# Inversion Modeling of Dam-Zoning Elasticity Modulus for Heightened Concrete Dam Using ICS-IPSO Algorithm

Authors: Yijun Chen; Chongshi Gu; Bangbin Wu; Chenfei Shao; Zhongru Wu; Bo Dai
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

A new approach was developed for the inversion modeling of dam-zoning elasticity modulus for heightened concrete dam, with old and new concrete zones. The proposed inversion modeling procedure takes advantage of the improved cuckoo search (ICS) algorithm and improved particle swarm optimization (IPSO) algorithm to adjust the mechanical parameters, which are used as input. An objective function is constructed based on the horizontal displacement increment by using the finite element method (FEM) and statistical analysis of the prototype monitoring data. One ideal arch dam model and one actual heightened concrete dam were taken as examples. The proposed method was used to implement the optimal selection of the dam-zoning elasticity modulus. The inversion analysis results indicate that the mechanical parameters identification method for heightened concrete gravity dams proposed in this article is accurate and has a fast convergence rate. Consequently, it can be applied as a reliable model to identify the dam-zoning elasticity modulus in practical engineering applications.


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## Body

## 1. Introduction

Due to the uneven geographical distribution of water resources and the need to alleviate the metropolitan water shortage in China, the water supply capacity of reservoirs is often increased by heightening dams or transferring water across regions. The new concrete dams are poured on the foundations of old dams that have been in service for many years, and thus there will exist new and old concrete zones in the dam body. A greater water pressure acts on the dam body as a result of the increasing reservoir water level. Furthermore, the structure response of a concrete dam is affected by many factors during its operation, including the dynamic and static cyclic water loading, ambient temperature, erosion, corrosion, crack, and so on. These influences are irreversible over the life cycle under complex environmental conditions [1, 2], which is mainly defined as the aging component in statistical regression analysis [3]. Therefore, the material property evolution of new and old concrete dam has an important impact on the working behavior of the heightened dam [4].The material properties are generally reflected by the mechanical parameters, among which the reasonable determination of the elasticity modulus is very important to understanding the working state and evaluating the safety of a concrete dam [5]. The elasticity modulus can well represent the stiffness and resistance of the material, which can be applied to characterize the performance evolution of concrete structures [6]. Particularly in damaged dam structure, the identified elastic modulus can effectively reflect the presence of a structure deterioration process, but the damaged zone cannot be localized [1]. Thus, the elasticity modulus is the basis and prerequisite for the health diagnosis of a concrete dam. It is of great importance to adopt reasonable and effective methods to identify the elasticity modulus in old and new concrete zones during the normal operation period.The elasticity modulus can be measured directly using core and rebound tests. Classical rebound tests are reliable but offer only a “pointwise” measurement in practice over concrete surfaces areas of a few square millimeters. Moreover, the mechanical parameters are very discrete for concrete dams that have been in service for many years, due to the cement paste, water cement ratio, cracks, corrosion, and carbonization in the concrete inside the dam body. The exact elasticity modulus of the concrete dam zones thus cannot be fully represented by the test method [7, 8].Therefore, it is necessary to conduct an inversion analysis method to identify the mechanical parameters of the dam body, dam foundation, and reservoir basin on the basis of the dam prototype monitoring data analysis and structure simulation [9, 10]. However, the conventional inversion method can only identify the integrated elasticity modulus of the dam body, dam foundation, and reservoir basin. The dam body can be divided into different zones according to the material composition [11]. In recent years, various intelligent optimization algorithms have been applied to identify the mechanical parameters of concrete dams. Xiang introduced the genetic algorithm to obtain the elasticity modulus of the concrete dam and foundation [12]. Gu utilized the chaos genetic optimization algorithm [11] and the adjoint method combined with the DFP quasi-Newton method [13] to conduct the inversion analysis of the dam-zoning modulus in concrete dam, respectively. Fei proposed a parameter identification method for concrete dam-foundation systems based on statistical analysis of recorded dam displacements with a hybrid simplex artificial bee colony algorithm [14]. Dou adopted a novel adaptive fireworks algorithm to identify Young modulus of a concrete gravity dam and a concrete arch dam [15]. Nevertheless, from the inversion literature, it is inconsistent with the actual situation by using the absolute displacement of the prototype monitoring to conduct inversion analysis. Therefore, the inversion modeling of dam-zoning elasticity modulus has been proposed based upon the horizontal displacement increment.Moreover, there have been very few studies addressing inversion problems on the zoning elasticity modulus of a heightened concrete dam. Even employing the cuckoo search (CS) optimization algorithm to obtain the mechanical parameters for the inversion analysis is currently one of the major gaps in the literature. In particularly, the deformation response of the heightened concrete dam is characterized by multiple extrema and nonlinearity, due to the many influencing factors and complicated influencing relations in the dam-water-foundation interaction. The CS algorithm has the advantages of simple and convenient operation, strong and stable global convergence ability, and good numerical stability in the process of extreme value optimization [16]. In this research, a new inversion modeling method was proposed to identify the dam-zoning elasticity modulus using the ICS-IPSO hybrid algorithm in a heightened concrete dam.The paper is organized as follows: In Section1, the background knowledge of the inversion analysis method is provided, including the CS optimization algorithm. Section 2 elaborates upon the separation method of the water pressure components using the hydraulic-thermal-time model, and an objective function is established. Section 3 describes the improved cuckoo search (ICS) algorithm and improved particle swarm optimization (IPSO) algorithm. The proposed inversion modeling is verified and illustrated by one ideal arch dam model and one actual heightened concrete dam project in Section 4. Finally, in Section 5, we draw some main conclusions.


## 2. Inversion Analysis Method of the Dam-Zoning Elasticity Modulus

### 2.1. Overview of the Hydraulic-Thermal-Time Statistical Model

The displacement field of the dam body generated at an arbitrary point is a vector fieldδ⃑ under the action of the water pressure (H), ambient temperature (T), uplift pressure, sediment pressure, and other factors that can be decomposed into the radial horizontal displacement δx, side horizontal displacement δy, and vertical displacement δz, as shown in Figure 1. According to the mechanical theory analysis of the dam structure, the prototype measured displacement includes a reversible displacement and irreversible displacement. (1) The water pressure component δH and temperature component δT are the main components of the displacement, which are included in the reversible displacement. The water pressure component is mainly composed of the displacement caused by the reservoir water. The temperature component is caused by the temperature variation of the concrete hydration heat or the outside air temperature. (2) The irreversible displacement is mainly the aging component δθ. Furthermore, the water pressure component δH of the radial horizontal displacement δx can be used to identify the dam-zoning elasticity modulus. Therefore, the hydraulic-thermal-time statistical model of each component for the concrete dam radial horizontal displacement monitoring data is depicted in [17](1)δx=δH+δT+δθ=a0+∑i=1naiHi+∑i=1mb1isin⁡2πit365+b2icos⁡2πit365+c1θ+c2ln⁡θwhere a0 is a constant term considering the influence of the initial state, ai denotes the fitting coefficient of the water pressure, and H denotes the upstream water depth. According to the dam type, the value of n is different; for a concrete gravity dam, it is generally set to 3, while for an arch dam, it is 4 or 5. m denotes the period; m=1 for an annual cycle, and m=2 for a half-year cycle. b1i and b2i are fitting coefficients for the temperature. t is the cumulative day from the initial day to the present day. c1 and c2 are the fitting coefficients for the aging factor. θ takes a value of t/100.Figure 1
The displacement field of a concrete dam body.


### 2.2. The Objective Function Description

In an actual project, according to the structural function and material performance of a concrete dam, the dam body can be divided into different zones. According to the mathematical statistical model in (1), the water pressure component δpwmeasure (p=1,2,…,P, where P is the number of measuring points. w=1,2,…,W, where W is the number of selected characteristic water levels) of the measured displacement in the dam prototype observation can be separated by adopting statistical regression analysis. The water pressure component increment Δδpwmeasure at the pth measuring point and wth water level stage of the measured displacement can be obtained in (2). δp0measure is the water pressure component at the initial water level. The calculated displacement δpwinverse can be obtained by the finite element method under different characteristic water levels Hw (w=1,2,…,W, where W is the number of characteristic water levels) in (3). The calculated displacement increment Δδpwinverse at the pth measuring point and wth water level stage can be obtained by (4), where δp0inverse is the calculated displacement at the initial water level. Eiinverse=(ei1inverse,ei2inverse,⋯,eizinverse)∈Rz (i=1,2,⋯,n, where n is the number of nests in the CS algorithm or particles in the PSO algorithm and the parameter z is the number of zoning elasticity moduli on the basis of practical problems) is the design variable vector in z-dimensional space applied in the FEM analysis as calculation parameters, which represent the dam-zoning elasticity modulus updated by using the ICS-IPSO optimization algorithm.(2)Δδpwmeasure=δpwmeasure-δp0measure(3)δpwinverse=fHw,Eiinverse(4)Δδpwinverse=δpwinverse-δp0inverseThe horizontal displacement increments by using the finite element method (FEM) and statistical analysis of the prototype monitoring data are applied in inversion modeling to construct an objective function. A schematic diagram is shown in Figure2. The water pressure component varying with the water level can be represented by line L2, and the measured displacement can be collected by direct and reversed plumbs. However, the concrete dam is already under water loading Hiniial when the plumbs were installed in the dam body. Thus, the horizontal displacements monitored by the plumb system can only represent the relative displacement of the dam, rather than the absolute displacement. Hence, the horizontal displacement increment can be used in inversion analysis to identify the dam-zoning elasticity modulus.Figure 2
Schematic diagram of pointp deformation in dam body with reservoir water level (L1: calculated displacement by FEM; L2: water pressure component of measured displacement).The objective of the inversion model is to identify the optimal representation elasticity modulus of the dams. Consequently, the inversion problem can be transformed into the extremum optimization problem [18]. The objective function is constructed as follows:(5)minimizeJ=∑p=1P∑w=1WΔδpwmeasure-Δδpwinverse2P×WsuchthatEmin≤Eiinverse≤Emaxwhere J denotes an objective function. Emin and Emax denote the lower and upper side constraints to the design variables, respectively, and the other symbols mean the same as above.The concrete dam system composed of the dam body, rock foundation, and reservoir basin is divided into elements applying the finite element method. An equilibrium equation [19, 20] is established between the nodal displacement and nodal load as(6)Kδ=RHwhere K denotes the overall stiffness matrix, δ denotes the nodal displacement array, and RH denotes the nodal force array. The expression [K] can be represented as(7)K=∑ej∈Ω1CejTKejCej+∑ej∈Ω2CejTKejCej+∑ej∈Ω3CejTKejCejwhere Ω1,Ω2,Ω3 are the computational domains of the dam body, rock foundation, and reservoir basin respectively. [C]ej denotes the stiffness transformation matrix of element ej. [K]ejdenotes the stiffness matrix of elementej, which can be represented as(8)Kej=∭ΩjBTDBdΩwhere [D] denotes the elastic matrix, which is related to the elasticity modulus E and Poisson ratio μ, and [B] denotes the unit of the geometric characteristics, which are related to the element shape and size.In this work, in particular, the difference in the objective function value between two adjacent iteration steps is defined to judge the convergence. Otherwise, the solution of the inversion analysis is complete when the number of iteration steps meets the requirements. The algorithm is iteratively checked until the specified convergence criterion of (9) is satisfied. (9)JT-JT-1JT≤εorT>Tmaxwhere T denotes the iterative step, JT denotes the objective function value during the Tth iteration, ε denotes the specified convergence tolerance, and Tmax denotes the maximum number of iterations of the ICS-IPSO algorithm.


## 2.1. Overview of the Hydraulic-Thermal-Time Statistical Model

The displacement field of the dam body generated at an arbitrary point is a vector fieldδ⃑ under the action of the water pressure (H), ambient temperature (T), uplift pressure, sediment pressure, and other factors that can be decomposed into the radial horizontal displacement δx, side horizontal displacement δy, and vertical displacement δz, as shown in Figure 1. According to the mechanical theory analysis of the dam structure, the prototype measured displacement includes a reversible displacement and irreversible displacement. (1) The water pressure component δH and temperature component δT are the main components of the displacement, which are included in the reversible displacement. The water pressure component is mainly composed of the displacement caused by the reservoir water. The temperature component is caused by the temperature variation of the concrete hydration heat or the outside air temperature. (2) The irreversible displacement is mainly the aging component δθ. Furthermore, the water pressure component δH of the radial horizontal displacement δx can be used to identify the dam-zoning elasticity modulus. Therefore, the hydraulic-thermal-time statistical model of each component for the concrete dam radial horizontal displacement monitoring data is depicted in [17](1)δx=δH+δT+δθ=a0+∑i=1naiHi+∑i=1mb1isin⁡2πit365+b2icos⁡2πit365+c1θ+c2ln⁡θwhere a0 is a constant term considering the influence of the initial state, ai denotes the fitting coefficient of the water pressure, and H denotes the upstream water depth. According to the dam type, the value of n is different; for a concrete gravity dam, it is generally set to 3, while for an arch dam, it is 4 or 5. m denotes the period; m=1 for an annual cycle, and m=2 for a half-year cycle. b1i and b2i are fitting coefficients for the temperature. t is the cumulative day from the initial day to the present day. c1 and c2 are the fitting coefficients for the aging factor. θ takes a value of t/100.Figure 1
The displacement field of a concrete dam body.


## 2.2. The Objective Function Description

In an actual project, according to the structural function and material performance of a concrete dam, the dam body can be divided into different zones. According to the mathematical statistical model in (1), the water pressure component δpwmeasure (p=1,2,…,P, where P is the number of measuring points. w=1,2,…,W, where W is the number of selected characteristic water levels) of the measured displacement in the dam prototype observation can be separated by adopting statistical regression analysis. The water pressure component increment Δδpwmeasure at the pth measuring point and wth water level stage of the measured displacement can be obtained in (2). δp0measure is the water pressure component at the initial water level. The calculated displacement δpwinverse can be obtained by the finite element method under different characteristic water levels Hw (w=1,2,…,W, where W is the number of characteristic water levels) in (3). The calculated displacement increment Δδpwinverse at the pth measuring point and wth water level stage can be obtained by (4), where δp0inverse is the calculated displacement at the initial water level. Eiinverse=(ei1inverse,ei2inverse,⋯,eizinverse)∈Rz (i=1,2,⋯,n, where n is the number of nests in the CS algorithm or particles in the PSO algorithm and the parameter z is the number of zoning elasticity moduli on the basis of practical problems) is the design variable vector in z-dimensional space applied in the FEM analysis as calculation parameters, which represent the dam-zoning elasticity modulus updated by using the ICS-IPSO optimization algorithm.(2)Δδpwmeasure=δpwmeasure-δp0measure(3)δpwinverse=fHw,Eiinverse(4)Δδpwinverse=δpwinverse-δp0inverseThe horizontal displacement increments by using the finite element method (FEM) and statistical analysis of the prototype monitoring data are applied in inversion modeling to construct an objective function. A schematic diagram is shown in Figure2. The water pressure component varying with the water level can be represented by line L2, and the measured displacement can be collected by direct and reversed plumbs. However, the concrete dam is already under water loading Hiniial when the plumbs were installed in the dam body. Thus, the horizontal displacements monitored by the plumb system can only represent the relative displacement of the dam, rather than the absolute displacement. Hence, the horizontal displacement increment can be used in inversion analysis to identify the dam-zoning elasticity modulus.Figure 2
Schematic diagram of pointp deformation in dam body with reservoir water level (L1: calculated displacement by FEM; L2: water pressure component of measured displacement).The objective of the inversion model is to identify the optimal representation elasticity modulus of the dams. Consequently, the inversion problem can be transformed into the extremum optimization problem [18]. The objective function is constructed as follows:(5)minimizeJ=∑p=1P∑w=1WΔδpwmeasure-Δδpwinverse2P×WsuchthatEmin≤Eiinverse≤Emaxwhere J denotes an objective function. Emin and Emax denote the lower and upper side constraints to the design variables, respectively, and the other symbols mean the same as above.The concrete dam system composed of the dam body, rock foundation, and reservoir basin is divided into elements applying the finite element method. An equilibrium equation [19, 20] is established between the nodal displacement and nodal load as(6)Kδ=RHwhere K denotes the overall stiffness matrix, δ denotes the nodal displacement array, and RH denotes the nodal force array. The expression [K] can be represented as(7)K=∑ej∈Ω1CejTKejCej+∑ej∈Ω2CejTKejCej+∑ej∈Ω3CejTKejCejwhere Ω1,Ω2,Ω3 are the computational domains of the dam body, rock foundation, and reservoir basin respectively. [C]ej denotes the stiffness transformation matrix of element ej. [K]ejdenotes the stiffness matrix of elementej, which can be represented as(8)Kej=∭ΩjBTDBdΩwhere [D] denotes the elastic matrix, which is related to the elasticity modulus E and Poisson ratio μ, and [B] denotes the unit of the geometric characteristics, which are related to the element shape and size.In this work, in particular, the difference in the objective function value between two adjacent iteration steps is defined to judge the convergence. Otherwise, the solution of the inversion analysis is complete when the number of iteration steps meets the requirements. The algorithm is iteratively checked until the specified convergence criterion of (9) is satisfied. (9)JT-JT-1JT≤εorT>Tmaxwhere T denotes the iterative step, JT denotes the objective function value during the Tth iteration, ε denotes the specified convergence tolerance, and Tmax denotes the maximum number of iterations of the ICS-IPSO algorithm.


## 3. ICS-IPSO Hybrid Optimization Algorithm

### 3.1. Cuckoo Search (CS) Standard Algorithm

In this section, the cuckoo search algorithm is briefly reviewed. The CS is one of the latest nature-inspired metaheuristic algorithms, developed in 2009 by Xin-She Yang and Suash Deb [16]. The CS is based on the brood parasitism of some cuckoo species. Cuckoos lay their eggs in the nests of other host birds and remove the eggs of the host birds. The host birds may discover that the eggs are not its own, and either remove the foreign eggs or abandon the nest and build a new nest somewhere [21]. This algorithm is enhanced by using the so-called Lévy flights [22], rather than simple isotropic random walks. The long jumps may increase the search efficiency of the cuckoo search significantly in some cases, especially for multimodal, nonlinear problems. Each nest xi is treated as a solution in a z-dimensional space, and the performance of each nest is evaluated according to a predefined fitness function, which is related to the solved problem.For simplicity in describing the standard cuckoo search, the following three idealized rules were set:(i) Each cuckoo lays one egg at a time and dumps it in a randomly chosen nest.(ii) The best nests with high-quality eggs will be carried over to the next generations.(iii) The number of available host nests is fixed, and the egg laid by a cuckoo is discovered by the host bird with a probabilitypa∈(0,1). In this case, the host bird either gets rid of the eggs or simply abandons the nest and build a completely new nest.Furthermore, the CS algorithm uses a balanced combination of a local random walk and the global explorative random walk, controlled by a switching parameterpa [16, 21]. The local random walk can be written as(10)xiT+1=xiT+αs⊗Hpa-ε⊗xjT-xkTwhere the superscriptT denotes the Tth iteration. α,ε∈(0,1) are random numbers drawn from a uniform distribution, s is the step size, and H(pa-ε) is a Heaviside function. xjT and xkT are two different solutions selected randomly by random permutation in the Tth iteration, and the product ⊗ represents entry-wise multiplication.On the other hand, the global random walk is carried out by adopting Lévy flights.(11)xiT+1=xiT+αLs,λwhere (12)Ls,λ=λΓλsinπλ/2π1s1+λ,s⟶+∞Here,α>0 denotes the step size scaling factor, which should be related to the scale of the problem of interest. In most cases, we can use α=O(L/10), where L is the characteristic scale of the problem of interest, while in some cases, α=O(L/10) can be more effective to avoid flying too far. Obviously, the α value in these two updating equations can be different, thus leading to two different parameters α1 and α2. Here, we use α1=α2=α for simplicity. L(s,λ) denotes the step-lengths that are distributed according to the probability distribution shown in (12), which has an infinite variance with an infinite mean.However, generating a pseudorandom step sizes using Lévy flights is not trivial. The most effective and direct method is to adopt the Mantegna algorithm to achieve symmetric Lévy flights, and the step size can be calculated in (13). The two variables U and V are drawn from Gaussian distributions that are depicted in (14) and (15) [21, 22].(13)s=UV1/λ,1<λ≤3where(14)U~N0,σ2,V~N0,1(15)σ2=Γ1+λλΓ1+λ/2·sin⁡πλ/22λ-1/21/λ


### 3.2. Improved Cuckoo Search (ICS) Algorithm

The parameterpa introduced in the CS algorithm helps the local random walk to update solutions, which can influence the convergence rate and the precision of the solutions [23]. In general, the CS standard algorithm uses a fixed value for pa, and the value is set in the initialization step and cannot be changed during new generations, which may result in premature convergence.To keep the balance between global and local searches and increase the population evolution intensity, a variable parameterPa is introduced to adjust the value of the switching parameter. In addition, the value of Pa should be decreased with the search progress to more easily produce new individuals at the later iteration stage to avoid falling into local optima, which can be presented as follows:(16)PaT=Pa,max-Pa,max-Pa.minTmax×Twhere Tmax denotes the maximum iterative steps, T denotes the current iteration step, and Pa,max and Pa,min denote the control probability of the switching parameter Pa.


### 3.3. Improved Particle Swarm Optimization (IPSO) Algorithm

To further enhance the global and local exploration capability and improve the convergence rate, as well as avoid the premature convergence of the algorithm, the IPSO algorithm is integrated into the ICS algorithm. Particle swarm optimization (PSO) is a parallel evolutional computation technique developed by Kennedy and Eberhart based on the simulation of the social behavior of birds in a flock [24, 25]. The new velocity vi of particle i is determined by adjusting its previous velocity based on the distance of its current position from its best position in (17). Then, the position xi of particle i is updated according to its companion and own flying experience, which is treated as a point in z-dimensional space according to the material zoning of the practical project in (18). pi denotes the best previous position that particle i has ever visited, and the value of pg denotes the best particle among all the particles in the population. The performance of each particle is evaluated according to a predefined fitness function, which is related to the solved problem.(17)viT+1=wviT+c1r1piT-xiT+c2r2pgT-xiT(18)xiT+1=xiT+αviT+1where the superscript T denotes the Tth iteration. w denotes the inertia weight, which is employed to control the impact of the previous velocities on the current velocity. c1 and c2 are positive constants, called the cognitive and social parameters, respectively. r1 and r2 are random numbers uniformly distributed in the range (0,1). α represents a scaling factor that controls the velocity of the particles.The appropriate inertia weightw can influence the trade-off between the global and local exploration abilities of the “flying points”. A larger inertia weight can facilitate the global exploration ability, while a smaller inertia weight can effectively improve the ability of local exploration, thereby improving the search precision of the algorithm and finding the optimal solution more quickly and accurately. Particularly, in this research, the inertia weight w is dynamically changed along with the algorithm optimization process to provide a balance between the global and local exploration abilities and increase the convergence rate [25]. The inertia weight is expressed in (19). The advantages of the function are as follows. (1) The function is monotonically decreasing in the region, which meets the requirement of a large inertia weight at the beginning and a small inertia weight at the end. (2) The decreasing speed of this function is relatively slow at the beginning, which is conducive to exploring the optimal solution and avoid getting stuck in local optima. (3) The function maintains a relatively flat decreasing speed at the end, which is conducive to facilitating the local search and improving the search precision [26]. (19)wT=wmax-wmin2×cos⁡π×TTmax+wmax+wmin2where wmax and wmin are the maximum and minimum inertia weight w, respectively, Tmax is the maximum number of iteration steps, and T denotes the current iteration step.


### 3.4. Steps for Implementing the Inversion Analysis Method

According to the above descriptions of the ICS algorithm and IPSO algorithm, the execution process of the dam-zoning elasticity modulus inversion analysis method using the ICS-IPSO optimization algorithm is mainly composed of three parts. First, the global random walk is carried out by adopting Lévy flights, instead of simple isotropic random walks. Then, the switching parameterPa is dynamically changed with the CS algorithm optimization progress to update the local position of the nest. Third, a monotonically decreasing inertia weight w is applied in the PSO algorithm to facilitate global exploration in the early stage, while a smaller inertia weight enhances the local exploration ability in the later stage. The model structure and operating process are shown in Figure 3 and have been implemented on the MATLAB platform. The strategy for implementing the dam-zoning elasticity modulus inversion analysis method is described as follows.Figure 3
Flowchart of the inversion analysis method of dam-zoning elasticity modulus using the ICS-IPSO algorithm.Step 1.
Set the parameters of the ICS-IPSO optimization algorithm. Namely,ε, Tmax, Emin, Emax, n, Pa,max, Pa,min, wmax, wmin, c1, and c2 are set to 1×10-32, 40, 20, 40, 10, 0.25, 0.005, 0.8, 0.4, 1.5, and 1.5, respectively. The parameter z represents the dimensional space of each nest or particle, which is determined according to the number of material partitions in the actual project. Input the measured displacement increment dataset Δδpwmeasure obtained from the dam body, which will be applied in the objective function to evaluate the performance of each nest or particle.Step 2.
Generate the random initial vector values of the nestsEi0=(ei10,ei20,⋯,eiz0)∈Rz (i=1,2,…,n) in the constraint condition equation (5) and the iteration step T=0. The displacement of typical points in the FEM model will be recomputed using each nest Ei0 as elasticity modulus in the calculation parameters in (3). Evaluate the fitness value Ji0 of each nest in (5), and record the best individual Eg0 with the best fitness value Jg0.Step 3.
Update the iteration numberT=T+1. Each nest EiT=(ei1T,ei2T,⋯,eizT)∈Rz is updated by adopting Lévy flights in (8). The displacement of typical points in the FEM model will be recomputed using each nest EiT as elasticity modulus in the calculation parameters in (3). Evaluate the fitness value JiT of each nest in (5), and determine the best individual EgT with the best fitness value JgT.Step 4.
Equation (16) is used to calculate the dynamic switching parameter pa. Equation (10) is applied to abandon the worst nests and build new nests EiT=(ei1T,ei2T,⋯,eizT)∈Rz. The displacement of typical points in the FEM model will be recomputed using each nest EiT as elasticity modulus in the calculation parameters in (3). Evaluate the fitness value JiT of each nest in (5), and record the best individual EgT with the best fitness value JgT.Step 5.
Equation (19) is utilized to calculate the dynamic inertia weight w and change the velocity and position EiT=(ei1T,ei2T,⋯,eizT)∈Rz of the particles according to (17) and (18), respectively. The displacement of the typical points in the FEM model will be recomputed using each nest EiT as elasticity modulus in the calculation parameters in (3). Evaluate the fitness value JiT of each nest in (5), and record the best individual EgT with the best fitness value JgT.Step 6.
If the convergence criterion in (9) is satisfied, then output the optimal solution EgT. Otherwise, loop to Step 3 for the next iteration.


## 3.1. Cuckoo Search (CS) Standard Algorithm

In this section, the cuckoo search algorithm is briefly reviewed. The CS is one of the latest nature-inspired metaheuristic algorithms, developed in 2009 by Xin-She Yang and Suash Deb [16]. The CS is based on the brood parasitism of some cuckoo species. Cuckoos lay their eggs in the nests of other host birds and remove the eggs of the host birds. The host birds may discover that the eggs are not its own, and either remove the foreign eggs or abandon the nest and build a new nest somewhere [21]. This algorithm is enhanced by using the so-called Lévy flights [22], rather than simple isotropic random walks. The long jumps may increase the search efficiency of the cuckoo search significantly in some cases, especially for multimodal, nonlinear problems. Each nest xi is treated as a solution in a z-dimensional space, and the performance of each nest is evaluated according to a predefined fitness function, which is related to the solved problem.For simplicity in describing the standard cuckoo search, the following three idealized rules were set:(i) Each cuckoo lays one egg at a time and dumps it in a randomly chosen nest.(ii) The best nests with high-quality eggs will be carried over to the next generations.(iii) The number of available host nests is fixed, and the egg laid by a cuckoo is discovered by the host bird with a probabilitypa∈(0,1). In this case, the host bird either gets rid of the eggs or simply abandons the nest and build a completely new nest.Furthermore, the CS algorithm uses a balanced combination of a local random walk and the global explorative random walk, controlled by a switching parameterpa [16, 21]. The local random walk can be written as(10)xiT+1=xiT+αs⊗Hpa-ε⊗xjT-xkTwhere the superscriptT denotes the Tth iteration. α,ε∈(0,1) are random numbers drawn from a uniform distribution, s is the step size, and H(pa-ε) is a Heaviside function. xjT and xkT are two different solutions selected randomly by random permutation in the Tth iteration, and the product ⊗ represents entry-wise multiplication.On the other hand, the global random walk is carried out by adopting Lévy flights.(11)xiT+1=xiT+αLs,λwhere (12)Ls,λ=λΓλsinπλ/2π1s1+λ,s⟶+∞Here,α>0 denotes the step size scaling factor, which should be related to the scale of the problem of interest. In most cases, we can use α=O(L/10), where L is the characteristic scale of the problem of interest, while in some cases, α=O(L/10) can be more effective to avoid flying too far. Obviously, the α value in these two updating equations can be different, thus leading to two different parameters α1 and α2. Here, we use α1=α2=α for simplicity. L(s,λ) denotes the step-lengths that are distributed according to the probability distribution shown in (12), which has an infinite variance with an infinite mean.However, generating a pseudorandom step sizes using Lévy flights is not trivial. The most effective and direct method is to adopt the Mantegna algorithm to achieve symmetric Lévy flights, and the step size can be calculated in (13). The two variables U and V are drawn from Gaussian distributions that are depicted in (14) and (15) [21, 22].(13)s=UV1/λ,1<λ≤3where(14)U~N0,σ2,V~N0,1(15)σ2=Γ1+λλΓ1+λ/2·sin⁡πλ/22λ-1/21/λ


## 3.2. Improved Cuckoo Search (ICS) Algorithm

The parameterpa introduced in the CS algorithm helps the local random walk to update solutions, which can influence the convergence rate and the precision of the solutions [23]. In general, the CS standard algorithm uses a fixed value for pa, and the value is set in the initialization step and cannot be changed during new generations, which may result in premature convergence.To keep the balance between global and local searches and increase the population evolution intensity, a variable parameterPa is introduced to adjust the value of the switching parameter. In addition, the value of Pa should be decreased with the search progress to more easily produce new individuals at the later iteration stage to avoid falling into local optima, which can be presented as follows:(16)PaT=Pa,max-Pa,max-Pa.minTmax×Twhere Tmax denotes the maximum iterative steps, T denotes the current iteration step, and Pa,max and Pa,min denote the control probability of the switching parameter Pa.


## 3.3. Improved Particle Swarm Optimization (IPSO) Algorithm

To further enhance the global and local exploration capability and improve the convergence rate, as well as avoid the premature convergence of the algorithm, the IPSO algorithm is integrated into the ICS algorithm. Particle swarm optimization (PSO) is a parallel evolutional computation technique developed by Kennedy and Eberhart based on the simulation of the social behavior of birds in a flock [24, 25]. The new velocity vi of particle i is determined by adjusting its previous velocity based on the distance of its current position from its best position in (17). Then, the position xi of particle i is updated according to its companion and own flying experience, which is treated as a point in z-dimensional space according to the material zoning of the practical project in (18). pi denotes the best previous position that particle i has ever visited, and the value of pg denotes the best particle among all the particles in the population. The performance of each particle is evaluated according to a predefined fitness function, which is related to the solved problem.(17)viT+1=wviT+c1r1piT-xiT+c2r2pgT-xiT(18)xiT+1=xiT+αviT+1where the superscript T denotes the Tth iteration. w denotes the inertia weight, which is employed to control the impact of the previous velocities on the current velocity. c1 and c2 are positive constants, called the cognitive and social parameters, respectively. r1 and r2 are random numbers uniformly distributed in the range (0,1). α represents a scaling factor that controls the velocity of the particles.The appropriate inertia weightw can influence the trade-off between the global and local exploration abilities of the “flying points”. A larger inertia weight can facilitate the global exploration ability, while a smaller inertia weight can effectively improve the ability of local exploration, thereby improving the search precision of the algorithm and finding the optimal solution more quickly and accurately. Particularly, in this research, the inertia weight w is dynamically changed along with the algorithm optimization process to provide a balance between the global and local exploration abilities and increase the convergence rate [25]. The inertia weight is expressed in (19). The advantages of the function are as follows. (1) The function is monotonically decreasing in the region, which meets the requirement of a large inertia weight at the beginning and a small inertia weight at the end. (2) The decreasing speed of this function is relatively slow at the beginning, which is conducive to exploring the optimal solution and avoid getting stuck in local optima. (3) The function maintains a relatively flat decreasing speed at the end, which is conducive to facilitating the local search and improving the search precision [26]. (19)wT=wmax-wmin2×cos⁡π×TTmax+wmax+wmin2where wmax and wmin are the maximum and minimum inertia weight w, respectively, Tmax is the maximum number of iteration steps, and T denotes the current iteration step.


## 3.4. Steps for Implementing the Inversion Analysis Method

According to the above descriptions of the ICS algorithm and IPSO algorithm, the execution process of the dam-zoning elasticity modulus inversion analysis method using the ICS-IPSO optimization algorithm is mainly composed of three parts. First, the global random walk is carried out by adopting Lévy flights, instead of simple isotropic random walks. Then, the switching parameterPa is dynamically changed with the CS algorithm optimization progress to update the local position of the nest. Third, a monotonically decreasing inertia weight w is applied in the PSO algorithm to facilitate global exploration in the early stage, while a smaller inertia weight enhances the local exploration ability in the later stage. The model structure and operating process are shown in Figure 3 and have been implemented on the MATLAB platform. The strategy for implementing the dam-zoning elasticity modulus inversion analysis method is described as follows.Figure 3
Flowchart of the inversion analysis method of dam-zoning elasticity modulus using the ICS-IPSO algorithm.Step 1.
Set the parameters of the ICS-IPSO optimization algorithm. Namely,ε, Tmax, Emin, Emax, n, Pa,max, Pa,min, wmax, wmin, c1, and c2 are set to 1×10-32, 40, 20, 40, 10, 0.25, 0.005, 0.8, 0.4, 1.5, and 1.5, respectively. The parameter z represents the dimensional space of each nest or particle, which is determined according to the number of material partitions in the actual project. Input the measured displacement increment dataset Δδpwmeasure obtained from the dam body, which will be applied in the objective function to evaluate the performance of each nest or particle.Step 2.
Generate the random initial vector values of the nestsEi0=(ei10,ei20,⋯,eiz0)∈Rz (i=1,2,…,n) in the constraint condition equation (5) and the iteration step T=0. The displacement of typical points in the FEM model will be recomputed using each nest Ei0 as elasticity modulus in the calculation parameters in (3). Evaluate the fitness value Ji0 of each nest in (5), and record the best individual Eg0 with the best fitness value Jg0.Step 3.
Update the iteration numberT=T+1. Each nest EiT=(ei1T,ei2T,⋯,eizT)∈Rz is updated by adopting Lévy flights in (8). The displacement of typical points in the FEM model will be recomputed using each nest EiT as elasticity modulus in the calculation parameters in (3). Evaluate the fitness value JiT of each nest in (5), and determine the best individual EgT with the best fitness value JgT.Step 4.
Equation (16) is used to calculate the dynamic switching parameter pa. Equation (10) is applied to abandon the worst nests and build new nests EiT=(ei1T,ei2T,⋯,eizT)∈Rz. The displacement of typical points in the FEM model will be recomputed using each nest EiT as elasticity modulus in the calculation parameters in (3). Evaluate the fitness value JiT of each nest in (5), and record the best individual EgT with the best fitness value JgT.Step 5.
Equation (19) is utilized to calculate the dynamic inertia weight w and change the velocity and position EiT=(ei1T,ei2T,⋯,eizT)∈Rz of the particles according to (17) and (18), respectively. The displacement of the typical points in the FEM model will be recomputed using each nest EiT as elasticity modulus in the calculation parameters in (3). Evaluate the fitness value JiT of each nest in (5), and record the best individual EgT with the best fitness value JgT.Step 6.
If the convergence criterion in (9) is satisfied, then output the optimal solution EgT. Otherwise, loop to Step 3 for the next iteration.


## 4. Numerical Examples

In this section, two examples are considered. In the first example, the target displacement of the ideal concrete arch dam prescribed elasticity modulus is carried out to verify the capability of the proposed dam-zoning elasticity modulus inversion analysis method using the ICS-IPSO algorithm. In the second example, to further verify the practicability of the inversion analysis method, a practical project is considered to inverse the elasticity modulus of the dam body and dam foundation under the actual working conditions.


### 4.1. Case 1: Concrete Arch Dam Subjected to Prescribed Elasticity Modulus

In the following example, the feasibility and effectiveness of the ICS-IPSO algorithm in the dam-zoning elasticity modulus inversion analysis are verified, and the strategy is detailed as follows.In this case, the prescribed dam-zoning elasticity modulusEspecify∈R2 for two materials zones and boundary conditions, as well as the water level, that are applied to produce a desired deformation of an ideal concrete arch dam, can be identified based on the proposed inversion modeling. The 3D finite element model is discretized by 22384 elements and 25836 nodes. The concrete arch dam is 305 m high and is subjected to 300 m of hydrostatic pressure. The finite element model and material partition are depicted in Figure 4. The prescribed elasticity modulus Especify=(30.55,24.55) GPa, Poisson’s ratio, and density for each material zones used in this analysis are listed in Table 1. A schematic diagram of the positions of the typical points in the arch dam body is depicted in Figure 5, and the displacement values δreal of selected typical points are shown in Table 2. The deformation of the dam body using the prescribed elasticity modulus is shown in Figure 6(d).Table 1
Real parameters for concrete material and the inversion elasticity modulus.
Zone γ ( K N / m 3 ) μ E s p e c i f y ( G P a ) E i n v e r s e ( G P a ) E s p e c i f y - E i n v e r s e 2  Zone B 2450 0.167 30.55 30.546 0.0064 Zone C 2700 0.220 24.55 24.545 Note:γ denotes water density. μ denotes Poisson’s ratio.Table 2
Displacement of a typical measuring point (mm).
Typical point Point 1 Point 2 Point 3 Point 4 Point 5 Point 6  δ r e a l 60.8940 64.4630 62.9560 55.5300 36.1290 7.8617 δ i n v e r s e 60.9030 64.4730 62.9650 55.5380 36.1350 7.8631Figure 4
3D finite element mesh for ideal concrete arch dam.Figure 5
Schematic diagram of the positions of typical points in the arch dam body.Figure 6
The Y-displacement over the inversion process. (a) The reconstructed deformation at the first iteration. (b) The reconstructed deformation at the 20th iteration. (c) The reconstructed deformation at the 40th iteration. (d) The deformation using the prescribed elasticity modulus.
(a)
Step=1 (b)
Step=20 (c)
Max step=40 (d)
RealAn inversion analysis on the basis of the target configuration is carried out. The reconstructed deformations of the representative iterations are shown in Figures6(a), 6(b), and 6(c). Figure 7 displays the zoning elasticity modulus variation over the inversion process. The moving process of optimizing the elasticity modulus is shown in Figure 8. The convergence of the objective functions and elasticity modulus L2 error norm determined by (20) over the iterations is illustrated in Figures 9 and 10, respectively. The inversion solutions Einverse=(30.546,24.545) GPa are obtained when the iteration steps reach 40, as shown in Table 1. The displacement values δinverse of selected typical points are shown in Table 2. As the results indicate, it is clear that the inversion elasticity modulus Einverse is in good agreement with Especify. Eerror is 0.0064, which implies that the proposed model is reasonable and precise. The objective function value between the real displacement δreal and the inversion displacement δinverse is 5.06×10-29 during iteration 40. Therefore, the results indicate that the dam-zoning elasticity modulus inversion analysis method using the ICS-IPSO algorithm is feasible and effective.(20)Eerror=∑r=1zErinverse-Erspecifywhere z is the number of material partitions. In this case, z=2.Figure 7
The zoning elasticity modulus variation during the inversion process.Figure 8
The moving process of optimizing the elasticity modulus.Figure 9
The objective function versus the number of iterations.Figure 10
Convergence ofL2 error norm in parameter space elasticity modulus versus the number of iterations.


### 4.2. Case 2: Inversion Analysis of Elasticity Modulus for Heightened Concrete Gravity Dams

#### 4.2.1. General Situation

A hydropower station is located at the junction of the Han river and its tributary Danjiangkou river, in Danjiangkou city, Hubei Province, China. It is a heightened concrete gravity dam project constructed in two phases. It is the key project to develop and control the Han river with comprehensive utilization benefits of flood control, water supply, power generation, and shipping. In addition, it is also the source of water for the middle route of the South-North Water Diversion Project.For the first phase, the concrete dam construction began in September 1958 and finished in December 1973. The crest elevation is 162.00 m, the crest length is 1141 m, the foundation surface is 81.90 m, and the dam height is 80.10 m. The width of each dam section is 24 m along the axis of the dam. The normal water level is 157.00 m. The concrete gravity dam is divided into 58 sections. According to the function of the concrete gravity dam, the dam body is divided into the right bank joins section, deep hole spillway section, shallow hole overflow section, powerhouse section, and left bank joins section. For the second phase, to implement cross-regional water diversion plans to solve the problem of water shortages, the concrete dam needed to be further poured to increase the storage capacity. In September 2005, the concrete dam project was constructed again, and the second phase finished in March 2010. The crest elevation is 176.60 m after the heightening, an increase of 14.60 m compared with the initial dam project. The normal reservoir level is 170.00 m above sea level. The total storage capacity of the reservoir is 33.91 billion m3, with its main purpose being water supply.This case focused on the inversion analysis method as applied to the heightened concrete dam project conducted in two phases. Therefore, the 13# dam section will be considered in this case, which is deep hole spillway section and located in the middle of the riverbed. The established model is composed of a total of 135232 elements and 146122 nodes, including 11376 elements and 12975 nodes for the new concrete dam (zone A), and 48328 elements and 53984 nodes for the old concrete dam (zone B); zone C is the dam foundation. The model and boundary conditions are illustrated in Figure11. The direct plumbs are installed in the dam body to measure the dam horizontal displacement, the measuring points PL1-1 and Pl1-2 are arranged in zone A, and the measuring points PL2-1, PL2-2, and PL2-3 are arranged in zone B, as shown in Figure 12.Figure 11
3D finite element mesh for 13# dam section of Danjiangkou project.Figure 12
Arrangement diagram of the measuring points and the direct plumbs in the 13# dam section.


#### 4.2.2. Inversion Analysis of Elasticity Modulus

The statistical model in (1) is used to separate the displacements of the water pressure component, temperature component, and aging component, based on the measured displacement of the corresponding measuring points. The water pressure component will be applied in this case to identify the dam-zoning elasticity modulus. Figure 13 shows the measured upstream water level and radial displacement separation results of measuring point PL2-3 in the old concrete dam (Zone B) from June 1, 2013, to December 31, 2017, during which a total of 72 groups of data were accumulated. Negative values denote the radial horizontal displacement upstream. The displacement separation results of the other measuring points are similar to those above and will not be described here.Figure 13
Time curves of monitored and separated displacements of the PL2-3 measuring point.According to the time series characteristics of the water level and the principle that the period of the sudden rise of the reservoir water level should be selected as the inversion analysis period, the period from August 24, 2017, to October 19, 2017, was selected as the inversion period in this case, as shown in Figure13. The water pressure component under different characteristic water levels of 157.79, 161.76, 164.56, and 166.34 m is input as δpwmeasure in (2). The ICS-IPSO algorithm is introduced to update the dam-zoning elasticity modulus, which is Eiinverse in (3). The displacements δpwinverse in (3) can be calculated under different characteristic water levels of 157.79, 161.76, 164.56, and 166.34 m by adopting the finite element method based on the updated elasticity modulus Eiinverse. In this case, the range values of the elasticity modulus inversion analysis are [20,40] GPa as the constraint condition in (5). The zoning elasticity modulus variation during the inversion process can be seen in Figure 14. The real parameters for the concrete material and the inversion result of the dam-zoning elasticity modulus Einverse=(36.46,30.53,24.17) GPa are shown in Table 3. The measured displacements increment Δδpwmeasure and calculated displacements increment Δδpwinverse by the inversion analysis of the measuring points under different characteristic water levels are displayed in Table 4. To verify the efficiency of the ICS-IPSO algorithm, the CS standard algorithm, PSO standard algorithm, and ICS, ICS-PSO were also used for the inversion analysis. The optimization objective function over the iterations is displayed in Figure 15, and the objective function value of the ICS-IPSO hybrid algorithm is 2.35×10-31 during iteration 40. Hence, the convergence rate of the ICS-IPSO hybrid algorithm is quicker than that of the other algorithms.Table 3
Real parameters for concrete material and the zoning elasticity modulus inversion.
Zone γ ( K N / m 3 ) μ E i n v e r s e ( G P a )  Zone A (new concrete dam) 2450 0.167 36.46 Zone B (old concrete dam) 2450 0.167 30.53 Zone C (rock) 2700 0.220 24.17Table 4
A comparison of the measured displacements and identified displacements obtained by the inversion analysis of typical measuring points at different water levels (mm).
Displacements Δ δ p w m e a s u r e Δ δ p w i n v e r s e  Date 2017/9/21 2017/10/11 2017/10/19 Water level 161.76 164.56 166.34 161.76 164.56 166.34  PL1-1 0.069 0.127 0.170 0.070 0.127 0.167 PL1-2 0.153 0.284 0.376 0.157 0.286 0.377 PL2-1 0.269 0.487 0.633 0.265 0.483 0.636 PL2-2 0.625 1.116 1.457 0.621 1.117 1.458 PL2-3 0.759 1.355 1.766 0.759 1.359 1.770 Note: in this case, the reservoir water level of 157.79 m on August 24, 2017, was used as the initial water levelH0.Figure 14
The zoning elasticity modulus variation during the inversion process.Figure 15
The objective function versus the number of iterations.To verify the correctness of the inversion analysis results, the period from October 15, 2017, to December 27, 2017, was selected as the verification stage, as shown in Figure13. Utilizing the elasticity modulus Einverse in Table 3 as the calculation parameters, the radial horizontal displacements were calculated by the finite element method under different characteristic water levels of 166.74, 166.01, and 165.52 m, as shown in Table 5. A comparison of the measured displacement increment Δδpwmeasure and calculated displacement increment Δδpwinverse can be seen in Table 5. The comparison of the calculated displacements and measured displacements of the measuring points under different characteristic water levels of 166.74, 166.01, and 165.52 m is shown in Figure 16. As seen from Table 5 and Figure 16, the calculated displacements have good correspondence with the measured displacements under the series of water loads. Consequently, the inversion results of the zoning elasticity modulus are rational in this case, and the practicability of the proposed inversion analysis method is verified.Table 5
A comparison of the measured displacements and calculated displacements of typical measuring points at different water levels (mm).
Displacements Δ δ p w m e a s u r e Δ δ p w i n v e r s e  Date 2017/10/25 2017/11/26 2017/12/27 Water level 166.74 166.01 165.52 166.74 166.01 165.52  PL1-1 0.173 0.157 0.145 0.177 0.160 0.148 PL1-2 0.403 0.359 0.336 0.399 0.360 0.334 PL2-1 0.677 0.610 0.566 0.672 0.607 0.564 PL2-2 1.538 1.388 1.300 1.537 1.393 1.298 PL2-3 1.861 1.687 1.578 1.865 1.692 1.578 Note: in this case, the reservoir water level of 157.79 m on August 24, 2017, was used as the initial water levelH0.Figure 16
Comparison between the measured and calculated displacement increments of typical measuring points at different water levels.


## 4.1. Case 1: Concrete Arch Dam Subjected to Prescribed Elasticity Modulus

In the following example, the feasibility and effectiveness of the ICS-IPSO algorithm in the dam-zoning elasticity modulus inversion analysis are verified, and the strategy is detailed as follows.In this case, the prescribed dam-zoning elasticity modulusEspecify∈R2 for two materials zones and boundary conditions, as well as the water level, that are applied to produce a desired deformation of an ideal concrete arch dam, can be identified based on the proposed inversion modeling. The 3D finite element model is discretized by 22384 elements and 25836 nodes. The concrete arch dam is 305 m high and is subjected to 300 m of hydrostatic pressure. The finite element model and material partition are depicted in Figure 4. The prescribed elasticity modulus Especify=(30.55,24.55) GPa, Poisson’s ratio, and density for each material zones used in this analysis are listed in Table 1. A schematic diagram of the positions of the typical points in the arch dam body is depicted in Figure 5, and the displacement values δreal of selected typical points are shown in Table 2. The deformation of the dam body using the prescribed elasticity modulus is shown in Figure 6(d).Table 1
Real parameters for concrete material and the inversion elasticity modulus.
Zone γ ( K N / m 3 ) μ E s p e c i f y ( G P a ) E i n v e r s e ( G P a ) E s p e c i f y - E i n v e r s e 2  Zone B 2450 0.167 30.55 30.546 0.0064 Zone C 2700 0.220 24.55 24.545 Note:γ denotes water density. μ denotes Poisson’s ratio.Table 2
Displacement of a typical measuring point (mm).
Typical point Point 1 Point 2 Point 3 Point 4 Point 5 Point 6  δ r e a l 60.8940 64.4630 62.9560 55.5300 36.1290 7.8617 δ i n v e r s e 60.9030 64.4730 62.9650 55.5380 36.1350 7.8631Figure 4
3D finite element mesh for ideal concrete arch dam.Figure 5
Schematic diagram of the positions of typical points in the arch dam body.Figure 6
The Y-displacement over the inversion process. (a) The reconstructed deformation at the first iteration. (b) The reconstructed deformation at the 20th iteration. (c) The reconstructed deformation at the 40th iteration. (d) The deformation using the prescribed elasticity modulus.
(a)
Step=1 (b)
Step=20 (c)
Max step=40 (d)
RealAn inversion analysis on the basis of the target configuration is carried out. The reconstructed deformations of the representative iterations are shown in Figures6(a), 6(b), and 6(c). Figure 7 displays the zoning elasticity modulus variation over the inversion process. The moving process of optimizing the elasticity modulus is shown in Figure 8. The convergence of the objective functions and elasticity modulus L2 error norm determined by (20) over the iterations is illustrated in Figures 9 and 10, respectively. The inversion solutions Einverse=(30.546,24.545) GPa are obtained when the iteration steps reach 40, as shown in Table 1. The displacement values δinverse of selected typical points are shown in Table 2. As the results indicate, it is clear that the inversion elasticity modulus Einverse is in good agreement with Especify. Eerror is 0.0064, which implies that the proposed model is reasonable and precise. The objective function value between the real displacement δreal and the inversion displacement δinverse is 5.06×10-29 during iteration 40. Therefore, the results indicate that the dam-zoning elasticity modulus inversion analysis method using the ICS-IPSO algorithm is feasible and effective.(20)Eerror=∑r=1zErinverse-Erspecifywhere z is the number of material partitions. In this case, z=2.Figure 7
The zoning elasticity modulus variation during the inversion process.Figure 8
The moving process of optimizing the elasticity modulus.Figure 9
The objective function versus the number of iterations.Figure 10
Convergence ofL2 error norm in parameter space elasticity modulus versus the number of iterations.


## 4.2. Case 2: Inversion Analysis of Elasticity Modulus for Heightened Concrete Gravity Dams

### 4.2.1. General Situation

A hydropower station is located at the junction of the Han river and its tributary Danjiangkou river, in Danjiangkou city, Hubei Province, China. It is a heightened concrete gravity dam project constructed in two phases. It is the key project to develop and control the Han river with comprehensive utilization benefits of flood control, water supply, power generation, and shipping. In addition, it is also the source of water for the middle route of the South-North Water Diversion Project.For the first phase, the concrete dam construction began in September 1958 and finished in December 1973. The crest elevation is 162.00 m, the crest length is 1141 m, the foundation surface is 81.90 m, and the dam height is 80.10 m. The width of each dam section is 24 m along the axis of the dam. The normal water level is 157.00 m. The concrete gravity dam is divided into 58 sections. According to the function of the concrete gravity dam, the dam body is divided into the right bank joins section, deep hole spillway section, shallow hole overflow section, powerhouse section, and left bank joins section. For the second phase, to implement cross-regional water diversion plans to solve the problem of water shortages, the concrete dam needed to be further poured to increase the storage capacity. In September 2005, the concrete dam project was constructed again, and the second phase finished in March 2010. The crest elevation is 176.60 m after the heightening, an increase of 14.60 m compared with the initial dam project. The normal reservoir level is 170.00 m above sea level. The total storage capacity of the reservoir is 33.91 billion m3, with its main purpose being water supply.This case focused on the inversion analysis method as applied to the heightened concrete dam project conducted in two phases. Therefore, the 13# dam section will be considered in this case, which is deep hole spillway section and located in the middle of the riverbed. The established model is composed of a total of 135232 elements and 146122 nodes, including 11376 elements and 12975 nodes for the new concrete dam (zone A), and 48328 elements and 53984 nodes for the old concrete dam (zone B); zone C is the dam foundation. The model and boundary conditions are illustrated in Figure11. The direct plumbs are installed in the dam body to measure the dam horizontal displacement, the measuring points PL1-1 and Pl1-2 are arranged in zone A, and the measuring points PL2-1, PL2-2, and PL2-3 are arranged in zone B, as shown in Figure 12.Figure 11
3D finite element mesh for 13# dam section of Danjiangkou project.Figure 12
Arrangement diagram of the measuring points and the direct plumbs in the 13# dam section.


### 4.2.2. Inversion Analysis of Elasticity Modulus

The statistical model in (1) is used to separate the displacements of the water pressure component, temperature component, and aging component, based on the measured displacement of the corresponding measuring points. The water pressure component will be applied in this case to identify the dam-zoning elasticity modulus. Figure 13 shows the measured upstream water level and radial displacement separation results of measuring point PL2-3 in the old concrete dam (Zone B) from June 1, 2013, to December 31, 2017, during which a total of 72 groups of data were accumulated. Negative values denote the radial horizontal displacement upstream. The displacement separation results of the other measuring points are similar to those above and will not be described here.Figure 13
Time curves of monitored and separated displacements of the PL2-3 measuring point.According to the time series characteristics of the water level and the principle that the period of the sudden rise of the reservoir water level should be selected as the inversion analysis period, the period from August 24, 2017, to October 19, 2017, was selected as the inversion period in this case, as shown in Figure13. The water pressure component under different characteristic water levels of 157.79, 161.76, 164.56, and 166.34 m is input as δpwmeasure in (2). The ICS-IPSO algorithm is introduced to update the dam-zoning elasticity modulus, which is Eiinverse in (3). The displacements δpwinverse in (3) can be calculated under different characteristic water levels of 157.79, 161.76, 164.56, and 166.34 m by adopting the finite element method based on the updated elasticity modulus Eiinverse. In this case, the range values of the elasticity modulus inversion analysis are [20,40] GPa as the constraint condition in (5). The zoning elasticity modulus variation during the inversion process can be seen in Figure 14. The real parameters for the concrete material and the inversion result of the dam-zoning elasticity modulus Einverse=(36.46,30.53,24.17) GPa are shown in Table 3. The measured displacements increment Δδpwmeasure and calculated displacements increment Δδpwinverse by the inversion analysis of the measuring points under different characteristic water levels are displayed in Table 4. To verify the efficiency of the ICS-IPSO algorithm, the CS standard algorithm, PSO standard algorithm, and ICS, ICS-PSO were also used for the inversion analysis. The optimization objective function over the iterations is displayed in Figure 15, and the objective function value of the ICS-IPSO hybrid algorithm is 2.35×10-31 during iteration 40. Hence, the convergence rate of the ICS-IPSO hybrid algorithm is quicker than that of the other algorithms.Table 3
Real parameters for concrete material and the zoning elasticity modulus inversion.
Zone γ ( K N / m 3 ) μ E i n v e r s e ( G P a )  Zone A (new concrete dam) 2450 0.167 36.46 Zone B (old concrete dam) 2450 0.167 30.53 Zone C (rock) 2700 0.220 24.17Table 4
A comparison of the measured displacements and identified displacements obtained by the inversion analysis of typical measuring points at different water levels (mm).
Displacements Δ δ p w m e a s u r e Δ δ p w i n v e r s e  Date 2017/9/21 2017/10/11 2017/10/19 Water level 161.76 164.56 166.34 161.76 164.56 166.34  PL1-1 0.069 0.127 0.170 0.070 0.127 0.167 PL1-2 0.153 0.284 0.376 0.157 0.286 0.377 PL2-1 0.269 0.487 0.633 0.265 0.483 0.636 PL2-2 0.625 1.116 1.457 0.621 1.117 1.458 PL2-3 0.759 1.355 1.766 0.759 1.359 1.770 Note: in this case, the reservoir water level of 157.79 m on August 24, 2017, was used as the initial water levelH0.Figure 14
The zoning elasticity modulus variation during the inversion process.Figure 15
The objective function versus the number of iterations.To verify the correctness of the inversion analysis results, the period from October 15, 2017, to December 27, 2017, was selected as the verification stage, as shown in Figure13. Utilizing the elasticity modulus Einverse in Table 3 as the calculation parameters, the radial horizontal displacements were calculated by the finite element method under different characteristic water levels of 166.74, 166.01, and 165.52 m, as shown in Table 5. A comparison of the measured displacement increment Δδpwmeasure and calculated displacement increment Δδpwinverse can be seen in Table 5. The comparison of the calculated displacements and measured displacements of the measuring points under different characteristic water levels of 166.74, 166.01, and 165.52 m is shown in Figure 16. As seen from Table 5 and Figure 16, the calculated displacements have good correspondence with the measured displacements under the series of water loads. Consequently, the inversion results of the zoning elasticity modulus are rational in this case, and the practicability of the proposed inversion analysis method is verified.Table 5
A comparison of the measured displacements and calculated displacements of typical measuring points at different water levels (mm).
Displacements Δ δ p w m e a s u r e Δ δ p w i n v e r s e  Date 2017/10/25 2017/11/26 2017/12/27 Water level 166.74 166.01 165.52 166.74 166.01 165.52  PL1-1 0.173 0.157 0.145 0.177 0.160 0.148 PL1-2 0.403 0.359 0.336 0.399 0.360 0.334 PL2-1 0.677 0.610 0.566 0.672 0.607 0.564 PL2-2 1.538 1.388 1.300 1.537 1.393 1.298 PL2-3 1.861 1.687 1.578 1.865 1.692 1.578 Note: in this case, the reservoir water level of 157.79 m on August 24, 2017, was used as the initial water levelH0.Figure 16
Comparison between the measured and calculated displacement increments of typical measuring points at different water levels.


## 4.2.1. General Situation

A hydropower station is located at the junction of the Han river and its tributary Danjiangkou river, in Danjiangkou city, Hubei Province, China. It is a heightened concrete gravity dam project constructed in two phases. It is the key project to develop and control the Han river with comprehensive utilization benefits of flood control, water supply, power generation, and shipping. In addition, it is also the source of water for the middle route of the South-North Water Diversion Project.For the first phase, the concrete dam construction began in September 1958 and finished in December 1973. The crest elevation is 162.00 m, the crest length is 1141 m, the foundation surface is 81.90 m, and the dam height is 80.10 m. The width of each dam section is 24 m along the axis of the dam. The normal water level is 157.00 m. The concrete gravity dam is divided into 58 sections. According to the function of the concrete gravity dam, the dam body is divided into the right bank joins section, deep hole spillway section, shallow hole overflow section, powerhouse section, and left bank joins section. For the second phase, to implement cross-regional water diversion plans to solve the problem of water shortages, the concrete dam needed to be further poured to increase the storage capacity. In September 2005, the concrete dam project was constructed again, and the second phase finished in March 2010. The crest elevation is 176.60 m after the heightening, an increase of 14.60 m compared with the initial dam project. The normal reservoir level is 170.00 m above sea level. The total storage capacity of the reservoir is 33.91 billion m3, with its main purpose being water supply.This case focused on the inversion analysis method as applied to the heightened concrete dam project conducted in two phases. Therefore, the 13# dam section will be considered in this case, which is deep hole spillway section and located in the middle of the riverbed. The established model is composed of a total of 135232 elements and 146122 nodes, including 11376 elements and 12975 nodes for the new concrete dam (zone A), and 48328 elements and 53984 nodes for the old concrete dam (zone B); zone C is the dam foundation. The model and boundary conditions are illustrated in Figure11. The direct plumbs are installed in the dam body to measure the dam horizontal displacement, the measuring points PL1-1 and Pl1-2 are arranged in zone A, and the measuring points PL2-1, PL2-2, and PL2-3 are arranged in zone B, as shown in Figure 12.Figure 11
3D finite element mesh for 13# dam section of Danjiangkou project.Figure 12
Arrangement diagram of the measuring points and the direct plumbs in the 13# dam section.


## 4.2.2. Inversion Analysis of Elasticity Modulus

The statistical model in (1) is used to separate the displacements of the water pressure component, temperature component, and aging component, based on the measured displacement of the corresponding measuring points. The water pressure component will be applied in this case to identify the dam-zoning elasticity modulus. Figure 13 shows the measured upstream water level and radial displacement separation results of measuring point PL2-3 in the old concrete dam (Zone B) from June 1, 2013, to December 31, 2017, during which a total of 72 groups of data were accumulated. Negative values denote the radial horizontal displacement upstream. The displacement separation results of the other measuring points are similar to those above and will not be described here.Figure 13
Time curves of monitored and separated displacements of the PL2-3 measuring point.According to the time series characteristics of the water level and the principle that the period of the sudden rise of the reservoir water level should be selected as the inversion analysis period, the period from August 24, 2017, to October 19, 2017, was selected as the inversion period in this case, as shown in Figure13. The water pressure component under different characteristic water levels of 157.79, 161.76, 164.56, and 166.34 m is input as δpwmeasure in (2). The ICS-IPSO algorithm is introduced to update the dam-zoning elasticity modulus, which is Eiinverse in (3). The displacements δpwinverse in (3) can be calculated under different characteristic water levels of 157.79, 161.76, 164.56, and 166.34 m by adopting the finite element method based on the updated elasticity modulus Eiinverse. In this case, the range values of the elasticity modulus inversion analysis are [20,40] GPa as the constraint condition in (5). The zoning elasticity modulus variation during the inversion process can be seen in Figure 14. The real parameters for the concrete material and the inversion result of the dam-zoning elasticity modulus Einverse=(36.46,30.53,24.17) GPa are shown in Table 3. The measured displacements increment Δδpwmeasure and calculated displacements increment Δδpwinverse by the inversion analysis of the measuring points under different characteristic water levels are displayed in Table 4. To verify the efficiency of the ICS-IPSO algorithm, the CS standard algorithm, PSO standard algorithm, and ICS, ICS-PSO were also used for the inversion analysis. The optimization objective function over the iterations is displayed in Figure 15, and the objective function value of the ICS-IPSO hybrid algorithm is 2.35×10-31 during iteration 40. Hence, the convergence rate of the ICS-IPSO hybrid algorithm is quicker than that of the other algorithms.Table 3
Real parameters for concrete material and the zoning elasticity modulus inversion.
Zone γ ( K N / m 3 ) μ E i n v e r s e ( G P a )  Zone A (new concrete dam) 2450 0.167 36.46 Zone B (old concrete dam) 2450 0.167 30.53 Zone C (rock) 2700 0.220 24.17Table 4
A comparison of the measured displacements and identified displacements obtained by the inversion analysis of typical measuring points at different water levels (mm).
Displacements Δ δ p w m e a s u r e Δ δ p w i n v e r s e  Date 2017/9/21 2017/10/11 2017/10/19 Water level 161.76 164.56 166.34 161.76 164.56 166.34  PL1-1 0.069 0.127 0.170 0.070 0.127 0.167 PL1-2 0.153 0.284 0.376 0.157 0.286 0.377 PL2-1 0.269 0.487 0.633 0.265 0.483 0.636 PL2-2 0.625 1.116 1.457 0.621 1.117 1.458 PL2-3 0.759 1.355 1.766 0.759 1.359 1.770 Note: in this case, the reservoir water level of 157.79 m on August 24, 2017, was used as the initial water levelH0.Figure 14
The zoning elasticity modulus variation during the inversion process.Figure 15
The objective function versus the number of iterations.To verify the correctness of the inversion analysis results, the period from October 15, 2017, to December 27, 2017, was selected as the verification stage, as shown in Figure13. Utilizing the elasticity modulus Einverse in Table 3 as the calculation parameters, the radial horizontal displacements were calculated by the finite element method under different characteristic water levels of 166.74, 166.01, and 165.52 m, as shown in Table 5. A comparison of the measured displacement increment Δδpwmeasure and calculated displacement increment Δδpwinverse can be seen in Table 5. The comparison of the calculated displacements and measured displacements of the measuring points under different characteristic water levels of 166.74, 166.01, and 165.52 m is shown in Figure 16. As seen from Table 5 and Figure 16, the calculated displacements have good correspondence with the measured displacements under the series of water loads. Consequently, the inversion results of the zoning elasticity modulus are rational in this case, and the practicability of the proposed inversion analysis method is verified.Table 5
A comparison of the measured displacements and calculated displacements of typical measuring points at different water levels (mm).
Displacements Δ δ p w m e a s u r e Δ δ p w i n v e r s e  Date 2017/10/25 2017/11/26 2017/12/27 Water level 166.74 166.01 165.52 166.74 166.01 165.52  PL1-1 0.173 0.157 0.145 0.177 0.160 0.148 PL1-2 0.403 0.359 0.336 0.399 0.360 0.334 PL2-1 0.677 0.610 0.566 0.672 0.607 0.564 PL2-2 1.538 1.388 1.300 1.537 1.393 1.298 PL2-3 1.861 1.687 1.578 1.865 1.692 1.578 Note: in this case, the reservoir water level of 157.79 m on August 24, 2017, was used as the initial water levelH0.Figure 16
Comparison between the measured and calculated displacement increments of typical measuring points at different water levels.


## 5. Conclusions

This paper presents the inversion analysis strategy to obtain the dam-zoning elasticity modulus in a heightened concrete dam combining statistical analysis of the prototype monitoring data and the ICS-IPSO hybrid optimization algorithm. Furthermore, an objective function is constructed based on the horizontal displacement increment. The conclusions obtained are as follows:(1) The proposed method is used to identify the dam-zoning elasticity modulus for the ideal arch dam model prescribed elasticity modulus in case 1. It has to be stated that the investigated example shows high performance for the inversion modeling. Consequently, it can be concluded that the ICS-IPSO hybrid algorithm has been successfully verified as an effective method for inverse analysis of dam-zoning elasticity modulus.(2) Taking a concrete gravity dam that is heightened in stages as an example in case 2, the inversion modeling on the basis of the ICS-IPSO is successfully adopted to identify the elasticity modulus in new and old concrete zones. In addition, the identified mechanical parameters can be utilized to accurately calculate the future displacement of the dam using the FEM.(3) The CS, PSO, ICS, and ICS-PSO algorithms are also employed to conduct an inversion analysis in case 2. The results show that the ICS-IPSO has the advantages of a fast convergence rate and more stable search results. The inversion modeling proposed in this paper can also be applied to other heightened concrete dam problems. It also has the power to deal with the problems with more zones of the concrete dam because of the global search ability.



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*Source: 9328326-2019-04-09.xml*