam height is 294.5 m. Vertical measuring points are set in different dam sections to measure dam horizontal displacement. Normal water level is 1240 m. The arch dam is divided into 43 sections. Among them, spillway sections are 22 m~26 m and the rest of dam sections are 20 m. Crest width changes gradually from the center to the arch from 12 m to 16 m. The maximum central angle of arch dam is 92.791°. The bottom width of crown cantilever is 73.124 m, arc-height ratio is 3.035, and thickness-height ratio is 0.248.In order to analyze the effect of reservoir basin deformation on the hyperbolic arch dam, a wide range finite element model of reservoir basin is established. The displacement boundary condition is extracted from finite element model of the near dam area, which is the basis for analyzing the hyperbolic arch dam working behavior under the influence of reservoir basin deformation. This case focuses on the zoning deformation modulus inversion of the near dam area. According to the characteristics of dam foundation, structure, and material, the dam body will be divided into the left bank A, river bed B, and right bank C. According to the geological condition, the dam foundation is divided into 9 zones. The established near dam area finite element model is composed of altogether 142455 elements and 154960 nodes. Among them, 19559 elements and 24889 nodes are for the dam body. The model is shown in Figure2.Figure 2
Finite element model of near dam area.In the process of inversion analysis, the values of measured data are shown as follows: the left bank of the dam (Zone A) utilizes the radial displacement measured data of number 1 vertical on dam center and crest. The river bed (Zone B) utilizes the radial displacement measured data of number 2 vertical on dam center and crest, and the right bank of the dam (Zone C) utilizes the radial displacement measured data of number 3 vertical on dam center and crest. The dam foundation is divided into 9 zones. Zone I utilizes the radial displacement measured data from a, b, and c measuring points disposed on Rock I. Zone II utilizes the radial displacement measured data from d, e measuring points disposed on Rock II. ZonesIIIa, IIIb1, IIIb2, and IVa utilize the radial displacement measured data from the f, g, h, and i measuring points disposed on the corresponding rocks. Zones F7, F11, and F10 utilize the radial displacement measured data from the j, k, and l measuring points disposed on the corresponding rocks. Selected measured data series is from June 1, 2010, to January 31, 2013. The material types for different zones can be seen in Table 1. Utilizing (9)–(13), based on the radial displacement measured data corresponding to the zoning, the actual initial zoning deformation modulus values obtained by inversion are shown in Table 1. The modulus value range for each zone is also shown in Table 1 according to the modulus obtained by inversion.Table 1
The actual initial modulus and modulus ranges corresponding to different material types.
Material types
Actual initial modulus(GPa)
Modulus range(GPa)
Zone A (dam body)
25.89
10.0–40.0
Zone B (dam body)
25.17
10.0–40.0
Zone C (dam body)
24.25
10.0–40.0
Zone I (rock)
24.56
10.0–40.0
Zone II (rock)
20.17
10.0–30.0
Zone IIIa (rock)
14.32
5.0–25.0
Zone IIIb1 (rock)
9.87
5.0–20.0
Zone IIIb2 (rock)
7.05
1.0–15.0
Zone IVa (rock)
5.21
1.0–15.0
Zone F7 (cataclastic rock)
2.02
1.0–10.0
Zone F11 (slightly weathered)
3.30
1.0–10.0
Zone F10 (slightly weathered)
3.42
1.0–10.0Combined with the actual situation of the case, considering the effect of initial value and utilizing radial displacement measured data of the selected representative measured point in each zone from June 1, 2010, to January 31, 2013, the displacement statistical model of each measuring point is established:(23)δ′=δH′+δT′+δθ′=a0+∑i=14a1iHui-Hu0i+∑i=12b1isin⁡2πit365-sin⁡2πit0365+b2icos⁡2πit365-cos⁡2πit0365+d1θ-θ0+d2ln⁡θ-ln⁡θ0.In (23), δ′ is the radial displacement of corresponding measuring point; δH′, δT′, and δθ′ are water pressure component, temperature component, and aging component, respectively; a0 is the constant term; a1i is the water pressure fitting coefficient; b1i, b2i are the temperature factor fitting coefficients; d1, d2 are the aging factor fitting coefficients; Hu is the upstream water depth of present measured data; Hu0 is the upstream water depth of the first measured data in the data series; t is the cumulative days from the initial day to the present day; θ takes t/100; θ0 takes 1/100.According to [1], ∑i=14a1i(Hui-Hu0i) is utilized to compute the radial displacements caused by the water pressure. Since it is an arch dam, δH′ has a linear relation with H4. During the operation period, the temperature of the dam body is affected by air temperature and water temperature, and the temperature of the dam presents a periodic variation. Therefore, the temperature variation of the dam is regarded as a simple harmonic variation, such that  ∑i=12b1isin⁡(2πit/365)-sin⁡(2πit0/365)+b2icos⁡(2πit/365)-cos⁡(2πit0/365) is utilized to compute the radial displacements caused by the temperature. According to the rule of aging variation, at the beginning of the impoundment, the aging changes greatly and then gradually to be steady. In this paper, d1(θ-θ0)+d2(lnθ-lnθ0) is introduced to compute the radial displacements caused by the aging.Make use of displacement monitoring mathematical model in (23) to separate water pressure component δH′, temperature component δT′, and aging component δθ′ from displacement value of corresponding measuring point. Figure 3 is the radial displacement separation results of number 2 vertical crest measuring point in river bed (Zone B) arch crown beam section. The rest of displacement separation results of measuring points are similar to the above and will not be described here.Figure 3
The radial displacement separation results of number 2 vertical crest measuring point.According to the above analysis, the water pressure components can be separated from actual radial displacement of each measuring point by (23), which are δipHl′ in (15). Utilizing the initial actual zoning modulus E0i in Figure 1, the finite element analysis model is established to obtain the water pressure components under different water loads, which are δipHl in (15). Then the zoning deformation modulus optimization objective function equation (15) is formed. Finally zoning deformation modulus in this case is obtained utilizing the deformation modulus value range of each zone in Table 1 as the optimization constraints and deformation modulus inversion method based on chaos genetic optimization algorithm. The results are in Table 2. The modulus inversion process of Zone A can be seen in Figures 4–6.Table 2
The zoning modulus inversion results.
Material types
Actual deformation modulus inversion (GPa)
Zone A (dam body)
27.91
Zone B (dam body)
27.15
Zone C (dam body)
26.31
Zone I (rock)
25.23
Zone II (rock)
20.73
Zone IIIa (rock)
14.59
Zone IIIb1 (rock)
10.09
Zone IIIb2 (rock)
7.58
Zone IVa (rock)
5.41
Zone F7 (cataclastic rock)
2.07
Zone F11 (slightly weathered)
3.52
Zone F10 (slightly weathered)
3.37Figure 4
The assumption modulusE′.Figure 5
The actual initial modulusE0.Figure 6
The final inversed modulusE.In order to verify the correctness of inversion of the zoning deformation modulus, utilizing zoning deformation modulus in Table2 as calculation parameters, water pressure component values are calculated by finite element method under different water loads in Table 3, and measured water pressure component values are also shown as the comparison.Table 3
Calculated values and measured values of radial displacements from vertical measuring points (unit: mm).
Water pressure component
Water pressure component calculated by finite element
Measured water pressure component
Water level (m)
1165.85
1181
1225
1233
1240
1165.85
1181
1225
1233
1240
Measuring  points
Number 1 vertical dam crest
20.09
30.42
60.58
70.81
79.69
21.61
31.17
61.60
71.34
80.25
Number 1 vertical dam center
15.45
21.96
34.97
38.65
41.21
15.93
22.20
35.52
39.10
41.92
Number 2 vertical dam crest
42.91
56.74
103.05
121.23
141.78
43.10
57.14
104.34
122.10
143.69
Number 2 vertical dam center
37.25
52.17
76.73
84.95
92.61
37.99
52.66
77.11
85.61
93.09
Number 3 vertical dam crest
18.31
25.54
54.68
63.79
72.81
18.74
26.08
55.03
64.11
73.13
Number 3 vertical dam center
14.11
19.05
31.96
35.03
38.14
14.48
19.93
32.41
35.94
38.91As can be seen from Table3, inversed water pressure components of vertical measuring point under different water loads are close to measured water pressure components. The relative errors are all below 5% which reflect the rationality of the zoning deformation modulus inversion results and show the effectiveness of the proposed inversion analysis method in this paper.


## 5. Conclusions

The inversion method of zoning deformation modulus is mainly focused on, based on the conventional inversion method for the deformation modulus of dam, dam foundation, and reservoir basin. The inversion objective function for zoning deformation modulus is established and the inversion analysis steps are shown. The conclusions are as follows.(1)
Dam displacement field characterization method is developed under the effect of water load based on the basic balance equations. On the basis of this, the method for inversing deformation modulus of dam, dam foundation, and reservoir basin is introduced by utilizing temporal and spatial distribution safety monitoring model of space displacement field. The inversion equations of average deformation modulus of dam, dam foundation, and reservoir basin are derived.(2)
The actual initial zoning deformation modulus of dam, dam foundation, and reservoir basin is put forward, based on the analysis for the defect of conventional inversion method and the comprehensive utilization of the actual displacement measured data and finite element calculation results. Furthermore, the inversion objective function for the actual zoning deformation modulus is determined.(3)
The method and implementation steps are proposed to inverse zoning deformation modulus of dam, dam foundation, and reservoir basin based on chaos genetic optimization algorithm. The effectiveness of the zoning inversion method in this paper is verified through the actual project application.



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*Source: 817241-2015-06-08.xml*