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

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.


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][3][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.

The Inversion Analysis Method
2.1.The Conventional Inversion Method.For the dam of ages, the elastic modulus of dam body   and deformation moduli of dam foundation and reservoir basin   ,   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 2 Mathematical Problems in Engineering 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 pressure , the balance equation [11] of dam system composed by dam, dam foundation, and reservoir basin is In this equation, [], {}, and {  } are the overall stiffness matrix, node displacement matrix, and node load matrix, respectively.One has In the above equation, Ω Analysis shows that Poisson ratios   ,   , and   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 [] in (2) can be expressed as follows [12]: In (3), []   is mainly concerned with Poisson ratio of corresponding part and geometric characteristics.Node displacement array according to (1) can be expressed as where {  } is a fixed value under water pressure .
In (5),  is the depth of the water; , , and  are the node coordinates.
Reference [1] shows that node displacement caused by the dam, dam foundation, and reservoir basin is In (6),   ,   , and   are fitting coefficients of node displacement component factors of dam, dam foundation, and reservoir basin under water pressure .
Dam displacement field can be determined by (6).

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.

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 into  zones, and the corresponding modulus is   ( = 1, 2, . . ., ).For the zone , 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   (  ) from the value of deformation measuring point  in zone  under the action of water head   .On the basis of above, assume that the deformation modulus of zone  is    and the rest of moduli of the other zones utilize the design values.Then calculate water pressure component    ( = 1, 2, . . ., , where  is the number of measuring points;  = 1, 2, . . ., , where  is the number of water pressure components) of measuring point  in zone  under different water level by structure analysis method such as finite element.The equation of    is Assume Minimize , without constraint conditions; then Assume that the initial value of the actual modulus is  0 in zone  and the equation of  according to its physical meaning is Equation ( 12) can be used to determine the actual initial modulus in zone  as

Method for Determining Actual Zoning Deformation
Modulus. 0 is the initial deformation modulus, which assumes that other zones utilize design values except zone .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][14][15][16].The inversion method is as follows.
Firstly, the zoning initial actual deformation modulus  0 ( = 1, 2, . . ., ) obtained above is utilized to calculate deformation value    of deformation measuring point under effect of different water head   with application of structural analysis method.For example, the calculated deformation value of measuring point  in zone  is    as where     is the measured deformation value of measuring point , which is separated by safety monitoring mathematical model from the measured data.Make In (15),  is the number of zones.   is the number of measuring points in zone . is deformation monitoring frequency.The meaning of    is the same as that in (12).   is the final deformation modulus adjusting coefficient of zone  when  in (15) reaches the minimum.Then the actual deformation modulus of zone  can be obtained by (13).Therefore, the following focuses on determining method of    , which is also the process of zoning modulus inversion of dam, dam foundation, and reservoir basin.

Zoning Modulus Inversion of Dam, Dam Foundation, and Reservoir Basin
in ( 15) is the undetermined parameter.Under the condition of satisfying constraints, get the minimum .Then inversing zoning deformation modulus of dam, dam foundation, and reservoir basin is actually an optimization problem [17,18].Zoning deformation modulus   ( = 1, 2, . . ., ) is denoted as  vector.That is, Assume To meet the end condition In (18),  min ,  max are the minimum and maximum value of   , 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 In (19),  * is the optimal chaotic vector of zoning deformation modulus obtained by mapping the current optimal solution ( * 1 ,  * 2 , . . .,  *  ) to the interval of [0, 1].  is the chaotic vector of zoning deformation modulus after iteration for  times.   is the corresponding chaotic vector of zoning deformation modulus after adding random disturbance ( 1 ,  2 , . . .,   ). takes value between 0 and 1.When ( 1 ,  2 , . . .,   ) are gradually close to the optimal solution,  takes the smaller value [21].
Under normal circumstances,  can be determined by the following equation: In (20),  is the integer according to the optimization objective function and  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.

𝑗,𝑖
and  is the number of chaotic iterations.
(5) Choose and calculate ( (0,) , ).Identify individuals in each zoning deformation modulus family, so that they have the opportunity to become the father generation.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 Figure 2.

Case Study
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

Modulus initialization
Coding Cross

Chaotic genetic optimization operation
Is Q small enough?
Yes No Generate the initial value X 0 j,i Produce chaotic sequence of this generation X (0,1) j,i Generate family members E (0,1) j,i Choose and calculate F(E (0,1) j,i ) utilize the radial displacement measured data from the f, g, h, and i measuring points disposed on the corresponding rocks.Zones F 7 , F 11 , and F 10 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.Combined 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: In ( 23),   is the radial displacement of corresponding measuring point;    ,    , and    are water pressure component, temperature component, and aging component, respectively;  0 is the constant term;  1 is the water pressure fitting coefficient;  1 ,  2 are the temperature factor fitting coefficients;  1 ,  2 are the aging factor fitting coefficients;   is the upstream water depth of present measured data;  0 is the upstream water depth of the first measured data in the data series;  is the cumulative days from the initial day to the present day;  takes /100;  0 takes 1/100.
According to [1], ] is utilized to compute the radial displacements caused by the water pressure.Since it is an arch dam,    has a linear relation with  4 .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 ∑ 2 =1 [ 1 (sin(2/365)−sin(2 0 /365))+ 2 (cos(2/365)− cos(2 0 /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,  1 ( −  0 ) +  2 (ln − ln 0 ) is introduced to compute the radial displacements caused by the aging.
Make   crown beam section.The rest of displacement separation results of measuring points are similar to the above and will not be described here.According to the above analysis, the water pressure components can be separated from actual radial displacement of each measuring point by (23), which are     in (15).Utilizing the initial actual zoning modulus  0 in Figure 1, the finite element analysis model is established to obtain the water pressure components under different water loads, which are    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.
In order to verify the correctness of inversion of the zoning deformation modulus, utilizing zoning deformation modulus in   As can be seen from Table 3, 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.

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.

( 6 )( 9 )
Coding: encode the selected father generation.(7)Cross: cross according to crossover probability.(8) Variation: produce variation according to the mutation probability.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), until  in (15) reaches the minimum.Then zoning deformation modulus inversion results are obtained.The process is shown in Figure 1.

Figure 1 :Figure 2 :
Figure 1: Flow diagram for chaos genetic algorithm to inverse deformation modulus.

Figure 3 :
Figure 3: The radial displacement separation results of number 2 vertical crest measuring point.
(5)2.Inversion of   ,   , and   .Under the assumption of average deformation moduli  0 ,  0 , and  0 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  as  0 (, , , ),  0 (, , , ), and  0 (, , , ) in(5).  ,   , and   in ( (8) ,   , and   are fitting coefficients after the , and  in(7)through the optimization analysis.Then average elastic modulus of dam body   and average deformation moduli of dam foundation and reservoir basin   ,   can be obtained by utilizing(8).

Table 1 :
The actual initial modulus and modulus ranges corresponding to different material types.
use of displacement monitoring mathematical model in (23) to separate water pressure component    , temperature component    , and aging component    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

Table 2 :
The zoning modulus inversion results.
Table 2 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 the comparison.

Table 3 :
Calculated values and measured values of radial displacements from vertical measuring points (unit: mm).