A dam ant colony optimization (DACO) analysis of the overall stability of high arch dams on complicated foundations is presented in this paper. A modified ant colony optimization (ACO) model is proposed for obtaining dam concrete and rock mechanical parameters. A typical dam parameter feedback problem is proposed for nonlinear backanalysis numerical model based on field monitoring deformation and ACO. The basic principle of the proposed model is the establishment of the objective function of optimizing real concrete and rock mechanical parameter. The feedback analysis is then implemented with a modified ant colony algorithm. The algorithm performance is satisfactory, and the accuracy is verified. The
Scale models [
Parameters such as elastic modulus, unit weight, Poisson’s ratio, friction coefficient, and cohesion are parameters in structural analysis intrinsic to the determination of stress distributions and displacements, especially when the design of the structure is based on elasticity considerations. In a damfoundation system, these parameters, for mass concrete, are hard to determine directly from tests due to the necessity for large specimens and large testing machines. The parameters for rock are also hard to determine because of the complicated nature of most geological situations. Currently, the inversed parameter is focused on Young's modulus of concrete and rock material, which may limit the variety of materials in the inversion Young's modulus. If we can find an algorithm with capability of inversion Young's modulus of more material, and even a variety of mechanical parameters of various materials, this will be a powerful tool for determining the mechanical parameters of damfoundation systems. Through inverse analysis, the exact parameter values can be determined, and a precise evaluation of dam cracking mechanism, the overall stability of dams, and underground excavations can be made [
In recent years, inverse analysis has been mainly based on the two approaches of neural networks [
The paper is organized as follows. Firstly, the modified ACO algorithm for inverse analysis is proposed. Secondly, based on actual operational conditions and the monitored Lijiaxia arch dam deformation data collected over decades [
Structural health monitoring of large concrete arch dams is based on the acquisition of displacement measurements. These displacements are interpreted to identify significant deviations from what could be considered as the normal response based on statistical or deterministic models of dam behavior.
The finite element analysis method is adopted for solving the damfoundation system. The analytical model in finite element formulation is
Actual deformations of arch dams can be obtained through monitoring. Given a group of concreterock mechanical parameters, displacements can be computed by (
This study is similar to the application of ACO in the case of travelling salesman problem (TSP) [
There are
An artificial ant is an agent which moves between parameter points. It chooses the next point by using a probabilistic function determined by both the pheromone value,
Without
Additionally,
The characteristics of high arch dams and dam foundations lead to 3D complex mesh model, highly nonhomogeneous material distributions, and very high loadings. All of these have detrimental impacts on the convergence of elasticplastic analyses. The convergence of FEM is a principal characteristic of a stable geotechnical structure, for example, the specific and widely used strength reduction method. In this study, the backanalysis adopted a nonlinear constitutive model based on the DruckerPrager (DP) criterion [
In this study, robustness of iteration and an integration policy based on the DP criterion increased the stability of calculation. This method improved computational convergence and ensured that computation converged to the correct solution. The yielding condition for the ideal elasticplastic model adopted the DP criterion:
In this study
Concrete and rock masses are materials low in tensile strength. Conditions of tension are
The literature [
To verify the accuracy of the method, a group of parameters are chosen to obtain computed displacements employing the FEM code [
In this modified ACO model, let artificial ants search such that each ant will have a tour which contain a group of parameters. An ant cycle has two halfmotions. Particularly, in first halfmotion, the artificial ants start at random points between
The key concept and parameters of the ACO model corresponding with the arch dam.
Concept  Define corresponding with the arch dam 

Parameter discrete  Each group of concrete or rock mechanical parameters corresponds to a group of displacements and these displacements are defined as computed data. As the value of each parameter is continuous, some discrete points are chosen within the range. 


Ant  An ant is an agent which moves between parameter discrete points. Each ant can choose a group of parameters during each search. 



Objective function is expressed as the sum of the squares of the errors between the computed displacements and the field monitored displacements. 


Pheromone update  The object of pheromone updating is to study the influence of material parameters on the modified ACO algorithm. 


Inversion parameter  To find a group of parameters that contains every parameter and minimizes the value of 


Edge 
Relationship of the discrete points between two adjacent parameters, for example, each of 50 discrete points of two adjacent parameters, there are 2,500 edges between parameters. 


One cycle  Including two searches: “moveahead” and “moveback,” and output is two times parameters 


One tour 
A group of parameters, the tour is optimized depending on 
Flow chart of DACO inverse analysis for deformation of large arch dam.
Lijiaxia hydropower station is located on the Yellow River, border between the villages of Jianzha and Hualong in Qinghai province, about 100 km southeast of Xining (Figure
Location map and snapshot of the Lijiaxia arch dam: (a) schematic map of the location; (b) a snapshot of Lijiaxia arch dam.
The dam lies across the middle of the Lijiaxia Valley, about 5 km long, a deep, narrow, and Vshaped gorge. The valley is also symmetrical with a slope angle for two side abutments of 45~50°. The dam foundation is very complicated, and the bedrock consists of simian black mica schist and chlorite schist in continuous bands interlocked with granite rock. The joints are comparatively well developed. The metamorphic rock at the damheel is influenced by multistage tectonic activity, and faults and joints are well developed; see Figure
Faults distribution in dam zone of Lijiaxia arch dam [
On the basis of the actual Lijiaxia project sites (Figure
The elastic modulus inversion results using various feedback methods (unit: GPa).
Number  Material  Design value  DPIPACO  Generalized least squares  Neural networks 

1  Dam concrete  20  27.2  30  28 
2  A2  20  23.3  28  26 (A) 
3  A4  12  11.2  17  
4  B2  8  10  11.6  22 (B) 
5  B4  12  16.4  18  
6  C3  5  7  7  13 (C) 
7  A1  15  16  22.5  26 (A) 
8  F20  0.62  —  —  — 
9  F201  0.31  —  —  — 
10  F26  0.52  —  —  — 
11  F27  0.57  —  —  — 
12  F32  0.243  —  —  — 
13  F50  0.21  —  —  — 
14  D  2.75  3.4  3.85  — 
15  f20  0.94  —  —  — 
16  f33  1.05  —  —  — 
17  f35  0.8  —  —  — 
18  Gravity pier  20  29.6  30  — 
19  Foundation reinforcement zone  10  27.2  15  — 
3D FEM mesh model of Lijiaxia arch dam: (a) 3D overall mesh model; (b) main faults distribution.
During analysis, the node displacements of the overall model are applied as the boundary conditions. The upstream/downstream surfaces of foundation are employed displacement constraints along the river direction (
The numerical analysis especially takes reinforcement parameters into account. Analysis and evaluation mainly concerns dam displacements, stresses, safety failure locations in the dam model and foundation, stability evaluation of the abutments, and the riverbed interface. In this study, the dam selfweight, water and silt loadings, and temperature loadings were taken into account in the analyses. The upstream water level is EL 2178 m and the downstream level is EL 2050 m. The main analysis cases are listed below.
Analysis case 1: dam selfweight + normal water load.
Analysis case 2: dam selfweight + normal water load + silt load + temperature dropped loading (the temperature loading determined by the average March temperature).
Analysis case 3: dam selfweight + normal water load + silt load + temperature increased load (the temperature loading determined by the average September temperature).
In order to prove the ant colony optimization analysis is effective in relation to the monitored feedback as far as concrete and rock mechanical parameters are concerned, a comparison analysis was carried out employing various material parameters obtained from design values, generalized least squares, and neural networks methods.
According to the proposed inverse analysis model in Section
This is the updating stage of the pheromone value,
Based on discussion of Section
Field monitored displacement values and numerical results for solving the objective function
Monitoring point  EL (m)  Dam monolith  Displacement of case 3 (mm)  Displacement of case 2 (mm)  

Monitoring  Numerical simulation  Monitoring  Numerical simulation  
2  2150  Number 6  19.52  17.75  24.33  25.1 
3  2114  Number 6  15.64  15.05  17.71  18.6 
4  2087  Number 6  7.29  10.75  8.45  12.2 
5  2150  Number 11  25.53  23.7  33.42  38 
6  2114  Number 11  26.84  24  29.74  32 
7  2087  Number 11  22.22  20  22.87  25 
8  2059  Number 11  13.65  14  15.30  16 
9  2035  Number 11  8.18  7.8  8.74  8.8 
10  2150  Number 16  12.91  19.1  14.05  25.8 
11  2114  Number 16  12.20  14.2  15.32  17.1 
12  2087  Number 16  8.08  8.3  9.05  9.3 
In this study, the elastic modulus,
The feedback results show (1) an elastic modulus of the dam concrete which is about 36% greater; (2) an elastic modulus of abutment and riverbed rock 4~85% which were not included feedback in the comparisons. Based on feedback results shown in Table
Based on the material parameters obtained from the various methods (original design value, generalized least squares), comparative analyses were conducted on dam displacements, stress characteristics, and overall stability of damfoundation system.
Table
Dam displacements in the direction along/perpendicular the river (analysis case 2, unit: mm).
Analysis case  EL (m)  Design value  Generalized least squares  DACO  

Right arch side  Arch crown  Left arch side  Right arch side  Arch crown  Left arch side  Right arch side  Arch crown  Left arch side  

















 
Case 2  2185  1.5  0.7  48.8  1.3  3.4  1.4  0.9  −0.4  35.0  0.9  2.2  0.9  0.9  −0.7  39.6  0.7  2.6  1.7 
2148  4.5  −2.6  48.6  1.2  5.3  2.4  3.0  −1.7  34.1  0.8  3.6  1.7  2.6  −1.8  37.9  0.6  3.9  2.5  
2119  7.3  −3.3  42.8  1.0  8.0  3.0  5.0  −2.2  30.1  0.7  5.6  2.2  4.5  −2.3  33.2  0.5  5.9  2.8  
2100  8.7  −3.5  37.3  0.7  9.3  3.2  6.1  −2.4  26.2  0.5  6.6  2.4  5.6  −2.6  28.7  0.3  7.0  3.0  
2075  10.4  −3.3  27.7  0.5  8.5  2.6  7.3  −2.3  19.4  0.4  6.1  2.0  7.4  −3.0  21.0  0.2  6.8  2.9  
2050  9.4  −2.0  17.9  0.3  8.4  2.5  6.7  −1.4  12.6  0.2  6.0  1.9  6.6  −2.0  13.4  0.1  6.5  2.6  
2035  7.7  −0.9  11.4  0.2  7.3  1.8  5.6  −0.6  8.2  0.2  5.3  1.4  5.5  −1.1  8.6  0.1  5.6  1.8  
2030  6.9  −0.4  9.4  0.2  6.8  1.5  5.0  −0.3  6.8  0.2  4.9  1.1  5.2  −0.9  7.2  0.1  5.3  1.4  
Max. value 


















The difference of downstream displacement in the direction perpendicular, along river obtained from various feedback analysis methods (analysis case 2, unit: mm).
Analysis case  EL (m)  Between generalized least squares and design  Between DACO and design  

Right arch side  Arch crown  Left arch side  Right arch side  Arch crown  Left arch side  













Case 2  2185  0.6  1.1  13.8  0.4  1.2  0.5  0.6  1.4  9.2  0.6  0.8  −0.3 
2148  1.5  −0.9  14.5  0.4  1.7  0.7  1.9  −0.8  10.7  0.6  1.4  −0.1  
2119  2.3  −1.1  12.7  0.3  2.4  0.8  2.8  −1  9.6  0.5  2.1  0.2  
2100  2.6  −1.1  11.1  0.2  2.7  0.8  3.1  −0.9  8.6  0.4  2.3  0.2  
2075  3.1  −1  8.3  0.1  2.4  0.6  3  −0.3  6.7  0.3  1.7  −0.3  
2050  2.7  −0.6  5.3  0.1  2.4  0.6  2.8  0  4.5  0.2  1.9  −0.1  
2035  2.1  −0.3  3.2  0  2  0.4  2.2  0.2  2.8  0.1  1.7  0  
2030  1.9  −0.1  2.6  0  1.9  0.4  1.7  0.5  2.2  0.1  1.5  0.1  
Max. value 












Under the same analysis case, at the upper EL 2050 m level, in the direction perpendicular to the river, the difference in displacements between design material parameters and optimized material parameters (DACO) is greater on the right side than on the left abutment. Below EL 2050 m, the difference in displacement of right side is greater than that of a shift toward the left side. The results are consistent with field monitoring results. The displacements monitoring of each dam monolith showed a shift to the right bank before 1998, while in 1998, and especially after 2000, except for the foundation displacement which continued to shift toward the right bank, the displacement of the upper part of the arch crown shifted towards the left bank. The shifted displacements were 1.53 mm at EL 2150 m and 2.45 mm at EL 2185 m.
Figure
Numerical and field survey results of dam displacement in the direction along river.
Crown cantilever
Right 1/4span
Left 1/4span
Contrasting curves of temperature rise and fall at different elevation levels for each dam monolith are shown in Figure
Field monitoring results and feedback analysis results fit particularly well in the arch crown beam and right hand arch (approximately 1/4 arch, number 6 dam monolith). Monitored results and feedback analysis for the temperature drop condition (case 2) are basically the same.
Errors exist between field monitoring results and feedback evaluation at EL 2150 m in the left hand arch (approximately 1/4 arch results, number 16 dam monolith).
Generally after ten years dam prototype observations, monitoring and reinforcement numerical analysis of the dam downstream are basically the same.
For analysis case
Characteristic value of dam stresses under analysis case 2.
Position  Stress type  Design (MPa)  Generalized least squares (MPa)  DACO (MPa) 

Upstream surface  Maximum tensile stress  1.24 (left side)  1.35 (left dam abutment)  2.56 (left side) 
Maximum compressive stress  −3.6 (EL 2100 m)  −3.67 (EL 2100 m)  −3.33 (EL 2093 m)  


Downstream surface  Maximum tensile stress  0.61 (EL 2150 m)  0.69 (EL2141 m)  0.7 (EL 2151 m) 
Maximum compressive stress  −7.52 (EL 2060 m)  −7.73 (EL 2060 m)  −8.12 (EL 2060 m)  


Interface  Maximum tensile stress  0.61 (dam heel)  0.38 (dam heal)  — 
Maximum compressive stress  −1.87 (EL 2030 m)  −1.88 (EL 2030 m)  — 
Principal stress vector contour of upstream/downstream surface under analysis case 2: (a) upstream surface; (b) downstream surface.
The comparison analysis is as follows.
Under analysis case 2, the dam downstream surface is predominantly in a compressive stress state adopting the optimal parameters evaluation. The maximum compressive stress when using DACO parameters is −7.22 MPa, greater than the cases of design parameters and generalized least squares.
Analyzing the two cases of temperature drop and temperature rise, there is a tensile zone in the same direction as the beam at the dam downstream surface. The stress level is below 1 MPa in both cases.
The dam is in a stable stress state. Comparing the results obtained with different numerical material parameters, the characteristic of dam stress is in agreement with that obtained by DACO.
The comparison results show that the improved ant colony algorithm (DACO) can effectively determine the material mechanical parameters and reflects well the actual dam deformation and dam stress distribution. The algorithm accuracy satisfies project safety evaluation requirements.
By employing the material parameter obtained from DACO and generalized least squares, the overall dam safety factors are shown in Table
Under analysis case 2, dam foundation safety factors are basically symmetric. After reinforcement, point safety factor at the elevation EL 2150 m at the left bank is a little lower than that at the right. No overall yielding appears.
The riverbed always has the lowest safety factors, and cracks first occur at the damfoundation interface. Once yielding zones and cracks occur, unbalanced thrust forces will transfer to those zones with comparatively higher safety factors at both abutments, where high bearing capacity levels develop. As the heightwidth ratio of Lijiaxia arch dam is high, and there is a large pedestal of thickness 30 m, the safety factor in the valley upstream is at least 2.0 in any case under normal load, and there is no cracking.
During dam overloading process, unbalanced forces in the arch dam will transfer to the two banks, under the water load of 3 times normal loads, the depth of the crack upstream is about 1/4 of the dam thickness, and the point safety factor is about 1.0~2.0. The carrying capacity of the riverbed will reduce and transfers to areas with higher safety factors at both banks. Under a water load of 3 times normal loads, the dam can still work and without any yielding zone at the downstream interface. Under a water load of 5 times, there is yielding at the downstream interface.
Based on Table
Point safety factor of abutments.
EL (m)  Generalized least squares  DACO  

Left abutment  Right abutment  Left abutment  Right abutment  
2185  3.0 (gravity pier), 1.5~2.0 (out of gravity pier)  2  2.0~3.0 (gravity pier); 1.1~1.5 (out of gravity pier)  2.0 


2148  1.5~3, 1.2~1.5 (fault)  2~3  1.5~3.0; 1.0~1.5 (fault)  1.5~2.0 


2100  2~3  2, 1.2~1.5 (fault)  1.5~3.0  1.5~2.0; 1.2~1.5 (fault) 


2050  2~3, 1.2~1.5 (fault)  2  2.0~3.0; 1.1~1.5 (fault)  1.5~2.0 


2030 

2~3  1.5~3.0  1.5~5.0 
Above all, modified ACO showed effectively distributed computing capabilities, strong robustness, and easiness to combine with other algorithms, or FEM numerical code, and can well avoid premature convergence phenomenon. The proposal model can effectively solve for feedback multiple parameters of dam concrete and rock material. Through inverse analysis, the exact parameter values can be determined, and a precise evaluation of dam cracking and deformation mechanism and the overall stability of damfoundation can be made.
Resulting from this study, a modified dam ant colony optimization (DACO) model is proposed for obtaining concrete and rock mechanical parameters of a large arch dam. Based on field monitored deformations and the ant colony optimization technique, a typical dam parameters feedback problem was solved using a nonlinear backanalysis numerical model. The basic DACO principle is introduction of distributed computing, DACO initial value determination, and DACO pheromone update, which is used to establish the objective function. The numerical analysis is implemented through proposed dam ant colony algorithm. The mechanical parameters are determined using this algorithm, and construct solutions combined with nonlinear constitutive relations.
By employing the proposed back analysis model, calculated deformations and a stability evaluation of the Lijiaxia arch dam were compared with the monitoring results taken over 10year supervision period of the dam prototype. The results demonstrated that the proposed model can effectively solve for feedback multiple parameters of dam concrete and rock material. Through inverse analysis, the exact parameter values can be determined, and a precise evaluation of dam cracking and deformation mechanism and the overall stability of damfoundation can be made.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research work was supported by the National Natural Science Foundation of China (Nos. 11272178 and 51339003), National Basic Research Program of China (973 Program) Grants Nos. 2011CB013503 and 2013CB035902, and Tsinghua University Initiative Scientific Research Program. The authors are very grateful to Professor W. Y. Zhou for supporting this study, and to Guest Editors of this special issue and three reviewers for their critical recommendations which helped the authors to improve this paper significantly.