Structural Optimization of Ship Lock Heads during Construction Period considering Concrete Creep

Traditional structural optimization is mainly based on the assumption that the materials are elastic, which cannot represent real stress fields in structures. In this study, the genetic algorithm, big bang-big crunch algorithm, and hybrid big bang-big crunch algorithm were employed to optimize the design factors of ship lock heads during concrete construction. )e optimization goal was to determine the minimum volume of concrete.)e factors considered included the hydration heat, the early-stage creep, and the transient deformation under external loads. In the finite element analysis, three types of boundary conditions were considered. )e whole construction process was simulated, and the maximum tensile and compressive stresses, the stability, and the overturning of the lock head were examined. Based on the finite element analysis, to reduce the consumption of memory, a set of implicit recursive equations were used to calculate the thermal creep stress. )irty-four design variables were distinguished for optimization. A case study on the optimization of a ship lock head was used to demonstrate the optimization process. )e optimization results showed that the hybrid big bang-big crunch algorithm was more effective, and some conclusions were derived.


Introduction
Ship locks are one of the most critical massive concrete structures used in inland waterways. Surmounting differences in the water level is the primary role of ship locks [1]. e lock head is a complex part of a lock. e finite element method (FEM) has been used to optimize lock heads [2]. Obtaining reliable safety with the best possible minimum volume of concrete is the optimization objective of this study. Due to the time-dependent properties of mass concrete structures [3], temperature and creep should also be studied. e hydration temperature effects are strong during the construction period [4]. ey often cause the mass concrete temperature to rise by 40-70°C, and thermal stressinduced cracking can be generated [5]. ermal stress-induced cracking poses a significant threat to mass concrete durability [6]. In addition, creep can relax the thermal stress of early-age concrete [7]. erefore, to understand the influence of the optimization on the structural properties during the construction period, structural optimization should consider the evolution of the temperature field and creep.
Genetic algorithms (GAs) are intelligent search algorithms.
ey are widely used due to their functional properties in different engineering application areas, such as structural optimization [8], automated management [9], safety evaluation [10], sensitivity analysis [11], and back analysis [12]. For optimization problems, GAs can typically generate high-quality solutions with an adaptive search mechanism.
e three main characteristics of GAs in structural optimization problems are as follows [13]: (1) they exhibit excellent searchability for the global optimal solution, (2) they discretize the design variables, and (3) the constraints are modifiable. Erol and Eksin [14] developed the big bang-big crunch (BB-BC) algorithm, which is a more robust method than GAs [15].
is algorithm converges quickly [16]. Kaveh and Talatahari [17] proposed the hybrid big bang-big crunch (HBB-BC) algorithm to improve the search capability of the BB-BC method. e HBB-BC algorithm integrates the particle swarm optimization (PSO) and BB-BC algorithms [18].
Mass concrete structures have long construction periods and complicated construction sequences, such as concreting, formwork, and dismantling formwork. Rita et al. [19] optimized the thick concrete foundation of a building to minimize the construction cost. ey analysed the temperature and stress fields of the structure during the construction period, and the results guaranteed that the structure would not generate thermal cracks. Fairbairn et al. [20] optimized a small hydropower plant dam to minimize the construction cost. ey used a coupled thermal-chemical-mechanical model to calculate the transient hydration, temperature, and stress fields during the construction period. e results indicated that the developed procedure is suitable for the design of dams. However, few optimization studies have simulated the real temperature fields and stress states of ship lock heads during the construction period.
e study of the temperature field of mass concrete structures is not a new issue. In the 1920s, the Schmidt method was developed [21].
is method is a numerical method for predicting temperatures throughout mass concrete elements, and it is convenient for manual calculations. It is a simplified finite difference formulation. However, it is more suitable for heat conduction problems with rectangular shapes. Due to the advances in computer science, Wilson et al. [22] applied the finite element method to the solution of the transient heat conduction problem.
is method is suitable for complex solids with arbitrary shapes. It also accounts for heat flux and temperature boundary conditions. Wilson et al. [23] subsequently presented techniques for solving large and complex three-dimensional heat conduction problems. On this basis, a series of research articles were published [4,24]. Yang et al. [25] discovered that a structure's geometric dimensions can affect the temperature distribution of the concrete. Structural optimization will change the sizes of the mass concrete structures. e rate and amount of temperature change are two crucial factors for evaluating the temperature field [26]. However, few studies have analysed the effects of structural optimization on the temperature fields of mass concrete structures. e creep deformation is of utmost importance in early-age concrete. erefore, to truly calculate the stress field of mass concrete structures, a creep model must be included in mechanical analyses [27]. Creep is a viscoelastic property of concrete [28]. In recent decades, several creep models have been developed, such as the B3 model [29], the M4 model [30], and the ACI 209R-92 model [31]. Bažant and Xi [32] found that the Kelvin chain was accurate enough to describe any linear viscoelastic behaviour, and its parameters were easily determined from creep tests. Creep is influenced by stress history [33]. Consequently, the entire history must be stored during the calculation process, which limits the application of creep models in mass concrete structures. Bažant and Xi [32] converted creep compliance functions to a Dirichlet series. Zhu [34] derived a set of implicit recursive equations without requiring the stress history to be stored. is method was used to calculate the thermal creep stress. However, most studies on structural optimization assumed that the materials were elastic [35]. Few optimization studies have calculated the stress fields of mass concrete structures using creep models and analysed the effects of structural optimization on the thermal creep stress.
is study aimed to determine the optimal shape of ship lock heads and analyse the influence of structural optimization on the temperature and stress fields during the construction period. e GA, BB-BC, and HBB-BC algorithms were used to solve the optimization problem of a ship lock head, and corresponding optimization procedures were proposed. In the optimization process, the optimization objective was to determine the minimum volume of concrete. e set of design variables contained 34 parameters. e remaining sections of this paper are organized as follows. In Section 2, the theoretical formulations of the temperature field and the thermal creep stress are described. In Section 3, the optimization problem of a ship lock head is established. In Section 4, the key points for applying the GA, BB-BC, and HBB-BC algorithms to optimize a ship lock head are established. In Section 5, a complete example with analysis is provided. Finally, the valuable conclusions drawn from the study are summarized in Section 6.

Calculation
Principle of Temperature Field. According to Fourier's law of heat conduction and the energy conservation principle, the governing equation of a three-dimensional transient temperature field is expressed as follows [36]: where ρ is the material density, c is the specific heat of the materials, T is the concrete temperature, k is the thermal conductivity coefficients, and Q is the heat of hydration introduced per unit volume. e initial condition is as follows [37]: ere are three types of boundary conditions for concrete structures, which are denoted as S 1 , S 2 , and S 3 . e boundary conditions are expressed as follows [37].
On boundary S 1 , the function of temperature (T) is known: On boundary S 2 , the thermal flux density is a function of time: On boundary S 3 , a convective-type condition is applied: In these boundary conditions, n represents the external normal direction to the boundary, β represents the surface exothermic coefficient, and T a is the atmospheric temperature.
e finite element method (FEM) can be used to solve the above problem using the following equation [38]: where [C] e represents the capacitance matrix, [K t ] e represents the heat stiffness matrix, and R t e represents the total load heat vector.
Using the finite difference approximation, equation (6) was solved in the time domain numerically. After assembling the stiffness matrices, the solution can be obtained as the following equation [39]: where T { } b and R t b represent T { } and R t at time (b), respectively, and T { } a and R t a represent T { } and R t at time (a), respectively. According to the Galerkin method, θ equals 2/3. Equation (7) can be written in the following general form [38]: where ΔT { } is the temperature change with respect to time Δt at nodal points. e equation for evaluating the temperatures at the new time point is as follows [38]:  [31], the compliance function is given as follows: For mass concrete structures, the creep compliance function can be expressed as follows [34]: In addition, where t represents the calculated age, τ represents the loading age, E(τ) represents the transient elastic modulus, E 0 represents the ultimate elastic modulus, m equals the number of Kelvin elements, and r i , α, λ, A i , B i , and C i are the experimental fitting parameters [4]. In a typical time increment Δτ n (τ n−1 ⟶ τ n ), the formulas of the creep strain increment Δε c n and the stress increment Δσ n in 3D are expressed as the following equation [34]: where In these equations, τ n � (τ n−1 + τ n )/2, [Q] represents Poisson's ratio matrix, and μ represents Poisson's ratio [40].

Equations of ermal Creep Stress.
In a time increment, the total strain increment is given in the following equation [7]: where Δε n is the total strain increment, Δε e n is the elastic strain increment, and Δε T n is the thermal strain increment. e thermal strain increment is calculated by the following equation: where ζ is the linear expansion coefficient. Based on equations (14)- (17), η n is not related to the current stress increment. erefore, the formulas of the stress and strain increment can be obtained as follows [4]: where in which [D n ] is the viscoelastic modulus.

Geometric and Mesh Models of Ship Lock
Head. e structural optimization of ship lock heads involves determining the size of the lock heads. erefore, a geometric model of a ship lock head should be established. Along the direction of the water flow, the lock head is left-right symmetric, and thus, half of the structure was selected for parametric modelling. Figure 1(a) shows a typical 3D lock head model, and Figure 1(b) also contains the bedrock and three layers of backfilled gravel.
As shown in Figure 2, the design parameters of the bottom plate are defined as X bi (i � 1-10), Y bi (i � 1-10), and Z bi (i � 1-3) in three dimensions. e coordinate origin is the point O. Figure 3 shows a typical section of the second-stage concrete. e design parameters include the following: R si (i � 1-2) are the arc radii, θ si (i � 1-7) are the flip angles, and S si (i � 1-5) are the oblique length parameters. Figure 4 shows the design of a typical section of the corridor layer. Combining the design of the bedrock and backfilled gravel, the design of the overall geometric model (as shown in Figure 1(b)) requires 170 parameters.
In the FEM simulation, manually meshing is a timeconsuming task. erefore, an automated gridding program written in Python was applied in ABAQUS. e workflow of this program was as follows. Using design parameters, the three structures (as shown in Figure 1(b)) were divided into regular bulks. Meshes of the regular bulks were generated by calling the meshing tool of ABAQUS. During this process, each bulk was called by the corresponding design parameters. Figure 5, the design variables cover 34 parameters of the corridor layer and the empty-box layer, which are denoted as x 1 , . . ., x 12 , y 1 , . . ., y 7 , and z 1 , . . ., z 15 in the x, y, and z directions, respectively. e coordinate origin of the parametric design is the point O.

Design Loads.
e ship lock head is affected by the following loads during the construction period.

Self-Weight.
e self-weight of a ship lock head includes the weight of the concrete and structures fixed to the lock head, such as the lock gate. Figure 1(a), the ship lock head is a gravity-type structure. erefore, the gravel backfill pressure can be calculated based on Rankine's theory [41]:

Gravel Backfill Pressure. As shown in
where E a is the horizontal unit active backfill pressure, c z is the unit weight of the backfill, h z is the thickness of the backfill, q z is the vertical surcharge pressure on the backfill surface, c a is the cohesion, K a is the active pressure coefficient of the backfill, and K a is the internal friction angle of the backfill.

Design Live Load.
In this study, the design live load was assumed to be 5 kPa, as specified in a previous publication [41].

Temperature
Load. e heat of hydration is time dependent and can be expressed by the following equation [34]: where Q(τ) is the accumulated heat at time τ, Q ∞ is the final value of the cement hydration heat, and A ∞ and B ∞ are the experimental fitting parameters.

Constraints.
ere are three types of constraints for optimizing a ship lock head to meet the engineering safety requirements: Geometric Constraints. To guarantee that the finite element program can automatically generate FEM meshes during optimization, design variables, as shown in Section 3.2, must be provided with value ranges. Stress Constraints. According to the Code for Design of Hydraulic Structures of Ship Locks, the principal stresses should satisfy σ 1 ≤ [σ 1 ] and σ 3 ≤ [σ 3 ], where σ 1 is the first principal stress, σ 3 is the third principal stress, [σ 1 ] is the allowable value of the principal tensile stress, and [σ 3 ] is the allowable value of the principal compressive stress. Stability Constraints. During the construction period, stability constraints contain antisliding stability and antioverturning stability. e safety factor of the antisliding stability constraint is as follows [41]: where K s represents the safety factor of the antisliding stability, f represents the friction coefficient, V represents the total sum of loads along the normal direction of the sliding surface, H represents the total sum of loads along the tangent direction of the sliding surface, and [K s ] represents the allowable value of the safety factor. e safety factor of the antioverturning stability constraint is as follows [41]: where K m represents the safety factor of the antioverturning stability, M R represents the total sum of steady moments on the cross section of the calculation, M m represents the total sum of overturning moments on the cross section of the calculation, and [K m ] represents the allowable value of the safety factor.

Objective Function.
In this study, the goal was to minimize the volume of concrete such that the stress and stability requirements were satisfied. e following equation defines the objective function m(x): where x � (x 1 , . . ., x 12 , y 1 , . . ., y 7 , z 1 , . . ., z 15 ) represents the updating parameters, v 0 represents the volume of the initial ship lock head, and v(x) represents the volume of the optimum lock head. e optimization problem proposed in this study can be summarized as a problem of finding the objective function to minimize. If a design does not satisfy the constraints (as shown in Section 3.4), the objective function of the design must be set to a very large number as a penalty to the optimization process so as to greatly reduce the impact of this scheme on the optimization search and improve the optimization efficiency. A penalized objective function is applied, which is defined as follows: Ship lock head The first layer of gravel The second layer of gravel The third layer of gravel Bedrock (b) Figure 1: Geometric models of (a) the ship lock head and (b) the ship lock head-bedrock-backfilled gravel system.  GAs are heuristic algorithms [42] that simulate the theory of natural selection in evolution proposed by Charles Darwin. A GA is an evolutionary optimization method [43]. It starts by converting initial solutions (design variables) to a population of chromosomes. e initial solutions are selected randomly in the ranges of values for the design variables. Selection, crossover, and mutation operators are the three basic genetic operators. ese operators improve initial parent solutions to generate better solutions. Briefly, in the selection stage, the probability of each chromosome being selected is determined by calculating the objective function (as shown in Section 3.5). e crossover and mutation operations are then applied to selected chromosomes to generate a better offspring group. Finally, the optimization procedure is terminated by reaching the stopping criterion. In this study, each chromosome represents a set of solutions for the design variables. e number of design variables is expressed as parameter N, and each design variable is Figure 3: Typical section of the second-stage concrete. Figure 4: Typical section of the corridor layer.
represented by a gene chain composed of four genes. erefore, the total number of genes is 4N. A chromosome with three design variables is displayed in Figure 6(a).

Binary Encoding and Decoding.
e value of each design variable is represented by a binary string in the optimization calculation. A modelling program was written in Python to complete the modelling work in ABAQUS. Because the design variables in parametric modelling must be real numbers, binary digits should be decoded into decimal numbers. ey are interconverted using the following equation: Furthermore, the following constraint must be satisfied: where X is the value of each design variable, X min is the minimum value of each design variable, F is the number of chromosome genes used for each design variable, where in this study, F � 4, I is the decimal number of each design variable, X max is the maximum value of each design variable, a j is the number of the jth gene (0 or 1), and Y is the parameter increment of each design variable.

Population Initialization.
e initial population is generated randomly after the binary coding. A two-dimensional array W is used to represent the initial population. e chromosome of a set of design variables is a row object. Meanwhile, the population size is the number of corresponding columns. W is expressed as follows: where g is the population size and p is the chromosome length, namely, the total number of genes in a chromosome. In this study, p � 4N.

Genetic Operators
Selection Operator. Selecting elite individuals to form new populations is the role of the selection operator. In this study, the selection operator is based on the roulette wheel selection method. As shown in Figure 6(b), the wheel represents the whole population from which the selection is made. e area of each individual is positively correlated with the probability of being selected.
is selection probability can be calculated as follows: where J i is the selection possibility of each individual and F i is the objective function value of each individual. Accordingly, the roulette wheel selection method consists of three steps. First, the selection possibility of each individual is calculated. Second, random numbers are generated and arranged from the smallest to the  Mathematical Problems in Engineering largest. Finally, the individuals greater than the random numbers are kept by comparing each pair of random numbers and selection possibilities. Crossover Operator. Generating two new chromosomes by exchanging genes between a pair of original chromosomes is the role of the crossover operator. e crossover probability p c is typically between 0.4 and 0.9. In this study, a two-point crossover was adopted. As shown in Figure 6(c), genes of the two chromosomes between two randomly selected crossover points are exchanged, and two new chromosomes are generated. Mutation Operator. e mutation operator increases the population diversity. As shown in Figure 6(d), the bitflip mutation operator is adopted. A 0.8% mutation rate p t is used. In the chromosome, the mutation operator can change one gene by randomly producing a minor perturbation. A new chromosome is then generated.

Concept of BB-BC.
e BB-BC algorithm consists of two stages: A big bang step and a big crunch step [44]. In the first phase, some candidate solutions are randomly generated to cover the entire search space. In this step, each candidate's objective function value is calculated according to equation (26). In the second phase, the big crunch is a contraction operation that uses the objective function value and the current position of each design variable to calculate the "centre of mass." e term mass is actually the inverse of the  Mathematical Problems in Engineering objective function value. e centre of mass is calculated by the following equation [17]: where x (j,G) i represents the i th design variable of candidate j in iteration G.
After the big crunch step, the big bang step of the next iteration can be calculated using the following equations: where X (j,G+1) i represents the new position of each design variable, k j represents a random number, n 1 represents a parameter that limits the size of the search space, X imax represents the maximum value of each design variable, and X imin represents the minimum value of the i th design variable. e above two steps are repeated until the stopping criteria are met.

Concept of HBB-BC.
Although the BB-BC algorithm has good performance in exploitation, it easily becomes trapped in local optimal solutions. e HBB-BC algorithm not only utilizes the "centre of mass" but also uses the global best solution X gbest(G) i and the best solution of the G th iteration X lbest(j,G) i to generate a new group of candidates in the big bang step [17]. Equation (33) is replaced by the following equation [45]: where n 2 and n 3 are the parameters that adjust the influence of the global and local optimal solutions for the next population. e value of X (j,G+1) i is still obtained by equations (34)-(36).

Termination Criteria.
In the optimization procedure, the search continues until the termination criterion is met [46]. e stopping criterion of this study is the maximal number of assigned iterations G max .

Optimization Procedure.
e optimization procedure of a ship lock head was realized by a self-developed Python script that was imported into ABAQUS. An ABAQUS user subroutine (HETVAL) was developed using equation (23) to calculate the heat of hydration. Meanwhile, to calculate the thermal creep stress of concrete, an ABAQUS user subroutine (CREEP) was also developed accordingly using equations shown in Section 2.2.

GA.
e optimization procedure is given as follows: (1) A two-dimensional array was generated, which was the initial population of 34 design variables. e chromosomes represent the design variables.
(2) e binary digits of the initial population were decoded into decimal values of 34 design variables. (3) Geometric and mesh models of a ship lock head were automatically established. e material properties, design loads, and boundary conditions were then automatically applied. (4) e heat of hydration was calculated using equation (23). e temperature field was calculated using the equations shown in Section 2.1. (5) ermal creep stress was calculated using the solutions of the temperature field and the equations described in Section 2.2.
(6) e constraints defined in Section 3.4 were checked. If the result of the ship lock head satisfied all the constraints, the solution of the objective function value was determined using equation (26). Otherwise, the objective function was obtained using equation (27). e flowchart summarizing all the steps is illustrated in Figure 7(a).

HBB-BC and BB-BC.
e optimization procedures of HBB-BC and BB-BC algorithms were similar. ey are shown in Figure 7(b).

Basic Information of Ship Lock Head.
In this study, a ship lock head was presented, which was established on a rock foundation. It had a height of 12.3 m and a width of 53.8 m.
e ranges of 34 design variables are listed in Table 1.

Mathematical Problems in Engineering 9
A creep model was adopted for this concrete structure. e elastic modulus of the concrete is as follows: e creep function is as follows: e other material properties of concrete are summarized in Table 2, and the properties of the bedrock and backfilled gravel are also listed in Table 2.
e total duration of the project was 369 days. As a longterm project, the detailed construction schedule is shown in Table 3. e initial temperature of the bedrock was 18.8°C, which was the multiyear average value. Subsequently, the other initial temperatures of each section are also listed in Table 3.
For concrete, the heat of hydration can be obtained as follows: e air temperatures during the construction period were obtained using the following fit equation: e GA parameters used in this project were set as follows. e population size g was 30, and the chromosome length p was 136. e maximum generation G max was 25. e crossover probability p c was 0.8, and the mutation probability p t was 0.008. e HBB-BC and BB-BC parameters are shown as follows: k j represents a random number between [−1, 1], n 1 � 1, n 2 � 0.4, and n 3 � 0.8 [47].

Optimization Results.
For this project, the optimization processes (as shown in Section 4.5) were completed using a workstation, which consisted of two fourteencore CPUs (Xeon E5-2680 v4, Intel Corporation). For 50 runs of the GA, BB-BC, and HBB-BC algorithms, the evolutionary optimization process of each algorithm's best design is presented in Figure 8. e optimization algorithms led to different results. e BB-BC and GA algorithms converged rapidly but fell into local optima. e HBB-BC algorithm produced the best design. e optimal values of 34 design variables are presented in Table 1. Table 4 compares the ship lock head's initial volume and the optimum design obtained by each algorithm.
e HBB-BC algorithm performed better in searching for the optimal solution. e structure volume was reduced by 9.4%. Figure 9 displays the effect of various population sizes g on the HBB-BC algorithm's convergence history during 25 iterations. e population sizes were 20, 25, 30, 35, and 40. When g was 20, the objective function value was stable at 0.908 after 17 iterations. Due to the small population sizes, the search failed to obtain the optimal solution. When the population size was 25, the objective function value converged to the optimal solution of 0.9065 at the 18 th iteration. e convergence rate was faster than that when the population size was 30. When the population sizes were 35 and 40, the objective function values did not converge at the 25 th iteration. is confirmed that due to the large population size, the efficiency of convergence to the optimal solution was relatively low.
is would result in a waste of

Analysis for Temperature Field of the Lock Head.
ree feature points are identified in Figure 10, which were used to analyse the temperature and stress changes caused by structural optimization during the construction period. To intuitively understand each construction step's influence, the three points were located at lateral walls of the bottom plate, corridor, and empty-box layers. Figure 11(a) shows the temperature changes of selected points during the construction period. e temperature increased rapidly when pouring the corridor and empty-box layers. e subsequent temperature changes almost corresponded to the change in the air temperature. e temperature rise reached 18°C, while the temperature rise was approximately 10.5°C when pouring the bottom plate. is result suggests that the implementation of temperature control measures should give priority to the thin-walled sections during construction.
Compared with the initial design, Figure 11(b) shows that the bottom plate's temperature field was virtually unaffected by the size changes of the empty-box layer and the     corridor layer. As shown in Figure 11(c), at the corridor layer, the optimum structure's maximum temperature decreased by 2.5°C. e thinning of the lateral walls would increase the rate of heat dissipation. Figure 11(d) shows that structural optimization significantly affected the empty-box layer. e maximum temperature of the empty-box layer decreased by 4.3°C. us, the structural optimization helped to reduce the temperature rise when pouring the concrete. Figure 12(a) shows the calculated maximum principle stresses of selected points during the construction period. e tensile stress was small after taking creep behaviour into account when pouring the concrete. e tensile stress then increased to the maximum value with the increase in the concrete elastic modulus. us, the stress field of the ship lock head was mainly affected by the thermal stress. e creep model could reflect the actual stress field. e tensile stresses of the bottom plate and the corridor layer reached 1.2 MPa, which was much larger than the value with only the empty-box layer. Comparing Figures 11(a) and 12(a), the tensile stress developed to a maximum value when the temperature dropped to the minimum value. It was concluded that temperature control measures should be applied during winter to avoid thermal cracks.

Analysis for Stress Field of Lock Head.
As shown in Figure 12(b), compared with the initial design, the bottom plate's tensile stress increased by 0.03 MPa due to the reduced volume of the corridor and the empty-box layers. Figure 12(c) shows that the optimum design reduced the corridor layer's tensile stress during the construction period. e maximum value could be reduced by 0.28 MPa. Figure 12(d) shows that the optimum design decreased the corridor layer's tensile stress when pouring the concrete. Due to the reduction of the structure self-weight, the tensile stress was subsequently larger than that of the initial design with the increase in the concrete elastic modulus. For the emptybox layer, the maximum values could be increased by 0.07 MPa. In general, structural optimization helped to decrease the tensile stress significantly when pouring concrete. However, for the bottom plate and the emptybox layer, the reduction of the structure's self-weight will have some increasing effect on the tensile stress in the subsequent construction period.

Conclusions
In this study, the structural optimization of a ship lock head during the construction period was considered.
ree optimization procedures based on the GA, BB-BC, and HBB-BC algorithms were developed. e optimization process aimed to determine the minimum volume of concrete. irty-four design variables of the corridor layer and the empty-box layer were chosen as updating parameters in the optimization processes. Based on the calculation results of the case study, the HBB-BC algorithms could obtain a better solution. e procedure comprised the calculation of the concrete temperature and stress fields, considering the time-varying properties of concrete. A set of implicit recursive equations proposed by Zhu were adopted to calculate the thermal creep stress. e storage of the stress history could be avoided, thereby reducing the memory consumption during the calculation process.
rough the analysis of the temperature and stress fields of the lock head, the following important conclusions are summarized: (1) e temperature increased rapidly in the thin-walled sections (such as the corridor and empty-box layers) when pouring the concrete. (2) Structural optimization helped to reduce the temperature rise when pouring the concrete, especially in the empty-box layer. (3) e stress field of the ship lock head was mainly affected by thermal stress. e tensile stress of the lock head was low at early ages. It subsequently increased with the improvement of the elastic modulus. e tensile stresses of the bottom plate and the corridor layer were at high levels in the winter. e linear viscoelastic creep model could reflect the actual stress field's variation rule of the ship lock head during the construction period. (4) Structural optimization helped to decrease the tensile stress previously when pouring concrete. In particular, at the corridor layer, it was around a 50% decrease. e reduction of the structure's self-weight had an increasing effect on the tensile stress in the subsequent construction period for the bottom plate and the empty-box layer.
Based on these four conclusions, it is appropriate to introduce creep behaviour into structural optimization during the construction period. Future work should consider solar radiation in the calculation of the mass concrete structure's temperature field.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.