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The physical and mechanical parameters of hydraulic structures in complicated operating conditions often change over time. Updating these parameters in a timely manner is important to comprehend the operating behaviors and monitor the safety of hydraulic structures. Conventional inverse analysis methods can only generate inversions on the comprehensive deformation modulus of concrete dam structures, which contradict practical conditions. Based on the researches on conventional reversion methods of the deformation modulus of the dam body, foundation, and reservoir basin, the objective fitness function is established in this paper according to engineering-measured data and finite element simulation results. The quantum genetic algorithm has high global search efficiency and population diversity. A mechanical parameter inversion of high-arch dams is built from the intelligent optimization of an established algorithm by applying the quantum genetic algorithm. The proposed algorithm is tested to be feasible and valid for practical engineering projects and therefore shows scientific and practical application values.

Hydraulic structural designs and the stress-strain and crack formation mechanisms of engineering structures are largely determined by the physical and mechanical parameters of the geotechnical engineering medium. Parameter selection can reflect and test the design and construction of hydraulic structures. The physical and mechanical parameters of high-arch dams often change with time. This variation is attributed to complicated and diversified service conditions. Hence, updating these parameters in a timely manner is highly important for the security monitoring of concrete gravity dams. In addition, the designed physical and mechanical parameters differ from the actual situations to some extent because of uncertain design parameters, calculation models, and boundary conditions. The difference can also be attributed to the calculation loads and working conditions, construction quality, and geological conditions. To comprehend the actual deformation behaviors of high-arch dams, the deformation of the physical and mechanical parameters under the actual state should be determined through parameter inversion based on measured data [

Based on the results of a positive analysis, some in situ monitoring information is used to trace back the physical and mechanical parameters of the dam. This process is called the inversion analysis of physical and mechanical parameters of the dam. Chen [

The inverse analysis is typically a complicated nonlinear function optimization problem. The use of a global optimization algorithm with rich theoretical experience is the approach to ideal intelligent optimization. Friswell et al. [

As an emerging computing model, quantum computing is a combination of quantum theory, information theory, and computer science. Based on the quantum state characteristic, quantum computing uses the qubit probability coding to represent the chromosomes. Quantum computing can achieve the objective of the objective function for optimization through the role of the quantum revolving door of constant update to update and optimize the population, so it has high efficiency in global searching and good species diversity. Genetic algorithm is a heuristic search algorithm that simulates the evolution mechanism of species in nature. However, since the traditional genetic algorithm involves too much computation to solve some problems, it is difficult to find their optimal solutions correspondingly [

In this paper, the intelligent optimization of the ideal objective function is realized using the quantum genetic algorithm (QGA), and the portioning QGA inversion model based on the finite element simulation results is constructed using the high-efficiency core solver, Abaqus. The fitness function is established based on the finite element simulation and measured results. The mechanical parameters of the high-arch dam are inverted using MATLAB programming. The proposed algorithm is verified feasible and valid through the inversion of the comprehensive concrete deformation modulus at different positions on the high-arch dam. The research results have certain scientific and practical application values.

The material parameters of a high-arch dam for the inversion analysis include the deformation modulus of the dam body (

In the linear deformation range, the overall stiffness matrix in the equilibrium equation of the dam and rock foundation is

If

If

Equation (

Given the inversion model of

Solving

During the inversion analysis of

Then, the objective function is given by

Given that

Therefore, the inversion value

Solving

Supposing the displacement at

Solving

The displacement data at the measuring points of the dam body include the displacements caused by the deformations of the dam body, dam foundation, and reservoir basin. Therefore, the displacements caused by the deformations of the rock foundation and reservoir basin are deducted from the hydraulic pressure (

The following equation is true when

QGA innovatively introduces quantum calculation into the genetic algorithm and overcomes the difficulties of traditional genetic algorithms in the selection of the evolution mode. The approach satisfies the accuracy requirements of the fitness function through a more efficient and accurate intelligent population optimization.

The basic information unit in quantum calculation is the quantum state. The quantum state has three major characteristics in quantum calculation: quantum superposition, quantum entanglement, and quantum interference. Quantum is characterized by the state vector, and encoding genetic chromosome is characterized by the probability amplitude of quantum bit such that the superposition of multiple state vectors can be expressed by one piece of chromosome. Chromosome updating is realized by the quantum logic gate. The basic principle of QGA is introduced in the following text [

The QGA uses quantum bit as the physical medium to store unit information. Quantum bit is often formed by the superposition of two quanta. Quantum bit is obtained by

Unlike traditional genetic algorithms, QGA expresses the gene state and stores gene state information by quantum bit encoding (0/1). For example, 1 means that the gene is at state “1,” and 0 means that the gene is at state “0.” If this gene contains uncertain information, this aspect could be expressed by the superposition of the “0” and “1” states. In other words, this gene state contains all possible information.

If an individual is composed of

Quantum gate is the most basic operation of the quantum bit. Specific quantum gate operations are often selected according to specific problems. Multiple operations are generally optional. Quantum revolving gate is selected in this paper. The adjustment operation of the quantum revolving gate is

Quantum gate updating is^{th} quantum of the chromosome and

Coordinates of the revolving quantum gate.

When the quantum bit is at the first and third quadrants,

The basic flowchart of QGA is shown in Figure

Population

Individuals in the initial population are measured singly, and the corresponding determined solutions of individuals are obtained as

The fitness degree of

The optimal individual and its fitness degree are recorded.

The individual is judged whether it satisfies the end condition. If the end condition is satisfied, the step is exited; otherwise, the calculation is continued.

If the individual does not satisfy the end condition, all the individuals in

The fitness degrees of the different determined solutions are assessed.

The individuals are adjusted by the quantum revolving gate

The optimal individual and its fitness degree are recorded.

The iterations are increased by 1, and the process returns to step (5).

Flowchart of QGA.

The commercial finite element software Abaqus involves a high-efficiency core solver and can establish the interface between Abaqus and MATLAB users through secondary development. Consequently, the inversion efficiency is increased. On the basis of the finite element simulation results, we construct an inversion model for QGA. The fitness function is established on the basis of the difference between the measured value and the finite element simulation result. The inversion of the mechanical parameters of hydraulic structural materials is realized by MATLAB programming. The specific steps are as follows:

The calling interface of the Abaqus finite element program is established using MATLAB. The INP document of the Abaqus model and the constitutive Fortran subprogram document of materials are called through system commands. The problem is solved using the Abaqus finite element high-efficiency core.

After the INP document is edited in step (1), the deformation data of the characteristic unit in the Abaqus finite element simulation document is generated using the Elprint command. The Fopen command in MATLAB is called, and the fitness function of the characteristic unit error is calculated.

The QGA subprogram is compiled, and the fitness function of characteristic unit in step (2) is optimized.

The inversion parameters are corrected in the command stream document using the Num2str command in the MATLAB language. Meanwhile, the revolving quantum gate is updated, and the next generation of population is obtained.

Steps (1)–(4) are repeated until the objective function reaches the desired accuracy. The calculation flowchart of the inversion model combining the QGA and finite element simulation is shown in Figure

Flowchart of the inversion model combining QGA and finite element simulation.

The hydropower station located in the Yi Autonomous Prefecture of the Liangshan, Sichuan Province, was used as the case study in this work. It is the “control” reservoir gradient for the hydroelectric development planning at the midstream and downstream of the Yalung River. The station is mainly utilized for electricity generation and holds an outstanding downstream gradient compensation adjustment effect with its annual adjustment capability. The project belongs to the first-type first-class project, and the permanent major hydraulic structure is a first-class construction. The normal pool level, level of dead water, and the capacity below the normal pool level of the reservoir are 1880 m, 1800 m, and 7.76 billion m^{3}, respectively (Figure

Layout chart of the high-arch dam.

The dam is a concrete double-arch dam with a crest elevation of 1885.0 m, and the elevation of the lowest foundation surface is 1580.0 m. The maximum dam height, top width, foundation thickness, and thickness-height ratio are 305.0 m, 16.0 m, 63.0 m, and 0.207, respectively. The project achieved the main channel closure on December 4, 2006, and has accomplished four stages of impounding since November 30, 2012.

The horizontal displacement of the dam body is monitored by normal/inverted vertical lines. A total of 57 measuring points were set at the left mountain; dam sections #1, #5, #9, #11, #13, #16, #19, and #23; and crest elevations (1885, 1829, 1778, 1730, 1664, and 1601 m) of the right mountain.

The three-dimensional finite element simulation model of the high-arch dam in the hydropower station was constructed in accordance with its engineering design and geological data. The model covers 923,737 finite elements and 957,221 nodes. The dam body accounted for 29,840 finite elements and 36,079 nodes (Figure

Three-dimensional finite element model of the dam (upstream view).

The calculation ranges of the model covered 900 m at the left and right banks from the central line of the arc dam; 750 m upstream and downstream from dam-crest origin; 700 m below the foundation surface; the scope from the rock slope above the crest elevation to the natural boundary elevation; the left and right mountains extended by 2.5 times the elevation toward upstream and downstream, respectively; and the crest elevation extended upward by one dam height. The rock mass conditions and mountain shapes at the two banks of the dam after excavation were simulated. The finite element model used hexahedral eight-node isoparametric elements. Considering the landform and geological conditions, some regions adopted pentahedron six-node isoparametric elements.

The three-dimensional finite element model renders the material portioning to the dam body and dam foundation consistent to those of actual conditions. The dam body was divided into A, B, and C regions. A total of 33 vertical-line measuring points were set in the #5, #9, #11, #13, #16, and #19 sections of the high-arch dam. The location of the measuring points and the corresponding material portioning is shown in Figure

Distribution of the vertical-line measuring points and corresponding material partitioning.

An inversion analysis of the comprehensive deformation modulus of this high-arch dam was performed. The water levels on August 12, 2014 (1858.34 m), and August 24, 2014 (1878.22 m), were used as the calculation load of the finite element model and substituted into the finite element model for simulation. The inversion of the deformation modulus caused by the horizontal displacement at 32 measuring points in the three regions of the dam body was accomplished. The deformation inversion ranges were as follows: the deformation modulus range of region A was 30–42 GPa, the deformation modulus range of region B was 30–42 GPa, and the deformation modulus range of region C was 25–37 GPa. The error fitness function was established on the basis of the differences between the measured horizontal displacement and the simulated horizontal displacement at different measuring points. The parameter inversion was implemented by QGA. Control parameters for the parameter inversion on the basis of the QGA were set as follows. The initial population was about 0.05–200 times than that of the inversion parameters. The population size was 6000, and the quantum bit was set to 25. The initial search superposed at equal probability, and the probability amplitude of the quantum bit was

Calculated results from inversion model by two optimization methods.

Method | Regions | Best searching value | Average searching value | Fastest searching time | Average searching time |
---|---|---|---|---|---|

GA | Region A | 39.37 | 37.11 | ||

Region B | 36.19 | 34.35 | 29326.9 | 34185.4 | |

Region C | 32.09 | 30.84 | |||

QGA | Region A | 39.17 | 36.94 | ||

Region B | 34.98 | 34.18 | 29105.7. | 25225.8 | |

Region C | 31.8 | 31.02 |

In order to check the computational efficiency and quality of quantum genetic algorithm (QGA), this paper also selects the traditional genetic algorithm (GA) to carry out inverse analysis on the deformation modulus of each partition of dam body. In order to compare the computation abilities of these two algorithms equally, the two methods are calculated for 50 times, respectively. The calculation results are shown in Table

Since the grouping quantum genetic algorithm uses quantum probability coding to enhance the population diversity, the search average of the grouping quantum genetic algorithm is slightly different from the GA average. However, the grouping quantum genetic algorithm can effectively avoid the GA method’s local minimum weakness.

In terms of the computational efficiency of the optimization algorithm, there is little difference between the two methods in the fastest search time, but the average searching time of using QGA is shorter, which greatly improves the operation efficiency.

To verify the accuracy of the inversion results, the inversion results of the deformation modulus at different regions were used as the calculation parameters. Given the finite element analysis, the calculated and measured radial displacements at the vertical-line measuring points under different hydraulic loads (the water level of 1858.34 m and 1878.22 m) were obtained through finite element analysis (Tables

Comparison of calculated and measured displacement values of the dam sections’ measure points under the water level of 1858.34 m.

Section | Measuring point | Altitude (m) | Measured value (mm) | Calculated value (mm) | |
---|---|---|---|---|---|

GA | QGA | ||||

5^{#} | PL5-4 | 1729 | 2.72 | 2.70 | 2.71 |

PL5-3 | 1778 | 4.86 | 4.79 | 4.82 | |

PL5-2 | 1829 | 5.29 | 5.21 | 5.23 | |

PL5-1 | 1885 | 18.46 | 18.04 | 18.14 | |

9^{#} | IP9-1 | 1601 | 2.13 | 2.13 | 2.13 |

PL9-5 | 1665 | 7.60 | 7.46 | 7.49 | |

PL9-4 | 1729 | 17.86 | 17.17 | 17.33 | |

PL9-3 | 1778 | 19.45 | 18.28 | 18.92 | |

PL9-2 | 1829 | 14.64 | 14.09 | 14.22 | |

PL9-1 | 1885 | 6.22 | 5.95 | 6.01 | |

11^{#} | IP11-1 | 1601 | 8.35 | 7.93 | 8.03 |

PL11-5 | 1664 | 23.14 | 22.48 | 22.63 | |

PL11-4 | 1730 | 33.49 | 30.12 | 31.96 | |

PL11-3 | 1778 | 30.88 | 28.83 | 29.95 | |

PL11-2 | 1829 | 22.42 | 21.33 | 21.58 | |

PL11-1 | 1885 | 14.67 | 13.84 | 14.03 | |

13^{#} | IP13-1 | 1601 | 6.83 | 6.67 | 6.71 |

PL13-5 | 1664 | 27.68 | 25.94 | 26.34 | |

PL13-4 | 1730 | 32.42 | 29.69 | 31.18 | |

PL13-3 | 1778 | 27.08 | 25.17 | 26.21 | |

PL13-2 | 1829 | 18.96 | 17.80 | 18.07 | |

PL13-1 | 1885 | 31.08 | 29.64 | 29.97 | |

16^{#} | IP16-1 | 1601 | 11.93 | 11.54 | 11.63 |

PL16-4 | 1664 | 29.61 | 26.88 | 28.37 | |

PL16-3 | 1730 | 24.61 | 22.15 | 23.49 | |

PL16-2 | 1778 | 16.26 | 15.45 | 15.89 | |

PL16-1 | 1829 | 9.28 | 8.94 | 9.02 | |

19^{#} | IP19-1 | 1601 | 3.64 | 3.62 | 3.63 |

PL19-5 | 1664 | 11.88 | 11.41 | 11.52 | |

PL19-4 | 1730 | 24.56 | 23.85 | 24.01 | |

PL19-3 | 1778 | 22.24 | 21.45 | 21.88 | |

PL19-2 | 1829 | 17.60 | 16.70 | 17.19 | |

PL19-1 | 1885 | 9.70 | 9.53 | 9.58 |

Comparison of calculated and measured displacement values of the dam sections’ measure points under the water level of 1878.22 m.

Section | Measuring point | Altitude (m) | Measured value (mm) | Calculated value (mm) | |
---|---|---|---|---|---|

GA | QGA | ||||

5^{#} | PL5-4 | 1729 | 4.39 | 4.31 | 4.33 |

PL5-3 | 1778 | 9.20 | 9.10 | 9.12 | |

PL5-2 | 1829 | 11.38 | 11.29 | 11.31 | |

PL5-1 | 1885 | 30.58 | 29.68 | 29.89 | |

9^{#} | IP9-1 | 1601 | 2.45 | 2.44 | 2.45 |

PL9-5 | 1665 | 9.03 | 9.00 | 9.01 | |

PL9-4 | 1729 | 22.13 | 21.81 | 21.88 | |

PL9-3 | 1778 | 27.25 | 26.08 | 26.72 | |

PL9-2 | 1829 | 27.05 | 25.83 | 26.11 | |

PL9-1 | 1885 | 23.00 | 22.68 | 22.75 | |

11^{#} | IP11-1 | 1601 | 9.02 | 8.85 | 8.89 |

PL11-5 | 1664 | 26.20 | 25.54 | 25.69 | |

PL11-4 | 1730 | 40.79 | 36.72 | 38.94 | |

PL11-3 | 1778 | 40.23 | 36.84 | 38.69 | |

PL11-2 | 1829 | 35.41 | 33.81 | 34.18 | |

PL11-1 | 1885 | 30.97 | 29.31 | 29.69 | |

13^{#} | IP13-1 | 1601 | 7.85 | 7.68 | 7.72 |

PL13-5 | 1664 | 31.36 | 29.74 | 30.11 | |

PL13-4 | 1730 | 39.63 | 36.24 | 38.09 | |

PL13-3 | 1778 | 37.40 | 35.18 | 36.39 | |

PL13-2 | 1829 | 30.88 | 29.96 | 30.17 | |

PL13-1 | 1885 | 31.35 | 30.88 | 30.99 | |

16^{#} | IP16-1 | 1601 | 12.98 | 12.46 | 12.58 |

PL16-4 | 1664 | 34.83 | 31.60 | 33.36 | |

PL16-3 | 1730 | 31.61 | 30.20 | 30.97 | |

PL16-2 | 1778 | 26.10 | 25.37 | 25.77 | |

PL16-1 | 1829 | 20.86 | 19.98 | 20.18 | |

19^{#} | IP19-1 | 1601 | 3.90 | 3.88 | 3.89 |

PL19-5 | 1664 | 12.58 | 12.07 | 12.19 | |

PL19-4 | 1730 | 26.61 | 25.77 | 25.96 | |

PL19-3 | 1778 | 25.79 | 24.07 | 25.01 | |

PL19-2 | 1829 | 22.36 | 21.13 | 21.80 | |

PL19-1 | 1885 | 16.28 | 15.77 | 15.89 |

The comparison of nephograms between the calculated (by QGA) displacement values and the measured values are as follows (Figures

The nephogram comparison between the calculated (QGA) displacement values and the measured values under 1858.34 m.

The nephogram comparison between the calculated (QGA) displacement values and the measured values under 1878.22 m.

Tables

From Figures

The calculated results of the dam body deformation modulus from the FEM based on QGA are closer to the measured dam body horizontal displacement and have higher computational accuracy than the inversion results of the traditional genetic algorithm (GA). The results show that the quantum genetic algorithm (QGA), with better accuracy and computational efficiency, can find the better individuals faster, and the calculation process does not depend on the initial population.

On the basis of the conventional inversion algorithms of the physical and mechanical parameters of dam deformation, this paper focuses on studying the partitioning inversion analysis algorithm of the deformation modulus of a high-arch dam and establishes the objective function for the corresponding partitioning inversion of the deformation modulus of the dam body. The inversion analysis steps are introduced. Several conclusions are drawn:

Given the most fundamental equilibrium equation, the characterization method of the dam displacement field under hydraulic loads is studied. On this basis, the inversion algorithm of the deformation modulus of the dam body, dam foundation, and reservoir basin is discussed using the spatial-temporal safety monitoring model of the spatial displacement field. The inversion formulas of the mean deformation modulus of the dam body, dam foundation, and reservoir basin are deduced.

Multioptimization of the solution space is accomplished through QGA, which increased the chromosome diversity and ensured the algorithm convergence at the global optimal solution. Compared with traditional genetic algorithm, this aspect holds the characteristics of quick convergence and high conversion accuracy.

The fitness function is established on the basis of the difference between the measured and simulated results by combining the QGA and finite element simulation. The comprehensive deformation modulus of the different regions of the high-arch dam is deduced in reverse. The inversion results are used in the three-dimensional finite element calculation of the dam body and are compared with the measured data. The result demonstrates that the inversion results can represent the comprehensive deformation modulus of different dam body regions. This finding verifies the validity and practical application value of the proposed partitioning inversion algorithm.

The monitoring data of the high-arc dam used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Yantao Zhu wrote the original draft and reviewed and edited the manuscript. Xinqiang Niu supervised the study. Chongshi Gu administrated the project and formulated methodology. Jimin Wang obtained resources. Erfeng Zhao performed data curation. Lixian Huang did formal analysis. All authors have read and agreed the published version of the manuscript.

This work was supported by the National Key R&D Program of China (Grant no. 2018YFC0407104), the National Natural Science Foundation of China (Grant nos. 51739003, 51909173, 52079046, 51979093, and 51779086), the Fund of National Dam Safety Research Center (CX2018Z38), Free exploration project of Hohai University (no. B200201058), Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (YS11001), Basic Research Project Funded by the National Key Laboratory (nos. 20195025912 and 520003812), Open Foundation of Changjiang Survey, Planning, Design and Research Co., Ltd. (no. CX2019K01).