Numerical Analysis of Temperature Deformation Characteristics for Super-High Arch Dams considering Solar Radiation Effects

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Introduction
With wide utilization of hydroenergy resources, super-high arch dam construction has aroused public attention in China [1][2][3].Due to the remarkable economy and security of the arch dam, it has already become the primary type of dam to be adopted in future hydraulic engineering construction.To our knowledge, the dam safety operation is vital in economic proft increase of the project [4,5].Terefore, the actual structural behaviour of the super-high arch dam exists as a traditional issue [6][7][8].Deformation is the most intuitive indicator of the dam performance, which is largely afected by temperature load.However, the efect of solar radiation seems to be commonly ignored when studying the dam temperature [9].In fact, in view of the mutual masking of mountains and seasonal changes in the solar azimuth angle, the temperature distribution of superhigh arch dam is likely to be uneven as a result of variation of solar radiation.Hence, in order to accurately and efciently analyze the dam temperature deformation characteristic, there remains an urgent need for considering solar radiation efects.
In recent years, a growing number of studies have revealed that solar radiation plays a crucial role in dam temperature and operation behaviour.Terefore, scholars have carried out numerical calculation tasks to analyze the solar radiation efects.Léger et al. [10,11] and Daoud et al. [12] fully took solar radiation into account when simulating the periodic temperature feld and stress distribution of dams, but the variation of solar radiation on some exposed surfaces was ignored.Agullo and Aguado [13] proposed a practical numerical model for the dam thermal analysis by considering the solar radiation.However, the calculation accuracy needs to be improved.Sheibany and Ghaemian [14] built a 3D fnite element model to analyze the thermal stresses of the Karaj arch dam.Te results demonstrated that solar radiation afects the thermal loads by raising the temperature of the arch dam surface.Santillan et al. [15] simulated the thermal feld of concrete dams with the comprehensive concern of the solar radiation, the long-wave radiation exchange, the night cooling efects, and the evaporative cooling.Meyer and Mouvet [16] found that solar radiation and up-stream water infuence the thermal behavior of the dam by using the fnite element method.Mirzabozorg et al. [17] investigated the efects of direct and indirect solar radiation on thermal distribution of the thin high arch dams.Castilho et al. [18] calculated the hydration process and temperature evolution of an arch dam and eventually analyzed the solar radiation efects.Soltani et al. [19] studied the efects of solar radiation on ambient temperature and investigated the probabilistic risk of arch dam failure under thermal loading.Although the above studies have investigated the solar radiation efects from diferent perspectives, the shading efects of mountains and the dam itself on the solar radiation, and the infuence of solar radiation on the reservoir water temperature have been less explored.Tis may lead to further problems in precise dam temperature feld numerical simulation.
To resolve this problem, much theoretical research has been developed.Žvanut et al. [20] analyzed the dam temperature by considering the shading efects, convection, and solar radiation.Jin et al. [21] proposed a model to calculate the ambient temperature feld of arch dam.However, the calculation process of the model was complicated due to the ray-tracing method involved.Zhu et al. [22] calculated the dam stress on the basis of the solar radiation shading algorithm and meteorological conditions at the location of high arch dams.Tey concluded that the uneven absorption of solar radiation in diferent regions of the arch dam may lead to the adverse stress of high arch dam.
In summary, the solar radiation efects on super-high arch dam deformation characteristics have rarely been studied directly.Terefore, in this paper, according to the geographical location, topographic features, and climatic conditions of the super-high arch dam, a scientifc and reasonable numerical method is presented to calculate the infuence of solar radiation on dam temperature deformation.With the aid of the ASHRAE (American Society of Heating, Refrigerating, and Air-Conditioning Engineers) clear sky model, ray-tracing method, and precomputation algorithm, the dam temperature and reservoir water temperature afected by the solar radiation are calculated (Section 2).Accordingly, the precomputation algorithm is introduced to enhance the calculation accuracy and efciency.Subsequently, the thermodynamic parameter inversion is performed by the hybrid genetic algorithm to improve the accuracy of the numerical analysis (Section 3).To verify the feasibility and validation of the proposed method, a case study based on the Jinping I super-high arch dam project located in Southwest China was carried out (Section 4).Eventually, some valuable conclusions and forward-looking discussions are drawn in Section 5.

Numerical Calculation Method for
Deformation Temperature of Super-High Arch Dam under the Influence of Solar Radiation  [23].
When the weather is clear, the intensity of solar radiation on the Earth's surface can be expressed as where G ND is the intensity of direct solar radiation perpendicular to the Earth's surface; A is the intensity of solar radiation at zero air mass; B is the extinction coefcient of the atmosphere; β is the solar altitude angle; and C N is the atmospheric cleanliness.Te total solar radiation intensity incident on a nonperpendicular surface can be given by ( Te formula for calculating the total solar radiation incident on a vertical surface is provided as follows [21]: where θ is the angle between the sun's rays and the normal to the inclined plane; C is the ratio of the difuse radiation on a horizontal surface to the direct radiation from vertical incidence; F ws is the angle coefcient between the surface and the sky; ρ g is the refectivity of the surroundings; F wg is the angle coefcient between the surface and the ground; G dV is the sky scattering radiation of the vertical surface; G dH is the sky scattering radiation of the horizontal surface.Te computational formulas of F ws and F wg are generally determined by 2 Structural Control and Health Monitoring where α is the angle of incidence.Te angles of solar altitude β, the angle of incidence α, and the angle between the sun's rays and the normal to the inclined plane θ are depicted in Figure 1, respectively.

Numerical Calculation Method for Solar Radiation
Received by Super-High Arch Dam Based on Ray-Tracing Algorithm.Owing to the shading efects of towering mountains on solar radiation, the ray-tracing algorithm [24] is introduced to calculate the solar radiation received by the dam body.Te ray-tracing algorithm is derived from the principles of geometric optics, and it determines refection, refraction, and shadow by simulating the propagation path of light rays.First, a ray is generated from the dam surface in the direction of sunlight.Ten, it is judged whether the ray intersects with other surfaces.If it intersects, the direct solar radiation is obscured.Otherwise, the solar radiation can be received directly.Considering that the ray-tracing algorithm requires real-time calculation of the shading efects, which involves large unavoidable computational costs.To deal with this problem, the cube acceleration algorithm is introduced.Te main steps of solar radiation calculation are introduced as follows [21]: (1) Based on the fnite element model of the super-high arch dam, the edges of dam surface elements which connect with the air, referred to as free edges, are determined.Te free edges are indicated by the red lines in Figure 2. (2) In view of the topological information and nodal coordinates of the model, relevant parameters of the free edges can be obtained, such as the normal vector n → , length, center point coordinates, surface azimuth, and inclination.
(3) Te fnite element model can be divided into square grids, and each grid needs to be numbered according to coordinates.A linked list is established on the basis of the free edge numbers contained in each square grid.Figure 3 is the schematic diagram of the established free side linked list (a ⟶ b ⟶ c ⟶ d).(4) Te solar altitude angle and solar surface azimuth on each specifc day during specifc time should be calculated.(5) Rays can be generated according to the centre point coordinates of each free edge and the result calculated in step (4).( 6) Te square grid numbers that the rays pass through can be calculated subsequently.As illustrated in Figure 4, the grid intersected by the ray emitted from free edge d can be denoted as (2, 5), (3,5), and (3,6).(7) When the rays intersect with the free edges of the square grids they pass through, it indicates that the solar radiation received by the free edges will be blocked.Otherwise, it will not be blocked.( 8) By following steps ( 5)-( 7), the shading of solar radiation on all free edges can be determined.On this basis, it is possible to determine the solar radiation received by the dam surface.

Numerical Calculation Method of the Solar Radiation
Received by Reservoir Water.Considering the large surface area of the upstream reservoir water is likely to be exposed to prolonged periods of solar radiation, which leads to a substantial variation of the water temperature.Terefore, it becomes vital to investigate the infuence of solar radiation on the reservoir water to modify its temperature.Te reservoir water and the atmosphere can be perceived as a twophase coupled thermal system, engaged in continuous heat exchange.Te heat fux at the surface of the reservoir can be estimated by (5) [25].
where SR stands for the shortwave solar radiation quantity; AR indicates atmospheric longwave radiation quantity; BR represents the longwave radiation quantity emitted by water bodies; L denotes the quantity of heat loss due to evaporation; R is the conductive heat fux.In general, the aforementioned heat exchange process occurs only at the water surface, while the water below the surface only receives solar shortwave radiation.6)-( 9) are the hydrodynamic control equations, and equation ( 10) is the water temperature convection difusion equation.If the solar radiation is separated from the heat source term, equation ( 10) can be written as follows [28]:

Numerical Calculation
where c p represents the specifc heat capacity of water, with the unit of J/(kg• ∘ C).

Numerical Calculation Method for Temperature Field of Super-High Arch Dam under the Infuence of Solar
Radiation.To accurately simulate the dam temperature feld, it is of great importance to take the solar radiation efects and shading efects into account.Because the air temperature and reservoir water temperature also vary with the duration of sunlight and seasons, the transient temperature feld is introduced to characterize the dam temperature changes.
Te fnite element format of the dam transient temperature feld can be represented as where M denotes the heat capacity matrix; K t represents the thermal conductivity matrix; P is the temperature load matrix; ϕ is the array of nodal temperatures; _ ϕ is the array of nodal temperature derivatives with respect to time, _ ϕ � dϕ/dt.

Te heat conduction equation is
where a is the temperature difusivity, a � λ/cρ c ; λ denotes the thermal conductivity, c denotes the specifc heat; ρ c is the density of dam concrete; t is time; T c represents dam temperature; x, y, and z represent the Cartesian coordinates.When t � 0, then Te boundary conditions include the following three types: (1) Te frst type of boundary conditions: the boundary temperature.
(2) Te second type of boundary conditions: the heat fux of the boundary.(3) Te third type of boundary conditions: the convective heat transfer condition on the given boundary.

Structural Control and Health Monitoring
On the basis of the above analysis, the calculation boundary of the dam temperature feld consists of four types (S 1 to S 4 ), as described in Figure 5.
(1) Boundary S 1 : dam surface under the water surface.
In addition, it is assumed that temperature and heat fux are continuous at the interface.Terefore, the diferent material parameters determination is fundamental to the calculation.

Numerical Calculation Method for Super-High Arch Dam Deformation under the Infuence of Solar Radiation.
Te comprehensive infuence of multiple factors such as solar radiation, shading efects, reservoir water temperature, and air temperature contribute a lot to the temperature deformation of the super-high arch dam.Te equilibrium equation for calculating the dam temperature deformation based on the fnite element method can be illustrated as follows: Te equivalent load vector of the variable temperature nodes is commonly expressed as where K is the material stifness matrix; δ T is the temperature deformation vector; E is the dam elastic modulus; υ is the coefcient of linear expansion; μ is Poisson's ratio; N is the shape function of the element; N is the element shape function; T is the element temperature; i denotes the number of the element.On the basis of the dam temperature feld, the temperature deformation at each node can be calculated according to equation (15).

Inversion Method for the Termodynamic Parameters of
Super-High Arch Dam Based on Hybrid Genetic Algorithm (HGA)

Termodynamic Parameters Inversion of Super-High
Arch Dam.Te validity of numerical calculation results depends on the precision of the calculation parameters to a large extent.Based on equations ( 12) and ( 13), the calculation parameters of the super-high arch dam include concrete heat capacity c, density ρ c , thermal conductivity λ, thermal difusivity a, and heat transfer coefcient τ, etc., [29] in the dam temperature feld calculation.Among them, heat capacity c and density ρ c can be directly measured by experiments and do not need to be inverted in the numerical simulation.Te thermal conductivity λ is afected by the concrete compaction, material properties, and aggregate type, so the above parameters usually need to be determined.Moreover, since λ � acρ c , only λ and a need to be inverted in general.Because the heat transfer coefcient τ is infuenced by factors such as the thermal insulating layer of the dam, τ also needs to be inverted.Te thermodynamic parameters' inversion is to determine the parameters based on the measured response of the dam to the temperature in fact.Terefore, the thermodynamic parameter inversion can be regarded as an optimization process.On the basis of the above analyses, the measured data of thermometers can be fully utilized to invert the dam thermodynamic parameters.By comparing the calculation values and measured values of the temperature at the same locations of the dam and minimizing the errors between the two values, the objective function of the thermodynamic parameter inversion can be established as follows: where T c ij is the calculation value of the dam temperature; T m ij is the measured value of the dam temperature; M represents the number of monitoring points; N denotes the number of monitoring times.
Consequently, equation ( 17) converts the parameter inversion problem into a parameter optimization problem.Terefore, HGA as a kind of intelligent optimization algorithm can be employed to invert parameters.

Termodynamic Parameter Inversion Process Based on
HGA. Genetic algorithm (GA) is a computational optimization method inspired by the process of natural selection and evolution.It is based on the principles of genetics and survival of the fttest.However, GA has some limitations, such as slow convergence, premature convergence, and difculty in handling constraints [30].To overcome these shortcomings, HGA is presented by combining the principles of GA with niche elimination operation, simplex search algorithm, and accelerating cyclic operation.Accordingly, the quality and efciency of the parameter inversion can be greatly improved by HGA.
(1) Niche Elimination Operation.A niche elimination operation is introduced after the mutation operation.First, the Hamming distance between any two individuals X i and X j in the new population can be calculated as follows: 6 Structural Control and Health Monitoring where N is the number of parameters included in each individual.
When the Hamming distance is within a predetermined distance (‖X i − X j ‖ < L lim ), the ftness of the individuals is compared and the penalty function F min(X i ,X j ) is applied to individuals with lower ftness to further reduce their ftness.
(2) Simplex Search Algorithm.Te simplex algorithm starts from an initial simplex in the solution space and can stably search for the optimal solution to the problem through operations such as refection, expansion, and contraction.After the evolution operation of the genetic algorithm, the frst NS best individuals in the ofspring population are selected as initial points to form initial simplices and a simplex local search is performed to search for the local optimal solution corresponding to each excellent individual in the current environment.Subsequently, the corresponding individuals in the ofspring population are replaced by the optimal solutions to form a new ofspring population.Consequently, the overall performance of the population and the local search ability of the algorithm can be enhanced.
(3) Accelerating Cyclic Operation.Te maximum possible change range of NA excellent individuals generated in the frst two evolution operations is utilized to regenerate the initial population of the genetic algorithm and re-evolve.
Based on the circulate iteration, the range of the initial population gradually shrinks.Meanwhile, the evolution process is accelerated, the number of iterations is reduced, and the computational cost of the algorithm is reduced.Te thermodynamic parameters inversion process of super-high arch dam based on HGA is as follows: Step 1: Encoding.Te parameters to be inverted (such as thermal conductivity) are genetically encoded, and a counter for the number of evolutionary cycles IT is set to be 1.
Step 2: Population initialization.A set of randomly generated parameter vectors, including psize individuals, is used as the initial population for the algorithm.
Step 3: Objective function calculation.Based on the solution set of the previous generation population, the corresponding objective functions are calculated using the fnite element method (FEM).All individuals are ranked according to the size of their objective function values, and the frst NP individuals are remembered.
Step 4: Fitness evaluation.Te ftness of each individual F i (i � 1, 2, • • • , psize) is calculated based on the above ranking results.
Step 5: Genetic evolution operation.Based on the method described in Section 3.3.2, the parent population is subjected to selection, crossover, mutation, and other operations to produce an ofspring population.
Step 6: Niche elimination operation.Te Hamming distance between any two individuals X i and X j in the new population is calculated based on equation (18).When the Hamming distance is within a predefned distance, the ftness values of the individuals are compared and the penalty function F min(X i ,X j ) is applied to further reduce the ftness of the individuals with lower ftness values.Te individuals are then reranked based on their new ftness values, the new progeny populations are formed by the frst psize individuals, and the frst NP individuals are remembered.Step 8: Evolutionary iteration.Return to Step 3 and iterate twice.
Step 9: Accelerated cycle.Te above steps constitute the evolutionary iteration process, and the maximum possible range of variation for the frst NS individuals in the ofspring population obtained after two iterations is used as the initial range for generating the initial population.Return to Step 2 and accelerate the cycle once.
Step 10: Convergence judgment.Te convergence is judged based on the criterion that the best objective function value in the population is less than EPS, and the maximum number of allowed evolutionary cycles is used as an auxiliary criterion.If the convergence is not achieved, update the counter IT � IT + 1 and return to Step 2. Otherwise, proceed to the following step.
Step 11: Output results.We output the termination iteration number IT and the global optimal value of the parameters to be inverted.
In the above process, the number of individuals in the population psize, crossover probability p c , mutation probability p m , and other parameters also afect the performance of the algorithm and should be selected based on the specifc problem.

Case Study
Te Jinping I super-high arch dam is located in the Yalong River, spanning the counties of Yanyuan and Muli in Sichuan Province, China.Te location of the Jinping I superhigh arch dam in Sichuan province is exhibited in Figure 6.It is a key project in the cascade development of hydropower resources in the middle and lower reaches of the Yalong River.Te dam is a concrete double-curvature arch dam with a maximum height of 305 meters, making it the highest arch dam in service.

Finite Element Model of Jinping I Super-High Arch Dam.
Taking Jinping I super-high arch as an example, a fnite element model of the dam is established.Te model extends 1500 m from the arch beam of the super-high arch dam towards the upstream, downstream, left bank, and right bank and extends 1200 m below the dam foundation.Due to the huge project scale of Jinping I super-high arch dam, the size of the fnite element model of the dam foundation should be large enough to refect the actual scale of the project.Moreover, the calculated dam temperature feld is more stable accompanied by the increasing depth of the foundation after many trials.Te model adopts eight-node quadratic tetrahedral elements, with 95255 elements and 110648 nodes in all.Te dam body is divided into 51083 elements.Te horizontal and vertical dimensions of the dam body element mesh are both 9∼10 m, and the horizontal and vertical dimensions of the foundation element mesh gradually change from 100 m to 10 m from bottom to top.In fact, the number of elements should be controlled to guarantee the efciency and efectiveness of numerical simulation.Terefore, the horizontal and vertical dimensions of the foundation element mesh gradually change from 100 m to 10 m from bottom to top to avoid generating too many elements.Furthermore, after many trials, the calculation accuracy cannot be infuenced by the large element dimension of the unit away from the dam foundation.
Te model accurately depicts the shape of the shape of the actual engineering so that the shading range can be calculated in real time based on the ray-tracing method.Te fnite element model of the dam is shown in Figures 7(a) and 7(b), and a commercial software named ABAQUS was used to perform the fnite element analysis.On account of the smaller area of the downstream water, the efects of the solar radiation on the downstream water temperature could be negligible.Terefore, the focus is mainly on the efects of solar radiation on the upstream water.

Determination of Termodynamic Parameters and Environmental Variables.
To calculate the actual solar radiation on the dam surface and reservoir water, the solar altitude angle and azimuth angle at any time should be calculated based on the latitude and longitude of the dam, and then the solar radiation intensity at any time is calculated based on the ASHRAE clear sky model.Combined with the actual situation of the super-high arch dam, the parameters were taken as follows: (1) Te maximum dam height is 305 m, the dam bottom elevation is 1580 m, the dam crest elevation is 1885 m, the thickness of crown beam top is 16 m, the thickness of crown beam bottom is 63 m, and the thickness of crown beam top is 63 m.Te normal water level is 1880 m, the total storage is 7.76 × 10 9 m 3 , the latitude and longitude of the dam site are 28 °10′49″N and 101 °37′59″E.Te upstream surface normal direction of arch crown beam is 25 °NE.(2) In the ASHRAE clear sky model, the estimated transmittance coefcient C N can be estimated according to the dam actual location as 0.9, the refectance coefcient ρ g is 0.25, and the concrete absorption coefcient ξ is 0.65.(3) Table 1 lists the parameter values for the thermodynamic parameters of dam body and foundation.
As the main focus is on the distribution of the dam temperature feld, a linear elastic constitutive model is used for both the concrete and the foundation, with material parameters as shown in Table 2. Te zoning diagram of the dam body is referred to the construction manual.According to the construction manual, the dam body can be divided into three  3 lists the average monthly air temperature and water temperature for multiple years at the dam site.
Te annual average number of sunny, cloudy, overcast, rainy, and snowy days at the dam site is illustrated in Table 4. Figure 9 sketches the vertical distribution of water temperature at dam section on typical days in spring, summer, autumn, and winter.

Boundary Conditions and Precomputation Information.
In the numerical simulation, fxed constraints are applied at the bottom of the model and normal constraints are applied around it.A water temperature convective transfer boundary is applied to the surface covered by the water body, a temperature convective transfer boundary is utilized to the remaining surface, and the solar radiation boundary is adopted to the unshaded area.Finally, the water temperature also needs to be corrected by taking into account the solar radiation efects.
On the basis of the aforementioned calculation steps, it is necessary to compute the shading efects of the dam and water body under typical cases, as well as the corresponding modifcations of water temperature under diferent seasons, water levels, and weather conditions, in order to be directly used in actual calculations.Figures 10 and 11 demonstrate the shading efects of the mountains at diferent times on the summer solstice and winter solstice, respectively.
From Figures 10 and 11, it can be observed that (1) On the summer solstice at 9:00 am, the right bank of the super-high arch dam is shielded, while the left bank continuously receives direct sunlight, resulting in uneven distribution of solar radiation on the dam surface.On the winter solstice at 9:00 am, the dam surface cannot receive direct sunlight due to the shading efects of the mountains and the solar radiation is mainly scattered and refected in the sky.(2) At noon on the summer solstice and the winter solstice, when the solar altitude angle is high, the solar radiation on the dam surface is evenly distributed.However, only a small part of the left bank (3) At 3:00 pm on both the summer solstice and the winter solstice, the shading efects of the mountains are not signifcant, but the super-high arch dam itself has shading efects on solar radiation.(4) On the summer solstice, the shading efects of the mountains on the water body are small and the upstream water body is exposed to direct sunlight from 9:00 am to 3:00 pm.On the winter solstice, the water body begins to be exposed to direct sunlight in the afternoon and the shading efects of the mountains are more signifcant.
After obtaining the shading information of the upstream water in typical scenarios, the corresponding water temperature correction information can be obtained through numerical simulation, as depicted in Figure 12.
Te impact of solar radiation on the temperature of reservoir water rapidly decreases with increasing depth, and it is generally assumed that it only afects the temperature of the reservoir surface.Because the measured reservoir

Position
Termal conductivity λ   Structural Control and Health Monitoring temperature is usually a daily average, this seems to be a common problem in accurately refecting the actual situation of the reservoir temperature under the infuence of solar radiation at each moment of the day.For this reason, it is necessary to make corrections to the reservoir temperature at each moment based on the measured reservoir temperature.Particularly, the results of the correction of the pendant reservoir temperature at a certain time based on the measured pendant reservoir temperature on the summer and winter solstices illustrated are shown in Figure 13.
Because the reservoir water is monitored mainly by the upstream and downstream thermometers embedded in the concrete surface of the dam body, the measured data of reservoir temperature are relatively few.Terefore, the measured results of reservoir temperature are not accurate enough.Moreover, the solar radiation is weaker in winter   Structural Control and Health Monitoring and the temperature of the reservoir is more stable without fuctuation.Terefore, there was no temperature stratifcation during winter time.
Te impact of solar radiation on the temperature feld of the dam is relatively small compared to the infuence of the ambient temperature.Hence, the basic temperature feld under conventional air and water temperature loads was calculated frstly.Ten, the boundary conditions S 1 ∼S 4 were determined based on measured temperature data.Among them, according to the analysis of the measured data in this area for many years, the basic surface temperature of the foundation S 3 can be taken as 11 °C.Te base temperature feld is the temperature feld when the arch dam is sealed.Te dam temperature feld tends to be stable after fve years of pouring, and the infuence of solar radiation can be added to this basic temperature feld.Te base temperature felds in summer and winter are shown in Figures 14 and 15, respectively.

Numerical Analysis of the Temperature Deformation for
the Super-High Arch Dam.On the basis of the shading information of precomputation, it could be determined that the dam downstream surface is only exposed to small areas on the left bank at noon in winter and the upstream reservoir water is only exposed to the sun for a short time.Calculation results indicate that the efects of solar radiation in winter could be negligible.In summer, the downstream face of the dam on the left bank is exposed to strong solar radiation and the water surface is exposed to the sunlight for a long time.
Figure 16 illustrates the profle of the dam temperature feld after being exposed to the sun on the summer solstice.It can be concluded that solar radiation causes the temperature of the dam surface to rise by 2∼3 °C and locally up to 4∼5 °C. Figure 17 depicts the dam temperature feld considering the efects of solar radiation on the summer solstice.It can be observed that the downstream face of the dam on the left bank increases in temperature signifcantly under the infuence of solar radiation.
After the sun sets, the dam heat releases, which causes a decrease in the surface temperature of the dam.However, during the hot summer months, the heat absorbed by the dam surface cannot be fully released in the evening and continuous sunny days make the dam surface temperature accumulate.Figure 17 depicts the temperature variation of 12 Structural Control and Health Monitoring the dam downstream left bank on fve continuous sunny days in summer.In Figure 17, the dam surface absolute temperature increases, which results in more heat being released at night, eventually reaching a balance.In summer, the upper reservoir is greatly infuenced by solar radiation and the water surface temperature increases on account of the absorption of heat, which afects the adjacent dam. Figure 18 illustrates the efects of reservoir water temperature correction on the numerical simulation results of dam temperature feld in summer.It can be found that after water temperature correction, the top position of the dam surface temperature increases.
When the sun goes down, as the temperature drops, the dam body begins to exotherm externally, making the dam surface temperature lower.However, during the summer months when the temperature is high, the heat absorbed at the dam surface in the day is not fully released in the evening.When subjected to the solar radiation on continuous sunny days, the dam surface temperature will continue to accumulate.As the absolute dam surface temperature increases, the diference in temperature with the environment increases and the heat released increases in the evening.Nevertheless, an equilibrium is formed to reach the maximum absolute temperature of the dam surface eventually.
Ultimately, the radial deformation feld generated when the absolute temperature of the dam reaches its maximum after being afected by solar radiation on multiple consecutive sunny days is exhibited in Figure 19.It can be observed that the maximum upstream deformation of the super-high arch dam infuenced by solar radiation was 3.84 mm after continuous sunny days in summer, which is greater compared to the radial deformation generated by the solar radiation on the summer solstice.Te dam radial deformation feld generated by the infuences of air temperature, water temperature, and solar radiation on continuous sunny days is illustrated in Figure 20.Te maximum deformation occurred at the crown beam of the arch, with an upstream deformation of 12.61 mm.Terefore, the radial deformation of the dam caused by solar radiation on continuous sunny days accounts for about one-third of the total deformation.It indicates that the infuence of solar radiation cannot be ignored when analyzing the super-high arch dam deformation characteristics.w , heat transfer coefcient between the dam body and foundation τ c r , thermal conductivity of the foundation λ r , thermal conductivity of the mountain λ s , heat transfer coefcient between the mountain and reservoir water τ s w , and heat transfer coefcient between the mountain and air τ s a were calculated and the sensitivity of each parameter was determined.Six monitoring points with high, medium, and low elevations on the upstream and downstream faces of dam section 13# were selected, and the ranges of temperature calculation values of all monitoring points were used to analyze the degree of parameter infuence.Te maximum range was utilized to determine the sensitivity of the parameter, and a larger value indicates a more signifcant impact on the temperature feld.In the inversion analysis, the measured data of thermometers at dam sections 9# and 13# were applied.Tere are 138 thermometers (T9-1∼T9-138) and 161 thermometers (T13−1∼T13-161) installed in the dam section 9# and dam section 13#, respectively.Te thermometer layouts of dam section 9# and 13# are illustrated in Figure 21. Figure 22 demonstrates the measured temperature-time curves of the thermometer T13-36, T13-35, and T13-33, which are located near the downstream face, slightly away from the downstream face, and in the middle of the dam body at dam section 13#, respectively.Te specifc locations of the above three thermometers are marked by yellow stars in Figure 21.As observed in Figure 22, the temperature close to the downstream face varies signifcantly, while the temperature somewhat distant from the downstream face frst increases before stabilizing.In contrast, the temperature in the centre of the dam body is comparatively consistent.23 shows the comparison between the measured and the calculated temperature process lines at a thermometer near the downstream side of dam section 13#.It can be seen that the     Structural Control and Health Monitoring calculated value is quite consistent with the measured value, indicating that the inversion results are accurate.Because the joint closure temperature of the super-high arch dam is low and the dam concrete is thick, it takes a long time to reach a quasistable temperature feld.Consequently, the temperature increases in the frst fve years and afterward keeps constant.
Afterwards, the numerical were conducted by incorporating and excluding the efects of solar radiation, and the computed temperature felds were compared with the measured temperature felds as shown in Figures and 25.It can be found that considering solar radiation results in a computed temperature feld that is closer to the measured temperature feld and better represents the actual temperature distribution of the dam, while the computed temperature feld obtained without considering solar radiation cannot accurately depict the actual temperature feld of the dam.In addition, the comparison between Figures 24 and 25 represents that the efects of solar radiation are more pronounced of dam section 9# than of dam section 13#, which is due to the fact that the left bank is less afected by the shading efects of the mountain.

Conclusions
Focusing on the increasing need for exploring the solar radiation efects on dam temperature deformation, in this study, the ASHRAE clear-sky model, ray-tracing algorithm, and precomputation technique are applied to calculate the temperature deformation of super-high arch dam considering solar efects.Furthermore, HGA is presented by introducing niche elimination operation, simplex search algorithm, and accelerating cyclic operation in GA.HGA is utilized to invert the thermodynamic parameters.Finally, the numerical analysis of Jinping I super-high arch dam temperature deformation was carried out based on geographical location, dam geometry, surrounding topography, and date-specifc information, which is taken as an engineering example to verify the efciency of the proposed method.Te conclusions can be drawn as follows: (1) Te reservoir water temperature and its changes play a vital role in the super-high arch dam temperature feld below the water level on the upstream dam surface.It results in the uniform temperature distribution at the same elevation of the dam.On the dam surface above the water level, solar radiation and air temperature exert main efects.Additionally, it demonstrates that the intensity of solar radiation and temperature distribution on the upstream dam surface is uneven due to the shading efects of the dam structure and adjacent mountains.
(2) Te temperature distribution of the downstream dam surface in the case study is mainly infuenced by solar radiation and air temperature.Moreover, the shading efects of the mountains and dam structure lead to the higher temperature rising on the left bank, causing overall upstream-oriented temperature deformation.Taken together, the proposed method provides a feasible way to study the temperature felds and deformation characteristics under solar radiation for super-high arch dams located in different positions and orientations.
(3) Te analysis results of the case study demonstrate that the maximum upstream dam deformation caused by solar radiation is 3.84 mm after continuous sunny days in summer.Terefore, the dam radial deformation caused by solar radiation on continuous sunny days accounts for about one-third of the total deformation.When considering the impact of solar radiation on the deformation characteristics of the selected super-high arch dam in the case study, the results consistently indicate that the radial deformation values calculated with consideration of solar radiation are closer to the measured values in summer.Terefore, it is necessary to take into account the impact of solar radiation while assessing deformation characteristics of super-high arch dams during the summer.

Figure 2 :
Figure 2: Te fnite model and free edges of the super-high arch dam.

Figure 5 :
Figure 5: Boundary conditions of the arch dam temperature feld.

8
Structural Control and Health Monitoring zones which are poured by concrete with varying strength.Te corresponding zoning diagram of the dam body is shown in Figure 8. (4) Table

Figure 6 :Figure 7 :
Figure 6: Schematic diagram of the location of Jinping I super-high arch dam in Sichuan province.

Figure 8 :
Figure 8: Te zoning diagram of the dam body.

Figure 9 :
Figure 9: Vertical distribution of reservoir water temperature in front of the super-high arch dam on the typical days in diferent seasons.

Figure 12 :Figure 13 :
Figure 12: Te modifcation curves of reservoir water temperature considering the solar radiation efects.

3. 5 .
Termodynamic Parameter Inversion of Jinping I Super-High Arch Dam 3.5.1.Sensitivity Analysis of Inversion Parameters.Using the fnite element model established in Section 3.1, the efects of thermal conductivity of the dam body λ c , heat transfer coefcient between the dam body and air τ c a , heat transfer coefcient between the dam body and reservoir water τ c

Figure 18 :
Figure 18: Dam temperature feld numerical simulation results before and after reservoir water temperature correction on the summer solstice.(a) Before reservoir water temperature correction.(b) After reservoir water temperature correction.

Figure 19 :
Figure 19: Dam radial deformation feld considering the solar radiation efects after continuous sunny days in summer.(a) Of the dam upstream surface.(b) Of the dam downstream surface.

Figure 20 :
Figure 20: Dam radial deformation feld considering the continuous infuence of air temperature, water temperature, and solar radiation in summer.(a) Of the dam upstream surface.(b) Of the dam downstream surface.

Figure 23 :Figure 24 :
Figure 23: Te measured and calculated temperature process line of a thermometer in the middle of dam section 13# near the downstream side.

Figure 25 :
Figure 25: Te measured and calculated temperature feld of 13# dam section.(a) Te measured temperature feld.(b) Te calculated temperature feld without consideration of solar radiation efects.(c) Te calculated temperature feld considering the infuence of solar radiation.

Table 1 :
Te thermodynamic parameters of dam body and foundation.

Table 2 :
Te material parameters of dam body and foundation.

Table 3 :
Te average temperature of air and water for multiple years at the dam site.

Table 4 :
Te annual mean days of diferent weathers at the dam site.

Table 5
lists the ranges of diferent monitoring points under the infuence of various parameters.Figure 17: Te temperature feld numerical simulation result of super-high arch dam on the summer solstice.

Table 5 :
Te range of diferent indicators under the infuence of various parameters.
Structural Control and Health Monitoring 3.5.4.Inversion Analysis Results.After determining the required data, calculation loads, boundary conditions, etc., for inversion analysis and using the hybrid genetic algorithm introduced in Section 2.3.2 for inversion, the inversion results are illustrated in Table6.Figure

Table 6 :
Inversion results of dam temperature feld thermodynamic parameters.