Inversion Analysis for Thermal Parameters of Mass Concrete Based on the Sparrow Search Algorithm Improved by Mixed Strategies

Abstract

In the traditional mass concrete temperature field calculation, the accuracy of the thermal parameters is extremely important. However, the actual thermal parameters of mass concrete may have some errors with the laboratory-measured values or specification values due to the site ambient temperature, concrete surface insulation measures, cooling water flow, etc. Therefore, it can be combined with the measured temperature of the field temperature sensors using the sparrow search algorithm (SSA) for the inverse analysis of thermal parameters. Firstly, to address the problem that SSA has low convergence accuracy and easily falls into local optimum, a mixed strategy was adopted to improve the algorithm, including Logistic Chaos mapping initialization of the population, the introduction of adaptive weighting factors, and the use of the Cauchy mutation strategy. Then, the performance test was carried out to compare the performance of the algorithm with three different intelligent algorithms and reflect the superiority of the SSA that was improved by mixed strategies (SSAIMSs). Finally, the proposed method was applied to the thermal parameter inversion of a mass concrete pile cap. The inversion results demonstrated that SSAIMSs can improve the accuracy and speed of thermal parameter inversion, and the calculated results of the thermal parameters and temperatures obtained using the SSAIMSs matched well with the measured results in the field, which can meet the accuracy requirements of the actual engineering.

1. Introduction

After the casting construction of concrete dams, bridge piers, pile caps, and other large-volume concrete, due to its internal heat of hydration and external environmental temperature and other factors, it is easy to produce a certain degree of temperature stress. If the tensile strength of the concrete is unable to resist the temperature stress, cracks will occur, which may seriously threaten the safety of concrete structures [1,2,3,4,5]. Although much research and effort have been made before the pouring construction stage, the cracking problem still exists. In addition to improving materials and construction methods, it is vital to predict and monitor the internal temperature of the concrete structures to reduce the generation of cracks after the pouring construction. Compared with the previous manual on-site thermometer measurements, there is now an increasing tendency to bury temperature sensors to read the temperature in real-time [6,7,8,9]. New fiber-optic sensors are emerging, but to comprehensively and accurately obtain the internal temperature changes of mass concrete, it is not sufficient to attain the current temperature data from temperature sensors. It is also necessary to use finite element software to simulate the mass concrete, predict the temperature changes throughout the pouring process, and combine it with the measured temperature data to prevent cracking due to excessive temperature stress. Finite element temperature field analysis of mass concrete requires concrete thermal parameters, which are generally measured in the laboratory or provided by the specification. However, due to the ambient temperature, concrete surface insulation measures, cooling water flow rate, and other conditions of the site, the measured data of these parameters are often in large error with the laboratory measurements or specification values [10]. Therefore, by combining the temperature measurements from temperature sensors and the measured values of thermal parameters in the field, a new intelligent algorithm can be used to invert the required thermal parameters.

As the inversion of thermal parameters requires repeated use of the temperature field calculation procedure, the traditional calculation method requires more manpower, energy, and time, and there is a possibility of large calculation errors, so there are many scholars who choose to use intelligent algorithms to carry out the inversion of thermal parameters of concrete [11,12,13,14]. Zhang et al. [15] used BP neural network algorithms to carry out the inversion of the thermal parameters of the pile cap mass concrete, and the calculated values were in good agreement with the field-measured values. Mao et al. [16] used a cross-global artificial bee colony algorithm to invert the thermal parameters of arch dams considering the multi-stage water passage of the cooling water pipe, and the results showed that the algorithm had good adaptability in arch dam species. Wang et al. [17] used hybrid particle swarm optimization (HPSO) and fiber optic temperature monitoring data to perform parameter inversion for concrete with different materials, and the results showed that the algorithm could improve the accuracy of parameter inversion. Hu et al. [18] proposed an intelligent inversion model using field-distributed monitoring data and numerical simulation and used an improved whale swarm algorithm to find the optimal solution. The effectiveness of the intelligent inversion model is verified in terms of noise reduction effect, calculation convergence speed, and inversion accuracy. Su et al. [19] proposed an inversion analysis method for thermal parameters of lock head based on the BP neural network due to engineering limitations or other reasons, and the results of the case analysis showed that using uniform design theory to generate the parameter set to be inverted could improve the convergence speed of neural network inverse analysis. Sun and He [20] introduced the Metropolis acceptance criterion from a simulated annealing algorithm to improve the genetic algorithm and perform parameter inversion. The results showed that it basically conformed to the temperature change process of the cushion concrete of the sixth dam section of the Hohhot Pumped Storage Power Station.

Many novel intelligent algorithms simulate a process in nature, generally with the goal of solving optimization problems, and are sometimes very useful in practical engineering applications. The sparrow search algorithm (SSA) is a novel intelligent algorithm proposed by Xue and Shen [21] in 2020, which simulates the predatory and anti-predatory behaviors of sparrows. The SSA has been widely acclaimed for its high efficiency and speed, but like most other intelligent algorithms, it is easy to fall into local optimal solutions and suffers from insufficient accuracy. Therefore, some scholars have improved the SSA and applied it to examples. Zheng et al. [22] proposed a Sine-SSA-BP mode to improve the SSA and optimize inland vessel trajectory prediction. Zhu et al. [23] proposed a new optimization algorithm called adaptive sparrow search algorithm (ASSA) to conduct three practical case studies, and the final results showed that the proposed ASSA was the most efficient. Yao et al. [24] introduced two improvement points to obtain the improved sparrow search algorithm (ISSA) to predict the river runoff, and the results showed that the proposed model was significantly better than other baseline models. Li et al. [25] proposed a novel framework using bidirectional gated recurrent unit (Bi-GRU) and SSA, and the observations showed that the proposed method performed better than other methods in terms of accuracy and robustness through three case studies. Behera and Saikia [26] proposed a system with an anti-windup mixed-order generalized integrator (AWMOGI) and an ISSA, which was validated by the simulation using MATLAB software and OPAL-RT real-time simulation testbed. Khedr et al. [27] proposed a Modified Sparrow Search Algorithm-based Mobile Sink Path Planning for WSNs (MSSPP) to generate shorter routes of travel for MS and minimize data collection delays, and the results indicated that MSSPP improved performance and was more effective than other related methods. Ma et al. [28] introduced a two-dimensional logistic chaotic system, a Levy flight strategy, and nonlinear adaptive weighting for the sparrow search algorithm, which was able to optimize the working action trajectory of industrial robots and increase efficiency.

In this paper, the SSA was used as the inverse for thermal parameters of mass concrete, and a mixed improvement strategy was adopted to improve the SSA for the shortcomings of the SSA algorithm. Firstly, the Logistic chaos mapping was used to initialize the population at the initialization stage. Then, an adaptive weighting factor was introduced to improve the SSA. Finally, the Cauchy mutation strategy was adopted for the location of the optimal individuals. To verify the effectiveness of the sparrow search algorithm improved by mixed strategies (SSAIMS), it was tested by 12 standard test functions with the other three algorithms. At the same time, it was applied to the actual engineering of mass concrete of a pile cap in Shandong, China, which can provide reference and inspiration for other mass concrete construction projects.

2. Principles and Steps of SSAIMSs

2.1. SSA

The SSA is inspired by the various behaviors of sparrows in the process of foraging and anti-predation. It is necessary to rank the sparrow population using fitness advantages and disadvantages after initializing the population. There are different divisions of labor within the sparrow population, which can be roughly classified into three types when conducting a spatial search: producers, scroungers, and alerters. The producers are the better-adapted individuals who search for food for the sparrow population; the scroungers are the less well-adapted individuals who may compete with the producers for food after they have found the location of the food; and the alerters are the randomly-adapted individuals that are on the lookout for the food and signal to flee if they find any dangers such as natural enemies.

The producers provide approximate foraging directions for the entire sparrow population, and the location update criterion for the producers is

$$X_{i,j}^{k+1}=\left\{\begin{array}{l}{X}_{i,j}^k\cdot \exp \left(-{\displaystyle \frac{i}{q{t}_{\max }}}\right), R_2<{\delta }_{st}\\ X_{i,j}^k+Q\cdot L, R_2\ge {\delta }_{st}\end{array}\right.$$

(1)

During the search for food, the scroungers snatch the food found by the producers. If the snatch fails, the location update criterion for the scroungers is

$$X_{i,j}^{k+1}=\left\{\begin{array}{l}Q\cdot \exp \left({\displaystyle \frac{X_{worst}-X_{i,j}^k}{i^2}}\right), i>{\displaystyle \frac{n}{2}}\\ X_P^{k+1}+\left|X_{i,j}^t-X_P^{k+1}\right|\cdot A^{+}\cdot L, i\le {\displaystyle \frac{n}{2}}\end{array}\right.$$

(2)

The alerters monitor the foraging area space of the entire population and immediately signal danger when there is danger, followed by rapid movement towards the safe area space. The location update criterion for the alerters is

$$X_{i,j}^{k+1}=\left\{\begin{array}{l}{X}_{best}^k+\eta \cdot \left|X_{i,j}^k-X_{best}^k\right|, f_i>f_g\\ X_{i,j}^k+K\cdot \left({\displaystyle \frac{\left|X_{i,j}^k-X_{worst}^k\right|}{\left(f_i-f_w\right)+\epsilon }}\right), f_i=f_g\end{array}\right.$$

(3)

2.2. Sparrow Search Algorithm Improved by Mixed Strategies (SSAIMS)

2.2.1. Initialization of Population by Using Logistic Chaos Mapping

The Sparrow Search Algorithm adopts a random strategy for the generation of sparrow populations. Although the random number method can randomly assign values to the populations in the search space, it usually cannot guarantee that the initial populations fully and evenly cover the search space. Otherwise, it is difficult to find out the various possibilities of the solution, which often leads to the generation of problems such as low diversity and uneven distribution of sparrow populations and ultimately reduces the quality of the solution due to premature convergence. Chaotic mapping can produce chaotic sequences that are more uniformly distributed than random generation and are therefore used in sparrow search algorithms to generate sparrow populations. Currently, the frequently used mappings are tent mapping, Chebyshev mapping, Singer mapping, logistic mapping, sine mapping, circle mapping, and so on. In this paper, Logistic chaos mapping is introduced to improve the Sparrow Search Algorithm, as shown in Equation (4).

$$X_{k+1}=\mu X_k\left(1-X_k\right)$$

(4)

2.2.2. Adaptive Weight Factor

The weighting factor plays a large role in objective function optimization, and appropriate weights can speed up the convergence speed of the algorithm and improve its accuracy. To solve the problem of slow convergence speed and low precision in the optimization process, the adaptive weight factor ω is introduced, as shown in Equation (5), according to the inspiration of Liu and He [29]. In the very first iteration stage of the SSA, the sparrow population can traverse the entire search space with a larger weight, which is conducive to the algorithm to improve global development capabilities and accelerate the convergence speed. In the middle and late stages of the SSA iteration, the algorithm converges gradually, and a smaller weight can be used to explore the small region finely to improve the convergence accuracy; finally, in the final stage of the sparrow algorithm iteration, it can be assigned a relatively large perturbation to solve the problem that SSA is easy to fall into local optimum.

$$\omega =\left\{\begin{array}{l}{\delta }_1\cdot \left(\cos \left(k\cdot {\displaystyle \frac{\pi }{{\delta }_2}}\right)+{\delta }_3\right), k\le k_s\\ {\rho }_1\cdot \sin \left({\rho }_2\cdot k\cdot \pi \right)+{\rho }_3, k>k_s\end{array}\right.$$

(5)

When k_max = 500, the adaptive weight factor curve is shown in Figure 1.

In the standard SSA, the value of the factor $\exp \left(-{\displaystyle \frac{i}{q{t}_{\max }}}\right)$ in Equation (1) is in the range of (0, 1), which is positive for the convergence of the algorithm. However, as the number of iterations continues to increase, the value of the factor approaches 0 more and more quickly, which makes the producers converge to the origin, and the algorithm is prone to fall into the local optimum with the low speed of the convergence procedure [30]. In this paper, the adaptive weight factor ω is used to replace this factor, which makes the SSA converge faster and does not easily fall into the local optimum. The improvement is shown in Equation (6).

$$X_{i,j}^{k+1}=\left\{\begin{array}{l}{X}_{i,j}^k\cdot \omega , R_2<{\delta }_{st}\\ X_{i,j}^k+Q\cdot L, R_2\ge {\delta }_{st}\end{array}\right.$$

(6)

2.2.3. Cauchy Mutation Strategy

The Cauchy mutation is derived from the Cauchy distribution of a continuous type probability distribution with a one-dimensional Cauchy density function concentrated near the origin, which is functionally defined as Equation (7).

$$f\left(x\right)={\displaystyle \frac{1}{\pi }}\cdot {\displaystyle \frac{a}{a+x^2}}, x\in \left(-\infty ,+\infty \right)$$

(7)

When a = 1, it is called the standard Cauchy distribution.

The one-dimensional Cauchy density distribution function is shown on the image to be concentrated near the origin. According to this feature, the current optimal individual is perturbed by Cauchy mutation to promote its development in a better direction, which is conducive to solving the problem of falling into local optimum. The equation of the Cauchy mutation strategy is

$$X_{new}=X_{best}\cdot \left[1+Cauchy\left(0, 1\right)\right]$$

(8)

After the Cauchy mutation, it has a positive effect on the SSA to jump out of the local optimum, but there is no comparison of whether the updated position is better than the original position. Therefore, in order to determine whether the position information should be updated or not, a greedy selection strategy is introduced to compare the fitness values of the old and the new. At the same time, it can enhance the speed and accuracy of the algorithm convergence and improve the algorithm’s performance in optimization seeking. The greedy selection strategy is shown in Equation (9).

$$X_{i,j}^{k+1}=\left\{\begin{array}{l}{X}_{new}, f\left(X_{new}\right)<f\left(X_{best}\right)\\ X_{best}, f\left(X_{new}\right)\ge f\left(X_{best}\right)\end{array}\right.$$

(9)

2.3. Performance Testing

To test the performance of the SSAIMSs, this paper uses 12 commonly used test functions to perform simulation experiments with the standard particle swarm optimization algorithm (PSO) [31,32], simulated annealing algorithm (SA) [33,34], grey wolf optimization (GWO) [35,36]. The experiment introduces 12 test functions with different optimization characteristics, as shown in

Table 1. Test Functions.

Function	Dimension	Search Scope	Theoretical Value
$F_1\left(x\right)={\displaystyle \sum _{i=1}^n{x}_i^2}$	30	[−100, 100]	0
$F_2\left(x\right)={\displaystyle \sum _{i=1}^n\left|x_i\right|}+{\displaystyle \prod _{i=1}^n\left|x_i\right|}$	30	[−10, 10]	0
$F_3\left(x\right)={\max }_i\left\{\hspace{0.17em}\left|x_i\right|,\hspace{0.17em}1\le i\le n\right\}$	30	[−100, 100]	0
$F_4\left(x\right)={\displaystyle \sum _{i=1}^{n}i{x}_i^4}+random\left[0, 1\right)$	30	[−1.28, 1.28]	0
$F_5\left(x\right)={\displaystyle \sum _{i=1}^n\left[x_i^2-10\cos \left(2\pi x_i+10\right)\right]}$	30	[−5.12, 5.12]	0
$F_6\left(x\right)=-20\exp \left(-0.2\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n{x}_i^2}}\right)-\exp \left({\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n\cos \left(2\pi x_i\right)}\right)+20+e$	30	[−32, 32]	0
$F_7\left(x\right)={\displaystyle \frac{1}{4000}}{\displaystyle \sum _{i=1}^n{x}_i^2}-{\displaystyle \prod _{i=1}^n\cos \left(\frac{x_i}{\sqrt{i}}\right)}+1$	30	[−600, 600]	0
$\begin{array}{c}{F}_8\left(x\right)={\displaystyle \frac{\pi }{n}}\left\{10\sin \left(\pi y_1\right)+{\displaystyle \sum _{i=1}^{n-1}{\left(y_i-1\right)}^2\left[1+10{\sin }^2\left(\pi y_{i+1}\right)\right]}\right.+\\ \left.{\left(y_n-1\right)}^2\right\}+{\displaystyle \sum _{i=1}^{n}u(x_i,10,100,4)}\hspace{1em}\\ \hspace{1em}{y}_i=1+{\displaystyle \frac{x_i+1}{4}}\hspace{0.17em}\hspace{0.33em}u(x_i,a,k,m)=\left\{\begin{array}{l}k{(x_i-a)}^m x_i>a\\ 0 -a<x_i<a\\ k{(-x_i-a)}^m x_i<-a\end{array}\right.\end{array}$	30	[−50, 50]	0
$F_9\left(x\right)={\left({\displaystyle \frac{1}{500}}+{\displaystyle \sum _{j=1}^{25}\frac{1}{j+{\displaystyle {\sum }_{i=1}^2{\left(x_i-a_{ij}\right)}^6}}}\right)}^{-1}$	2	[−65, 65]	1
$\begin{array}{l}{F}_{10}\left(x\right)=\left[1+{\left(x_1+x_2+1\right)}^2\left(19-14x_1+3x_1^2-14x_2+6x_1{x}_2+3x_2^2\right)\right]\times \\ \left[30+{\left(2x_1-3x_2\right)}^2\times \left(18-32x_1+12x_1^2+48x_2-36x_1{x}_2+27x_2^2\right)\right]\end{array}$	2	[−2, 2]	3
$F_{11}\left(x\right)=-{{\displaystyle \sum _{i=1}^5\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}}^{-1}$	4	[0, 10]	−10.1532
$F_{12}\left(x\right)={{\displaystyle \sum _{i=1}^{10}\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}}^{-1}$	4	[0, 10]	−10.5363

, of which the first four are high-dimensional single-peak functions, which have only one optimal solution and are used to test the convergence speed and accuracy of the algorithms; the middle four are high-dimensional multi-peak functions, and the last four are low-dimensional multi-peak functions, which have multiple local optimal solutions and can be used to test whether the algorithms are prone to fall into local optimum. At the same time, the tests are conducted on the same experimental environment of Windows 10 and the experimental simulation platform of MATLAB R2020a to fairly compare the performance of various algorithms. The number of generated populations n is 30, and the maximum number of specified iterations k_max is 500. To reduce the impact of randomness on various algorithms and make the results scientific and credible, 20 independent experiments are carried out for each algorithm, and then the mean value and variance of the results of the 20 experiments are compared.

The test results of the four algorithms are shown in

Table 2. Performance comparison of four algorithms.

Function	PSO		SA		GWO		SSAIMS
	Mean Value	Variance	Mean Value	Variance	Mean Value	Variance	Mean Value	Variance
F1	6.60 × 10−5	4.78 × 10−9	1.89 × 10−17	2.65 × 10−33	3.75 × 10−57	9.23 × 10−60	1.31 × 10−71	3.44 × 10−141
F2	4.16	1.40	5.60 × 10−5	1.16 × 10−9	6.39 × 10−17	2.01 × 10−33	2.84 × 10−28	1. 08 × 10−54
F3	5.70 × 10−2	7.27 × 10−4	8.96 × 10−3	6.25 × 10−5	4.55 × 10−18	7.55 × 10−35	6.63 × 10−32	8.32 × 10−62
F4	13.8	76.5	0.146	2.13 × 10−3	2.15 × 10−3	1.02 × 10−6	1.15 × 10−3	3.60 × 10−7
F5	161	1.17 × 103	3.38	2.31	2.62	12.4	0	0
F6	2.66	6.93 × 10−2	7.22 × 10−3	3.19 × 10−4	1.06 × 10−13	2.13 × 10−28	8.88 × 10−16	0
F7	0.117	1.82 × 10−3	4.97 × 10−2	2.46 × 10−3	4.51 × 10−3	7.37 × 10−5	0	0
F8	1.33 × 10−4	2.16 × 10−7	5.47 × 10−19	2.64 × 10−36	1.49 × 10−2	1.32 × 10−2	1.47 × 10−32	7.36 × 10−66
F9	3.17	5.77	0.998	6.20 × 10−26	3.06	9.88	0.998	2.51 × 10−22
F10	3	3.19 × 10−29	8.40	122	3	2.16 × 10−9	3	1.61 × 10−30
F11	−7.64	10.6	−4.01	5.54	−9.14	4.32	−9.52	3.90
F12	−9.86	4.50	−5.52	12.1	−10.1	3.29	−10.3	1.36

. It can be seen that for most of the test functions, SSAIMSs are superior to the other three algorithms, and the variance index is slightly worse than the SA only in F9. For the high-dimensional single-peak function of F1–F4, although SSAIMS does not find the theoretical optimal solution of 0, it has higher accuracy and better stability, and it is closer to 0 than the other algorithms. For the high-dimensional multi-peaked functions of F5–F8, SSAIMS can find the theoretical optimal solution of 0 on the F5 and F7 functions. While the calculated fitness value for F6 is not 0, the variance is 0, which is very stable. For the low-dimensional multi-peaked functions of F9–F12, SSAIMS is also very superior and almost never falls into the local optimum and finds the actual optimal solution. It can be concluded that the SSAIMS is more effective than the other three algorithms and can be applied to the inversion of thermal parameters of mass concrete.

3. Fundamentals of Inversion of Thermal Parameters of Mass Concrete

3.1. Calculation Criterion for Temperature Field of Concrete

In the days just after the concrete has been poured, the concrete undergoes a heat of hydration reaction when there is an internal heat source. The poor thermal conductivity of concrete and a large amount of internal heat of hydration generated cannot be propagated out quickly for the time being, resulting in rapid heating of the concrete at an early stage, which may result in cracking due to temperature stresses, which requires special attention in the case of mass concrete. The equation for the thermal conductivity of concrete is

$${\displaystyle \frac{𝜕T}{𝜕t}}={\displaystyle \frac{\lambda }{c\rho }}({\displaystyle \frac{{𝜕}^{2}T}{𝜕x^2}}+{\displaystyle \frac{{𝜕}^{2}T}{𝜕y^2}}+{\displaystyle \frac{{𝜕}^{2}T}{𝜕z^2}})+{\displaystyle \frac{𝜕{\theta }_0}{𝜕t}}$$

(10)

The equation for the adiabatic temperature rise of concrete is expressed by Equation (11), which is a more intuitive representation of the heat generated by the heat of hydration of concrete.

$${\displaystyle \frac{𝜕T}{𝜕t}}={\displaystyle \frac{\lambda }{c\rho }}({\displaystyle \frac{{𝜕}^{2}T}{𝜕x^2}}+{\displaystyle \frac{{𝜕}^{2}T}{𝜕y^2}}+{\displaystyle \frac{{𝜕}^{2}T}{𝜕z^2}})+{\displaystyle \frac{𝜕{\theta }_0}{𝜕t}}$$

(11)

The equation for calculating the equivalent exothermic coefficient of a concrete structure covered with an insulating material is expressed by Equation (12), which can be measured and then calculated in the field. Concrete surfaces are susceptible to the site environment as they are exposed to frequent direct sunlight.

$${\displaystyle \frac{𝜕T}{𝜕t}}={\displaystyle \frac{\lambda }{c\rho }}({\displaystyle \frac{{𝜕}^{2}T}{𝜕x^2}}+{\displaystyle \frac{{𝜕}^{2}T}{𝜕y^2}}+{\displaystyle \frac{{𝜕}^{2}T}{𝜕z^2}})+{\displaystyle \frac{𝜕{\theta }_0}{𝜕t}}$$

(12)

The concrete boundary condition is

$$-\lambda {\displaystyle \frac{𝜕T_c}{𝜕m}}=\beta (T_c-T_a)$$

(13)

Mass concrete in the pouring is often used to cool water pipes for water cooling to prevent its internal heating from being too fast and too high. According to academician Zhu [37] and Zhou et al. [38], Equations (14)–(17) can be used to calculate the temperature of the concrete under the simultaneous consideration of the effects of the cooling water pipe and the external ambient temperature.

$$T\left(t\right)=T_{ij}+\left(T_i-T_{ij}\right){𝜙}_i\left(t\right)+{\theta }_0{\psi }_i\left(t\right)+{\eta }_i\left(t\right)$$

(14)

$${𝜙}_i\left(t\right)=\exp \left(-p_{i}t\right)$$

(15)

$${\psi }_i\left(t\right)={\displaystyle \frac{r}{r-p_i}}\left(\exp \left(-p_{i}t\right)-\exp \left(-rt\right)\right)$$

(16)

$${\eta }_i\left(t\right)=\left(T_{ia}-T_{ij}\right){\displaystyle \sum _k\left\{\exp \left[-p_i\left(t-t_k\right)\right]-1\right\}}\cdot \Delta \mathrm{erf}\left({\displaystyle \frac{h}{2\sqrt{a{t}_k}}}\right)$$

(17)

3.2. Selection of Thermal Parameters of Concrete

The heat of hydration of concrete is an important factor in the generation of temperature stresses; therefore, it is crucial to obtain the thermal parameters of concrete. Calculation of thermal parameters of concrete is generally related to parameters such as thermal conductivity α, thermal conductivity λ, adiabatic temperature rise θ₀, equivalent surface heat dissipation coefficient β_s, rate of reaction of heat of hydration r, density ρ and specific heat capacity c. For the density ρ and specific heat capacity c of concrete, the changes are very small during the pouring process, which can be obtained by laboratory and field measurements without inversion, while the thermal conductivity coefficient α can be calculated based on $\alpha ={\displaystyle \frac{\lambda }{c\rho }}$ to derive α. In general, the four parameters of λ, θ₀, β_s, and r after concrete pouring are greatly affected by the environment, which is not easy to obtain in the laboratory and can be used for the inversion of thermal parameters.

3.3. Selection of the Objective Function

The objective function is established as shown in Equations (18) and (19).

$$F\left(X\right)={\displaystyle \frac{\sqrt{{{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^N\left(T_{mn}^{\prime }-T_{mn}\right)}}}^2}}{MN}}$$

(18)

$$X=\left[x_1,x_2,x_3,x_4\right]$$

(19)

The objective function is established so that the parametric inversion problem becomes an optimization problem, and the objective function F(X) is converged to the minimum by constant computation. Then, the inversion parameters are continuously updated in MATLAB software. When the objective function reaches the maximum number of iterations t_max of the algorithm, the calculation is stopped, and the parameter inversion result is the final output. The flowchart of the SSAIMSs combined with the temperature field calculation procedure is shown in Figure 2.

4. Example Application

A twin-tower, single cable-stayed, pre-stressed concrete, large-span suspension cable-stayed bridge was constructed at a site in Shandong, China. The total length of the main bridge was 394.6 m, with a two-way six-lane carriageway, a roadbed width of 34.5 m, and a roadway width of 30 m. The main road was arranged as a 0.75 m wide earth shoulder, 3 m wide hard shoulder, 3.75 m wide carriageway, 0.75 m wide curb strip, and 3.0 m wide central divider, with symmetrical left and right widths. In addition, the main bridge was located on the straight section with +1.930% and −1.70% corresponding longitudinal slopes, respectively, and 2% cross slope of the bridge deck. The structural system was a tower–beam consolidation and tower–pier separation system. The structural schematic of the main bridge is shown in Figure 3.

The main materials used in this bridge include concrete, rebars, pre-stressed steel strands, stay cables, asphalt, and so on. Among them, all kinds of concrete were tested to meet the Chinese standard specifications, and the concrete grades used in each part of the bridge are shown in

Table 3. Concrete grades are at different positions on the bridge.

Main Girder	Main Tower	Main Pier and Side Pier	Pile Cap	Pile Foundation	Bearing Cushion Layer	Bottom Sealing of Pile Cap
C55	C50	C40	C35	C30	C50	C25

. The rebars used in the design were HPB300 and HRB400 grades. The performance of the pre-stressed steel strands showed high strength and low relaxation, with a nominal diameter of 15.2 mm, a nominal area of 139 mm², a standard value of tensile strength of 1860 MPa, a modulus of elasticity of 1.95 × 10⁵ MPa, and a relaxation rate of 2.5 percent. The stay cables adopted epoxy steel strands. Besides, there were 28 pairs of stay cables on each tower, with a total of 56 pairs on the whole bridge, with a longitudinal standard spacing of 4 m on the main girder and a vertical spacing of 1.2 m on the tower. After the bridge was completed, the bridge deck was paved with 4 cm SMA-13 fine-grain asphalt.

The main construction process of the main pier foundation is as follows: (1) insertion and drilling of steel shields; (2) construction of bored piles; (3) assembling of steel sheet pile cofferdams; (4) positioning of cofferdams; (5) pouring of bottom sealing concrete; (6) pumping and bearing platform construction; (7) construction of the pier body. The main pier pile cap adopts an octagonal pile cap, with a plan dimension of 25.5 m × 18.588 m, 5 m thick, using C35 concrete, which was a kind of mass concrete. The main pier foundation adopted a pile foundation, which was designed as a friction pile. There were 27 bored piles with a diameter of 1.8 m and a length of 83 m underneath.

When the ambient temperature is too high, the hydration reaction of the concrete accelerates, resulting in the internal temperature of the concrete. Because the internal heat of hydration heat can not be dissipated in time, the internal temperature stress will be generated. When the temperature stress exceeds the tensile strength of the concrete, a large number of cracks will be generated to reduce the performance of the concrete, so it is necessary to use cooling water and re-measure the thermal parameters of the concrete. When the ambient temperature is too low, the rate of hydration of the concrete decreases, which affects the strength development of the concrete and may lead to expansion of the concrete, irreversible cracks, and loss of strength. It is, therefore, necessary to insulate the concrete and re-evaluate the thermal parameters in a timely manner. The project site has moderate temperatures and abundant thermal energy resources, with average annual temperatures ranging from 11.0 °C to 14.3 °C. In addition, the lowest temperatures are found in January and February, and the highest temperatures are found in July. In addition, the temperatures are the lowest in January and February and highest in July. The mass concrete pile caps were placed in early August. Due to the high temperature in summer, the thermal parameters of the concrete to be inverted, such as thermal conductivity λ, adiabatic temperature rise θ₀, equivalent surface heat dissipation coefficient β_s, and heat of hydration reaction rate r, are easily affected by the ambient temperature, which plays an important role in the temperature change inside the concrete. The pile cap information is shown in Figure 4.

To measure the temperature, temperature sensors were tied on the surface of the reinforcement bars on site before pouring. A total of three layers were strapped, with the bottom layer of layer A at 1 m above the ground level, the middle layer of layer B at 2.5 m above the ground level, and layer C at 4 m above the ground level. Nine sensors were tied in each layer, with points 7, 6, 3, 8, and 9 spaced 3 m apart and points 1, 2, 3, 4, and 5 spaced 4 m apart, all on axes. In addition, to ensure the accuracy and reliability of the temperature data, the inlet and outlet temperatures, the concrete top surface temperature, and the atmospheric temperature were measured simultaneously. The pile cap was poured with three rows of cooling water pipes with the same height position as the temperature sensors, 60 mm diameter, 3 mm thickness, zigzagging back and forth horizontally, and 1.5 m spacing. After the pouring of the pile cap, the temperature was continuously measured by using the temperature sensor. It was feasible to measure the temperature on site, but it was easier and more accurate to take automatic readings and record the temperature data on a computer by using the temperature sensor software Vircom 10.9. Then, it was necessary to manually check the data and draw graphs for comparative analysis in the following steps. The temperature sensor information is shown in Figure 5.

4.1. Finite Element Modelling

According to the construction plan of the site, this paper establishes a finite element model by Midas FEA NX 2022 v1.1 as in Figure 6. It used an eight-node hexahedral cell, the upper half of which was the pile cap and the lower half of which was the foundation, and the temperature sensor measurement points were distributed in the model cell nodes. As the mesh near the cooling water pipes might have a large temperature difference, the cell nodes were modified to second-order cells after modeling to facilitate the analysis of the temperature magnitude and refine the cells. Finally, the total number of cells was 3100, and the total number of nodes was 41,305.

4.2. Inversion of Thermal Parameters and Comparison of Results

To verify the actual performance of the various algorithms in the engineering example, the thermal parameter ranges, as in

Table 4. Range of thermal parameters of C35 concrete.

Thermal Parameters of C35 Concrete	Unit	Range
Thermal conductivity λ	KJ/(m·h·°C)	[8, 16]
Adiabatic temperature rise θ0	°C	[50, 80]
Equivalent surface heat dissipation coefficient βs	KJ/(m2·h·°C)	[10, 100]
Hydration heat reaction rate r	h−1	[0.01, 0.04]

, were selected for inversion in MATLAB software according to the preliminary parameter trial calculation and the actual requirements of the site. As can be seen from Figure 7, the fitness values of all algorithms decreased as the number of iterations increased, but different algorithms had different speeds of decreasing fitness values and different final values. The PSO had the highest first-generation fitness value and the slowest convergence, falling into a local optimum around the 10th generation, with a final fitness value of 0.4977. The SA converged around the 16th generation, with a final fitness value of 0.4966; the GWO was also susceptible to falling into a local optimum but with a lower final fitness value of 0.4965; in contrast, the SSAIMSs performed the best, converging around generation 12, with the lowest fitness value of 0.4955. Overall, it can be seen that the SSAIMSs converge significantly faster, with the highest convergence accuracy, which can reduce the number of iterations required to meet the accuracy requirement, save the arithmetic time, power, and time, and perform thermal parameter inversion analysis more effectively.

After iterative operation, in addition to the final fitness value of the four algorithms, the thermal parameter values corresponding to the fitness value could also be obtained. To compare the validity and reliability of the thermal parameter values of the four algorithms, the actual measured and calculated thermal parameter values in the field were compared with them, as shown in

Table 5. Comparison of inversion results of thermal parameters of four algorithms and field measurements.

Comparison of Algorithms	Thermal Conductivity λ/[KJ·(m·h·°C)−1]	Adiabatic Temperature Rise θ0/°C	Equivalent Surface Heat Dissipation Coefficient βs/ [KJ·(m2·h·°C)−1]	Hydration Heat Reaction Rate r/(h−1)
On-site measurement	10.6	74.5	49.8	0.0162
PSO	12.2 (15.1%)	85.7 (15.0%)	86.2 (73.9%)	0.0153 (5.56%)
SA	9.7 (8.49%)	72.8 (2.28%)	48.3 (3.01%)	0.0194 (19.8%)
GWO	10.3 (2.83%)	76.8 (3.09%)	53.8 (8.03%)	0.0179 (10.5%)
SSAIMS	10.4 (1.89%)	76.6 (2.82%)	50.9 (2.21%)	0.0171 (5.56%)

. It lists the absolute values of the relative errors between the thermal parameter values calculated by the four algorithms and the actual measured values in the field. As can be seen in Table 5, the relative error between the thermal parameters calculated by the SSAIMSs and the actual measured values in the field was smaller in absolute value than the other three algorithms. Compared with the other three algorithms, the thermal parameters derived from the SSAIMSs were closer to reality, and the absolute values of the relative errors were all within 6% so that the accuracy could meet the practical requirements.

The four thermal parameters obtained using the SSAIMS were brought into the finite element modeling software for analysis, where the cooling water temperature was left unchanged at the same round of 10 °C. When observing and analyzing the temperature results, the 1/4 model was used to visually obtain the temperature magnitude. The temperature distribution inside and on the surface of the model at the age of 5 days is shown in Figure 8.

Temperature sensor readings at locations B-1, B-2, B-3, B-6, and B-7 in layer B (the middle layer) were compared with the temperatures derived from finite element simulations. Due to the rapid temperature change due to the internal heat of hydration after casting, the two were compared daily for the first 9 days and then every 2 days thereafter. The comparison results are shown in Figure 9. As can be seen from Figure 9, on day 3, the maximum absolute value of the difference between the two temperatures at position B-3 was 4.6 °C, and the corresponding relative error was 6.18%, which was relatively small. In the early stage, with cooling water, the temperature errors at all locations were larger than in the later stage, when there was no heat or water supply. In addition to the slight errors between the thermal parameters calculated by the algorithm and the actual situation, as well as the accuracy issues of finite element software calculations, the main reasons may be the unstable water supply in the early stage and the large temperature difference between day and night on site. When the ambient temperature is too high in the middle of the day, it is important to combine the finite element software to calculate the need to temporarily increase the amount of water supply. When the atmospheric temperature is too low at night, the concrete should be covered. Overall, the temperature sensor readings measured on site were in good agreement with the temperature calculated by the finite element model, and the trend was basically the same, which can better reflect the temperature changes on the construction site. Therefore, the thermal parameters of the inversion performance obtained using the SSAIMSs were more effective and can be used in the actual engineering sites.

5. Conclusions

(1)

To speed up the convergence of the SSA algorithm, reduce the possibility of falling into a local optimum as well and improve the computational accuracy, three mixed improvement strategies, Logistic chaos mapping initialization of the population, the adaptive weighting factor, and Cauchy mutation, were used to improve the SSA, which was called;

(2)

The performance test was carried out to compare the performance of the algorithm with three different intelligent algorithms (PSO, SA, and GWO) and reflected the superiority of the SSA improved by mixed strategies (SSAIMS). The results show that the SSAIMS was better than the other three algorithms in general. Therefore, it can be used to invert the thermal parameters of mass concrete;

(3)

The SSAIMS was used to invert the thermal parameters of an octagonal mass concrete pile cap. The error between the results of the SSAIMS and the field-measured values was no more than 6%, which was better than the other three algorithms and verified the accuracy of the inversion of thermal parameters inversion of the SSAIMS;

(4)

The inverse thermal parameters were brought into the finite element software for analysis. The points of the B layer were selected to compare with the measured temperature on site. The results indicated that the maximum error was 4.6 °C at the initial stage of using cooling water, which may be due to the unstable water throughput and the large temperature difference between day and night at the site. Therefore, we should combine finite element software to increase the water supply at noon and take insulation measures at night. Meanwhile, the error was smaller at the later stage without cooling water, and the overall trend was close. Therefore, the SSAIMSs can be used for practical engineering applications in the field.