Asphalt concrete panel structure multi-objective optimization method based on CA-MOEA algorithm

Abstract

The invention discloses an asphalt concrete panel structure multi-objective optimization method based on a CA-MOEA algorithm, and the method specifically comprises the following steps: 1, determining an optimization problem, and determining an optimization objective, optimization parameters and constraint conditions; step 2, generating a parameter combination about optimization parameters based on a Box-Behnken test design method; step 3, establishing a finite element numerical model, obtaining response values under each combination through finite element numerical simulation, and constructing a parameterized prediction model of the safety indexes; step 4, solving the model by using a CA-MOEA algorithm, and obtaining a Pareto optimal solution set representing the optimal tradeoff relationship; and 5, selecting a final scheme from the Pareto optimal solution set according to actual engineering preference. The method aims at efficiently and automatically obtaining the Pareto optimal solution set of economy and safety of the asphalt concrete panel structure parameters, and scientific decision support is provided for engineering design.

Description

Technical Field

[0001] This invention belongs to the technical field of water conservancy and hydropower engineering and civil engineering, specifically involving a multi-objective optimization method for asphalt concrete panel structures based on the CA-MOEA algorithm. Background Technology

[0002] Asphalt concrete panels are a preferred type of seepage prevention solution for pumped storage power stations due to their excellent seepage control, strong adaptability to deformation, and ease of construction. However, due to complex terrain conditions, there is often a cut-fill boundary zone at the bottom of the reservoir. Uneven settlement and deformation of the bedrock and backfill in this zone can easily cause significant tensile and shear strain in the panels, even leading to cracking. The structural parameters of the panels directly determine the seepage prevention safety, economy, and long-term durability of the project. Therefore, optimizing the panel structural design parameters is crucial to ensuring the safety of the panel structure. In recent years, although there has been research on the structural morphology of asphalt concrete panels for rockfill dams, there is currently a lack of scientifically sound and reasonable optimization design methods for asphalt concrete panel structures in pumped storage power station reservoirs. The design of panel structural parameters often relies on empirical formulas, recommended values ​​from specifications, or finite element numerical simulations, and adjustments are made through trial calculations. Therefore, it is necessary to conduct research on optimization design methods for asphalt concrete panel structures at the reservoir bottom. Summary of the Invention

[0003] The purpose of this invention is to provide a multi-objective optimization method for asphalt concrete panel structures based on the CA-MOEA algorithm, which aims to efficiently and automatically obtain the Pareto optimal solution set of the economic and safety parameters of asphalt concrete panel structures, and provide scientific decision support for engineering design.

[0004] The technical solution adopted in this invention is a multi-objective optimization method for asphalt concrete panel structures based on the CA-MOEA algorithm, which is implemented according to the following steps: Step 1: Define the optimization problem, and clarify the optimization objective, optimization parameters, and constraints; Step 2: Generate parameter combinations for optimization parameters based on the Box-Behnken experimental design method; Step 3: Establish a finite element numerical model, obtain the response values ​​under each combination through finite element numerical simulation, and construct a parameterized prediction model for safety indicators; Step 4: Solve the model using the CA-MOEA algorithm to obtain the Pareto optimal solution set representing the optimal trade-off relationship; Step 5: Select the final solution from the Pareto optimal solution set based on actual engineering preferences.

[0005] The invention is further characterized in that, Step 1 is implemented in the following steps: To address the need for multi-objective optimization of the economy and safety of asphalt concrete panel cut-fill interface structures, the system defines the set of design variables. b, h, i and constraint space, b Indicates the increased width of the panel. h This indicates the thickness of the panel. i Indicates the excavation slope. To represent the compaction density, we establish an economic objective function based on the increased panel thickness and a safety performance objective function based on the tensile strain index. The specific structure of the cut-fill interface zone of the asphalt concrete panel is as follows: from top to bottom, it consists of a sealing layer, an impermeable layer, a trapezoidal thickened zone, a leveling cementing layer, a crushed stone drainage cushion layer, and the bottom layer is bedrock and backfill area. The structure of the cut-fill interface zone of the asphalt concrete panel involves four key parameters, therefore, the following parameters are selected: b, h, i Four parameters are used as structural optimization design variables for the cut-fill interface area of ​​asphalt concrete panels. For the optimized deformation geometry parameters, the panel thickening width is determined through relevant engineering design specifications and engineering experience. b and panel thickness h The range of values ​​is determined by statistically analyzing the excavation slope at the cut-fill boundary of existing pumped storage power stations both domestically and internationally, and by appropriately expanding the excavation slope ratio to determine the appropriate value. i Based on the "Design Code for Rolled Earth-Rock Dams", the range of compaction density values ​​is obtained, and the range of optimized parameter values ​​is shown in Table 1. Table 1 Range of Optimization Parameters

[0006] The maximum compressive strain, maximum tensile strain, and maximum settlement gradient were selected as stress-deformation constraints. Based on extensive test results and engineering design experience of asphalt concrete materials, relevant specifications stipulate that the maximum compressive strain of asphalt concrete panels should be <0.08%, the maximum tensile strain should be <0.1%, and the maximum settlement gradient should be <0.1%.

[0007] In step 1, based on the structure of the cut-fill interface zone of the asphalt concrete panel, the geometric mapping relationship between the volume of the thickened area of ​​the panel and the structural optimization parameters is analyzed to construct an explicit expression between the volume of the thickened area of ​​the panel and the structural parameters. The specific expression is as follows: (1) In the formula: V Optimize the volume of the asphalt concrete panel; L The length of the asphalt concrete panel at the bottom of the reservoir around the cut-and-fill junction; b Increase the width of the panel; h To increase the thickness of the panel, i The slope for excavation at the junction of cut and fill.

[0008] In step 1, the maximum tensile strain of the panel is used as the target for optimizing the structural safety of the panel. Explicit correlations between the tensile strain safety index and the optimization parameters are constructed using the response surface methodology. (2) In the formula: E For tensile strain safety indicators; b Increase the width of the panel; h Increase the thickness of the panel; i The excavation slope ratio for the cut-fill boundary area; This represents the compaction density of the backfill area.

[0009] Step 2 is implemented in the following steps: The following parameters were selected for optimization: panel thickening width, panel thickening thickness, excavation slope ratio at the cut-fill junction, and backfill compaction density: Table 2 Results of Structural Optimization Tests

[0010] The Box-Behnken experimental design method generated 29 sets of parameter combinations for the four optimization design variables, as shown in Table 2 above.

[0011] Step 3 is implemented in the following steps: A refined finite element numerical model of the upper reservoir basin was established. The model defines the positive x-axis as the direction downstream of the water flow, the positive y-axis as the direction perpendicular to the water flow pointing to the left bank, and the positive z-axis as the vertical direction pointing upwards. To ensure the model's computational domain covers the main structures of the dam and its surrounding soil and rock mass, and to minimize the influence of the boundary on the calculation results of the reservoir basin's cut-fill interface, the model's computational range extends upwards 170-178 m along the water flow direction from the upper boundary of the reservoir basin to 123-131 m downwards at the downstream dam slope toe. Along the axial direction, it extends 226-234 m at the left and right bank boundaries of the reservoir basin, and downwards 196-204 m at the bottom of the dam body. The model was meshed using tetrahedral four-node C3D4 elements, containing a total of 489,358 nodes and 123,842 elements. Fixed constraints were applied to the bottom boundary of the model, and corresponding normal constraints were applied to the four sides of the model. Gravity loads were applied to the overall model to simulate the effect of the gravity field. The upstream reservoir water pressure was simulated as the surface force on the panel, and applied along the gradient of the panel surface according to the hydrostatic pressure formula to simulate the reservoir water pressure borne by the seepage barrier. The dam filling process was simulated in 22 analysis steps, and the water storage process was simulated in 5 analysis steps, for a total of 27 analysis steps. The maximum settlement deformation gradient, maximum compressive strain, and maximum tensile strain response values ​​of the panel under each combination were obtained through finite element numerical simulation. A second-order polynomial prediction model between the maximum compressive strain, maximum tensile strain, maximum settlement deformation gradient, and optimized parameters was established based on the response surface methodology. After variance analysis and model credibility verification, a parameterized prediction model for the safety index was constructed. The objective function for the panel tensile stress safety index is shown in the following equation: (3) In the formula: E For tensile strain safety indicators; b Increase the width of the panel; h Increase the thickness of the panel; i The excavation slope ratio for the cut-fill boundary area; This represents the compaction density of the backfill area.

[0012] Step 4 is implemented in the following steps: Based on the determined objective function, the CA-MOEA algorithm is used to solve the multi-objective optimization problem. First, the feasible solution space of the multi-objective optimization mathematical model is constructed according to the mapping relationship of the panel stress-deformation constraint index. Then, the CA-MOEA algorithm is used to solve the model within the determined feasible solution space. The CA-MOEA algorithm adopts a bottom-up hierarchical clustering strategy. First, each data point is treated as an independent cluster. Then, two clusters are merged according to the Ward link criterion until all clusters are merged into one. The Ward link criterion minimizes the sum of squared errors within the same group while maximizing the errors between different groups. The sum of squared errors is equivalent to the distance metric d(r, s) calculated by equation (4): (4) In the formula: and The center of classes r and s; for and The Euclidean distance between them; and Let be the number of elements in classes r and s; Assuming there are 6 candidate solutions A, B, C, D, E, and F in the target space, in the initial stage, each individual is regarded as an independent cluster, and the coordinates of its cluster center directly correspond to the coordinates of the individual in the target space, thus forming 6 initial clusters. Subsequently, based on the distance metric defined by formula (4) d ( r，s ), calculate the sum of squared errors between every two clusters, and obtain the sum of squared errors between 15 cluster pairs, assuming d If (A,B) is the smallest, then A and B are merged into a new cluster H, and the center of H is calculated using the following formula: (5) In the formula: Representing cluster centers a In the m Coordinate values ​​in each target dimension For class a The Middle j The first individual m The component values ​​of each target space. n This represents the total number of individuals in the current class a. The specific operation process of the CA-MOEA algorithm is as follows: First, based on the specific parameters of the multi-objective problem to be solved, a random number of parameters of size is generated. N The initial parent population P The number of decision variables is d The number of targets is set to M Constraint B, where constraint B is the upper and lower bounds of the decision variable, is then used to simulate binary crossover SBX and polynomial mutation, respectively, generating variables of similar scale. N Based on the offspring population Q, an elite preservation mechanism is introduced to merge the parent population P and the offspring population Q into a population of size 2N. Subsequently, selection was made from this mixed population. N The best-performing individuals constitute the next generation of the parent population. P ; When performing environmental selection, the fast non-dominated ordination method is first used to assign individuals in the population that are not dominated by any other individual to the first non-dominated frontier. Next, individuals that are not dominated by other individuals are selected from the remaining population and placed on the non-dominated frontier. And so on, 2 from the merged population N Individuals are divided into L Non-dominated frontier , , ..., From the front arrive The process of gradually selecting solutions continues until N solutions are chosen. The process of generating offspring and selecting environments is repeated until the preset termination condition is met. Finally, the individuals selected from the last generation of the population will be the final set of solutions to the problem. The initialization parameters of the CA-MOEA algorithm are set to determine the attribute values ​​of the generated population particles during the initialization phase. The key parameter settings of the CA-MOEA algorithm are shown in Table 3. Table 3 CA-MOEA Algorithm Parameters

[0013] After iterative optimization using the CA-MOEA algorithm, 50 sets of candidate optimization parameters were obtained. When the tensile strain value is at a relatively low level, the panel thickening volume tends to be in a higher range. Conversely, when the panel thickening volume is at a relatively low value, the tensile strain value will be relatively high.

[0014] Step 5 is implemented in the following steps: Given that the optimal solution obtained by the multi-objective optimization algorithm is not unique, it is necessary to closely combine it with the actual engineering needs and select the best-fitting solution from the Pareto front as the optimal panel structure scheme. The optimal structure configuration scheme based on differentiated decision preferences is extracted from the Pareto non-dominated solution set using the linear weighted sum method, which includes the following two steps: First, eliminate the incomparability caused by the difference in dimensions of different objective functions by transforming the panel thickness volume value and the panel safety index tensile strain into a single objective function form that can be uniformly compared. The specific normalization process is shown in the following equation: (6) Secondly, using the linear weighting method, based on the importance of each objective in the project, corresponding weight coefficients are assigned to the panel thickness volume value and the panel safety index tensile strain. By linearly combining the normalized values, structural optimization schemes with different tendencies are obtained. The calculation formula of the linear weighting method is shown in the following formula: (7) In the formula: The value is a normalized value for the target value; , , These are the k-th target value, the minimum value, and the maximum value, respectively. Weighting coefficients for the thickness volume value of the panel; The weighting coefficient for the tensile strain value of the panel safety index; With a balanced and coordinated configuration W=(0.5,0.5), the panel structure optimization scheme with a balanced and coordinated configuration is obtained by following the above steps.

[0015] The beneficial effects of this invention are that the multi-objective optimization method for asphalt concrete panel structures based on the CA-MOEA algorithm, through finite element analysis and combined with the two-layer evolutionary framework of the CA-MOEA algorithm, significantly accelerates the optimization process and improves computational efficiency. The adaptive mechanism effectively avoids premature convergence, enhances global exploration and local exploitation capabilities, and can obtain a higher-quality Pareto optimal solution set. Algorithm parameters (such as crossover rate and mutation rate) and search range can be adaptively adjusted based on knowledge extracted during the evolution process. This algorithm has strong decision support capabilities, providing a series of optimal solutions that balance economy and safety, rather than a single solution, thus providing a more comprehensive scientific basis for engineering decisions. Attached Figure Description

[0016] Figure 1 This is a flowchart of the multi-objective optimization calculation method for asphalt concrete panel structure parameters based on the CA-MOEA algorithm of this invention; Figure 2 It is the asphalt concrete panel structure at the bottom of the Zhen'an pumped storage power station; Figure 3 It is a diagram of a three-dimensional finite element calculation model; Figure 4 This is a schematic diagram illustrating the principle of clustering in the target space using the CA-MOEA algorithm. Figure 5 This is a scatter plot of the Pareto front for the optimization problem of asphalt concrete panel structure obtained according to the method of the present invention. Detailed Implementation

[0017] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0018] This invention presents a multi-objective optimization method for asphalt concrete panel structures based on the CA-MOEA algorithm, combined with... Figure 1 The specific steps are as follows: Step 1: Define the optimization problem, and clarify the optimization objective, optimization parameters, and constraints; Step 1 is implemented in the following steps: To address the need for multi-objective optimization of the economy and safety of asphalt concrete panel cut-fill interface structures, the system defines the set of design variables. b, h, i and constraint space, b Indicates the increased width of the panel. h This indicates the thickness of the panel. i Indicates the excavation slope. To represent the compaction density, we establish an economic objective function based on the increased panel thickness and a safety performance objective function based on the tensile strain index. Figure 2 The asphalt concrete panel structure at the cut-fill boundary is as follows: from top to bottom, it consists of a sealing layer (2 mm), an impermeable layer (10 cm), and a trapezoidal thickened area (top width). b ,thickness h Slope i The structure consists of a leveling cementitious layer (10 cm), a crushed stone drainage cushion layer (60 cm), and the bottom layer is bedrock and backfill. The asphalt concrete panel excavation-fill interface structure involves four key parameters, therefore, [the following is selected]. b, h, i Four parameters are used as structural optimization design variables for the cut-fill interface area of ​​asphalt concrete panels. For the optimized deformation geometry parameters, the panel thickening width is determined through relevant engineering design specifications and engineering experience. b and panel thickness h The range of values ​​is determined by statistically analyzing the excavation slope at the cut-fill boundary of existing pumped storage power stations both domestically and internationally, and by appropriately expanding the excavation slope ratio to determine the appropriate value. i Based on the "Design Code for Rolled Earth-Rock Dams", the range of compaction density values ​​is obtained, and the range of optimized parameter values ​​is shown in Table 1. Table 1 Range of Optimization Parameters

[0019] The maximum compressive strain, maximum tensile strain, and maximum settlement gradient were selected as stress-deformation constraints. Based on extensive test results and engineering design experience of asphalt concrete materials, relevant specifications stipulate that the maximum compressive strain of asphalt concrete panels should be <0.08%, the maximum tensile strain should be <0.1%, and the maximum settlement gradient should be <0.1%.

[0020] In step 1, based on the structure of the cut-fill interface zone of the asphalt concrete panel, the geometric mapping relationship between the volume of the thickened area of ​​the panel and the structural optimization parameters is analyzed to construct an explicit expression between the volume of the thickened area of ​​the panel and the structural parameters. The specific expression is as follows: (1) In the formula: V Optimize the volume of the asphalt concrete panel; L The length of the asphalt concrete panel at the bottom of the reservoir around the cut-and-fill junction; b Increase the width of the panel; h To increase the thickness of the panel, i The slope for excavation at the junction of cut and fill.

[0021] The main cause of the asphalt concrete panel failure at the cut-fill interface in Step 1 was excessive tensile strain. Tensile strain is a comprehensive indicator suitable for evaluating the structural safety of asphalt concrete panels at the cut-fill interface. Therefore, the maximum tensile strain of the panel can be used as the optimization target for panel structural safety. Using response surface methodology, an explicit correlation between the tensile strain safety index and the optimization parameters is constructed: (2) In the formula: E For tensile strain safety indicators; b Increase the width of the panel; h Increase the thickness of the panel; i The excavation slope ratio for the cut-fill boundary area; This represents the compaction density of the backfill area.

[0022] Step 2: Generate parameter combinations for optimization parameters based on the Box-Behnken experimental design method; Step 2 is implemented in the following steps: The panel thickening width, panel thickening thickness, excavation slope ratio at the cut-fill junction, and backfill compaction density were selected as optimization parameters. Based on the Box-Behnken experimental design method, 29 sets of parameter combinations for the four optimization design variables were generated, as shown in Table 2. Table 2 Results of structural optimization tests

[0023] Step 3: Establish a finite element numerical model, obtain the response values ​​under each combination through finite element numerical simulation, and construct a parameterized prediction model for safety indicators; Step 3 is implemented in the following steps: A refined finite element numerical model of the upper reservoir basin was established. The model defines the positive x-axis as the direction downstream of the water flow, the positive y-axis as the direction perpendicular to the water flow pointing to the left bank, and the positive z-axis as the vertical direction pointing upwards. To ensure the model's computational domain covers the main structures of the dam and its surrounding soil and rock mass, and to minimize the influence of the boundary on the calculation results of the reservoir basin's cut-fill interface, the model's computational range extends upwards 170-178 m along the water flow direction from the upper boundary of the reservoir basin to 123-131 m downwards at the downstream dam slope toe. Along the axial direction, it extends 226-234 m at the left and right bank boundaries of the reservoir basin, and downwards 196-204 m at the bottom of the dam body. The model was meshed using tetrahedral four-node C3D4 elements, containing a total of 489,358 nodes and 123,842 elements. Fixed constraints were applied to the bottom boundary of the model, and corresponding normal constraints were applied to the four sides of the model. Gravity loads were applied to the overall model to simulate the effect of the gravity field. The upstream reservoir water pressure was simulated as the surface force on the panel, and applied along the gradient of the panel surface according to the hydrostatic pressure formula to simulate the reservoir water pressure borne by the seepage barrier. The dam filling process was simulated in 22 analysis steps, and the water storage process was simulated in 5 analysis steps, for a total of 27 analysis steps. The maximum settlement deformation gradient, maximum compressive strain, and maximum tensile strain response values ​​of the panel under each combination were obtained through finite element numerical simulation. A second-order polynomial prediction model between the maximum compressive strain, maximum tensile strain, maximum settlement deformation gradient, and optimized parameters was established based on the response surface methodology. After variance analysis and model credibility verification, a parameterized prediction model for the safety index was constructed. The objective function for the panel tensile stress safety index is shown in the following equation: (3) In the formula: E For tensile strain safety indicators; b Increase the width of the panel; h Increase the thickness of the panel;i The excavation slope ratio for the cut-fill boundary area; This represents the compaction density of the backfill area.

[0024] Step 4: Solve the model using the CA-MOEA algorithm to obtain the Pareto optimal solution set representing the optimal trade-off relationship; Step 4 is implemented in the following steps: Based on the defined objective function, the CA-MOEA algorithm is used for multi-objective optimization. First, the feasible solution space of the multi-objective optimization mathematical model is constructed according to the mapping relationship of panel stress-deformation constraint indices. Then, within the defined feasible solution space, the algorithm strictly follows... Figure 4 The CA-MOEA algorithm is used to solve the model. The CA-MOEA algorithm adopts a bottom-up hierarchical clustering strategy. First, each data point is treated as an independent cluster. Then, two clusters are merged according to the Ward link criterion until all clusters are merged into one. The Ward link criterion minimizes the sum of squared errors within the same group while maximizing the errors between different groups. The sum of squared errors is equivalent to the distance metric calculated by equation (4). d ( r，s ): (4) In the formula: and For class r and s The center; for and The Euclidean distance between them; and For class r and s The number of elements in; Taking the bi-objective optimization problem shown in Figure 4 as an example, assuming there are 6 candidate solutions A, B, C, D, E, and F in the objective space, in the initial stage, each individual is regarded as an independent cluster, and the coordinates of its cluster center directly correspond to the coordinates of the individual in the objective space, thus forming 6 initial clusters. Subsequently, based on the distance metric defined by formula (4) d ( r，s ), calculate the sum of squared errors between every two clusters, and obtain the sum of squared errors between 15 cluster pairs, assuming d If (A, B) is the smallest, then A and B are merged into a new cluster. H In the middle, the center of H is calculated using the following formula: (5) In the formula: Representing cluster centers a In the m Coordinate values ​​in each target dimension For class a The Middle j The first individual m The component values ​​of each target space. n This represents the total number of individuals in the current class a. The specific operation process of the CA-MOEA algorithm is as follows: First, based on the specific parameters of the multi-objective problem to be solved, a random number of parameters of size is generated. N The initial parent population P The number of decision variables is d The number of targets is set to M Constraint B is the upper and lower bounds of the decision variable. Subsequently, simulated binary crossover (SBX) (Deb et al., 2014) is applied (crossover probability...). =1.0) and polynomial mutation (probability of mutation) =1 / d) The generation scale is also... N Based on the offspring population Q, an elite preservation mechanism is introduced to merge the parent population P and the offspring population Q into a population of size 2N. Subsequently, N individuals with the best performance were selected from this mixed population to form the next generation of parent population. P ; When performing environmental selection, the fast non-dominated ordination method is first used to assign individuals in the population that are not dominated by any other individual to the first non-dominated frontier. Next, individuals that are not dominated by other individuals are selected from the remaining population and placed on the non-dominated frontier. And so on, 2 from the merged population N Individuals are divided into L Non-dominated frontier , , ..., From the front arrive The process of gradually selecting solutions continues until N solutions are chosen. The process of generating offspring and selecting environments is repeated until the preset termination condition is met. Finally, the individuals selected from the last generation of the population will be the final set of solutions to the problem. To ensure efficient algorithm operation, initialization parameters for the CA-MOEA algorithm were set based on algorithm principles and empirical methods. The attribute values ​​of the generated population particles during the initialization phase were determined. The key parameter settings for the CA-MOEA algorithm are shown in Table 3. Table 3 CA-MOEA Algorithm Parameters

[0025] After iterative optimization using the CA-MOEA algorithm, 50 sets of candidate optimization parameters were obtained. The correspondence between the panel thickening volume and the tensile strain value of the safety index is as follows: Figure 5 As shown in the figure, the distribution of points on the Pareto front is relatively uniform, indicating good convergence of the algorithm and its ability to fully explore the feasible solution space and find representative non-dominated solutions. Simultaneously, it reveals a contradictory relationship between the increase in panel volume and the panel safety index; the two are unlikely to simultaneously reach their optimal state in the same optimization scheme. When the tensile strain value is at a low level, the panel thickening volume tends to be in a higher range. Conversely, when the panel thickening volume is at a low value, the tensile strain value will be relatively high.

[0026] Step 5: Select the final solution from the Pareto optimal solution set based on actual engineering preferences.

[0027] Step 5 is implemented in the following steps: Given that the optimal solution obtained by the multi-objective optimization algorithm is not unique, it is necessary to closely combine it with the actual engineering needs and select the best-fitting solution from the Pareto front as the optimal panel structure scheme. The optimal structure configuration scheme based on differentiated decision preferences is extracted from the Pareto non-dominated solution set using the linear weighted sum method, which includes the following two steps: First, eliminate the incomparability caused by the difference in dimensions of different objective functions by transforming the panel thickness volume value and the panel safety index tensile strain into a single objective function form that can be uniformly compared. The specific normalization process is shown in the following equation: (6) Secondly, using the linear weighting method, based on the importance of each objective in the project, corresponding weight coefficients are assigned to the panel thickness volume value and the panel safety index tensile strain. By linearly combining the normalized values, structural optimization schemes with different tendencies are obtained. The calculation formula of the linear weighting method is shown in the following formula: (7) In the formula: The value is a normalized value for the target value; , , These are the k-th target value, the minimum value, and the maximum value, respectively. Weighting coefficients for the thickness volume value of the panel; The weighting coefficient for the tensile strain value of the panel safety index; With a balanced and coordinated configuration W=(0.5,0.5), the panel structure optimization scheme with a balanced and coordinated configuration is obtained by following the above steps.

[0028] Example 1 This invention presents a multi-objective optimization method for asphalt concrete panel structures based on the CA-MOEA algorithm, combined with... Figure 1 The specific steps are as follows: Step 1: Define the optimization problem, and clarify the optimization objective, optimization parameters, and constraints; Step 2: Generate parameter combinations for optimization parameters based on the Box-Behnken experimental design method; Step 3: Establish a finite element numerical model, obtain the response values ​​under each combination through finite element numerical simulation, and construct a parameterized prediction model for safety indicators; Step 4: Solve the model using the CA-MOEA algorithm to obtain the Pareto optimal solution set representing the optimal trade-off relationship; Step 5: Select the final solution from the Pareto optimal solution set based on actual engineering preferences.

[0029] Example 2 This invention presents a multi-objective optimization method for asphalt concrete panel structures based on the CA-MOEA algorithm, combined with... Figure 1 The specific steps are as follows: Step 1: Define the optimization problem, and clarify the optimization objective, optimization parameters, and constraints; Step 1 is implemented in the following steps: To address the need for multi-objective optimization of the economy and safety of asphalt concrete panel cut-fill interface structures, the system defines the set of design variables. b, h, i and constraint space, b Indicates the increased width of the panel. h This indicates the thickness of the panel. i Indicates the excavation slope. To represent the compaction density, we establish an economic objective function based on the increased panel thickness and a safety performance objective function based on the tensile strain index. Figure 2 The asphalt concrete panel structure at the cut-fill boundary is as follows: from top to bottom, it consists of a sealing layer (2 mm), an impermeable layer (10 cm), and a trapezoidal thickened area (top width). b ,thickness h Slope i The structure consists of a leveling cementitious layer (10 cm), a crushed stone drainage cushion layer (60 cm), and the bottom layer is bedrock and backfill. The asphalt concrete panel excavation-fill interface structure involves four key parameters, therefore, [the following is selected]. b, h, i Four parameters are used as structural optimization design variables for the cut-fill interface area of ​​asphalt concrete panels. For the optimized deformation geometry parameters, the panel thickening width is determined through relevant engineering design specifications and engineering experience. band panel thickness h The range of values ​​is determined by statistically analyzing the excavation slope at the cut-fill boundary of existing pumped storage power stations both domestically and internationally, and by appropriately expanding the excavation slope ratio to determine the appropriate value. i Based on the "Design Code for Rolled Earth-Rock Dams", the range of compaction density values ​​is obtained, and the range of optimized parameter values ​​is shown in Table 1. Table 1. Range of Optimization Parameter Values

[0030] The maximum compressive strain, maximum tensile strain, and maximum settlement gradient were selected as stress-deformation constraints. Based on extensive test results and engineering design experience of asphalt concrete materials, relevant specifications stipulate that the maximum compressive strain of asphalt concrete panels should be <0.08%, the maximum tensile strain should be <0.1%, and the maximum settlement gradient should be <0.1%.

[0031] In step 1, based on the structure of the cut-fill interface zone of the asphalt concrete panel, the geometric mapping relationship between the volume of the thickened area of ​​the panel and the structural optimization parameters is analyzed to construct an explicit expression between the volume of the thickened area of ​​the panel and the structural parameters. The specific expression is as follows: (1) In the formula: V Optimize the volume of the asphalt concrete panel; L The length of the asphalt concrete panel at the bottom of the reservoir around the cut-and-fill junction; b Increase the width of the panel; h To increase the thickness of the panel, i The slope for excavation at the junction of cut and fill.

[0032] Step 2: Generate parameter combinations for optimization parameters based on the Box-Behnken experimental design method; Step 3: Establish a finite element numerical model, obtain the response values ​​under each combination through finite element numerical simulation, and construct a parameterized prediction model for safety indicators; Step 4: Solve the model using the CA-MOEA algorithm to obtain the Pareto optimal solution set representing the optimal trade-off relationship; Step 5: Select the final solution from the Pareto optimal solution set based on actual engineering preferences.

[0033] Example 3 This invention presents a multi-objective optimization method for asphalt concrete panel structures based on the CA-MOEA algorithm, combined with... Figure 1 The specific steps are as follows: Step 1: Define the optimization problem, and clarify the optimization objective, optimization parameters, and constraints; Step 1 is implemented in the following steps: To address the need for multi-objective optimization of the economy and safety of asphalt concrete panel cut-fill interface structures, the system defines the set of design variables. b, h, i and constraint space, b Indicates the increased width of the panel. h This indicates the thickness of the panel. i Indicates the excavation slope. To represent the compaction density, we establish an economic objective function based on the increased panel thickness and a safety performance objective function based on the tensile strain index. Figure 2 The asphalt concrete panel structure at the cut-fill boundary is as follows: from top to bottom, it consists of a sealing layer (2 mm), an impermeable layer (10 cm), and a trapezoidal thickened area (top width). b ,thickness h Slope i The structure consists of a leveling cementitious layer (10 cm), a crushed stone drainage cushion layer (60 cm), and the bottom layer is bedrock and backfill. The asphalt concrete panel excavation-fill interface structure involves four key parameters, therefore, [the following is selected]. b, h, i Four parameters are used as structural optimization design variables for the cut-fill interface area of ​​asphalt concrete panels. For the optimized deformation geometry parameters, the panel thickening width is determined through relevant engineering design specifications and engineering experience. b and panel thickness h The range of values ​​is determined by statistically analyzing the excavation slope at the cut-fill boundary of existing pumped storage power stations both domestically and internationally, and by appropriately expanding the excavation slope ratio to determine the appropriate value. i Based on the "Design Code for Rolled Earth-Rock Dams", the range of compaction density values ​​is obtained, and the range of optimized parameter values ​​is shown in Table 1. Table 1. Range of Optimization Parameter Values

[0034] The maximum compressive strain, maximum tensile strain, and maximum settlement gradient were selected as stress-deformation constraints. Based on extensive test results and engineering design experience of asphalt concrete materials, relevant specifications stipulate that the maximum compressive strain of asphalt concrete panels should be <0.08%, the maximum tensile strain should be <0.1%, and the maximum settlement gradient should be <0.1%.

[0035] In step 1, based on the structure of the cut-fill interface zone of the asphalt concrete panel, the geometric mapping relationship between the volume of the thickened area of ​​the panel and the structural optimization parameters is analyzed to construct an explicit expression between the volume of the thickened area of ​​the panel and the structural parameters. The specific expression is as follows: (1) In the formula: V Optimize the volume of the asphalt concrete panel; LThe length of the asphalt concrete panel at the bottom of the reservoir around the cut-and-fill junction; b Increase the width of the panel; h To increase the thickness of the panel, i The slope for excavation at the junction of cut and fill.

[0036] The main cause of the asphalt concrete panel failure at the cut-fill interface in Step 1 was excessive tensile strain. Tensile strain is a comprehensive indicator suitable for evaluating the structural safety of asphalt concrete panels at the cut-fill interface. Therefore, the maximum tensile strain of the panel can be used as the optimization target for panel structural safety. Using response surface methodology, an explicit correlation between the tensile strain safety index and the optimization parameters is constructed: (2) In the formula: E For tensile strain safety indicators; b Increase the width of the panel; h Increase the thickness of the panel; i The excavation slope ratio for the cut-fill boundary area; This represents the compaction density of the backfill area.

[0037] Step 2: Generate parameter combinations for optimization parameters based on the Box-Behnken experimental design method; Step 3: Establish a finite element numerical model, obtain the response values ​​under each combination through finite element numerical simulation, and construct a parameterized prediction model for safety indicators; Step 4: Solve the model using the CA-MOEA algorithm to obtain the Pareto optimal solution set representing the optimal trade-off relationship; Step 5: Select the final solution from the Pareto optimal solution set based on actual engineering preferences.

[0038] Example 4 This invention presents a multi-objective optimization method for asphalt concrete panel structures based on the CA-MOEA algorithm, combined with... Figure 1 The specific steps are as follows: Step 1: Define the optimization problem, and clarify the optimization objective, optimization parameters, and constraints; Step 1 is implemented in the following steps: To address the need for multi-objective optimization of the economy and safety of asphalt concrete panel cut-fill interface structures, the system defines the set of design variables. b, h, i and constraint space, b Indicates the increased width of the panel. h This indicates the thickness of the panel. i Indicates the excavation slope. To represent the compaction density, we establish an economic objective function based on the increased panel thickness and a safety performance objective function based on the tensile strain index. Figure 2 The asphalt concrete panel structure at the cut-fill boundary is as follows: from top to bottom, it consists of a sealing layer (2 mm), an impermeable layer (10 cm), and a trapezoidal thickened area (top width). b ,thickness h Slope i The structure consists of a leveling cementitious layer (10 cm), a crushed stone drainage cushion layer (60 cm), and the bottom layer is bedrock and backfill. The asphalt concrete panel excavation-fill interface structure involves four key parameters, therefore, [the following is selected]. b, h, i Four parameters are used as structural optimization design variables for the cut-fill interface area of ​​asphalt concrete panels. For the optimized deformation geometry parameters, the panel thickening width is determined through relevant engineering design specifications and engineering experience. b and panel thickness h The range of values ​​is determined by statistically analyzing the excavation slope at the cut-fill boundary of existing pumped storage power stations both domestically and internationally, and by appropriately expanding the excavation slope ratio to determine the appropriate value. i Based on the "Design Code for Rolled Earth-Rock Dams", the range of compaction density values ​​is obtained, and the range of optimized parameter values ​​is shown in Table 1. Table 1. Range of Optimization Parameter Values

[0039] The maximum compressive strain, maximum tensile strain, and maximum settlement gradient were selected as stress-deformation constraints. Based on extensive test results and engineering design experience of asphalt concrete materials, relevant specifications stipulate that the maximum compressive strain of asphalt concrete panels should be <0.08%, the maximum tensile strain should be <0.1%, and the maximum settlement gradient should be <0.1%.

[0040] In step 1, based on the structure of the cut-fill interface zone of the asphalt concrete panel, the geometric mapping relationship between the volume of the thickened area of ​​the panel and the structural optimization parameters is analyzed to construct an explicit expression between the volume of the thickened area of ​​the panel and the structural parameters. The specific expression is as follows: (1) In the formula: V Optimize the volume of the asphalt concrete panel; L The length of the asphalt concrete panel at the bottom of the reservoir around the cut-and-fill junction; b Increase the width of the panel; h To increase the thickness of the panel, i The slope for excavation at the junction of cut and fill.

[0041] The main cause of the asphalt concrete panel failure at the cut-fill interface in Step 1 was excessive tensile strain. Tensile strain is a comprehensive indicator suitable for evaluating the structural safety of asphalt concrete panels at the cut-fill interface. Therefore, the maximum tensile strain of the panel can be used as the optimization target for panel structural safety. Using response surface methodology, an explicit correlation between the tensile strain safety index and the optimization parameters is constructed: (2) In the formula: E For tensile strain safety indicators; b Increase the width of the panel; h Increase the thickness of the panel; i The excavation slope ratio for the cut-fill boundary area; This represents the compaction density of the backfill area.

[0042] Step 2: Generate parameter combinations for optimization parameters based on the Box-Behnken experimental design method; Step 2 is implemented in the following steps: The panel thickening width, panel thickening thickness, excavation slope ratio at the cut-fill junction, and backfill compaction density were selected as optimization parameters. Based on the Box-Behnken experimental design method, 29 sets of parameter combinations for the four optimization design variables were generated, as shown in Table 2. Table 2 Results of structural optimization tests

[0043] Step 3: Establish a finite element numerical model, obtain the response values ​​under each combination through finite element numerical simulation, and construct a parameterized prediction model for safety indicators; Step 4: Solve the model using the CA-MOEA algorithm to obtain the Pareto optimal solution set representing the optimal trade-off relationship; Step 5: Select the final solution from the Pareto optimal solution set based on actual engineering preferences.

[0044] Example 5 This invention presents a multi-objective optimization method for asphalt concrete panel structures based on the CA-MOEA algorithm, combined with... Figure 1 The specific steps are as follows: Step 1: Define the optimization problem, and clarify the optimization objective, optimization parameters, and constraints; Step 1 is implemented in the following steps: To address the need for multi-objective optimization of the economy and safety of asphalt concrete panel cut-fill interface structures, the system defines the set of design variables. b, h, i and constraint space, b Indicates the increased width of the panel. h This indicates the thickness of the panel. iIndicates the excavation slope. To represent the compaction density, we establish an economic objective function based on the increased panel thickness and a safety performance objective function based on the tensile strain index. Figure 2 The asphalt concrete panel structure at the cut-fill boundary is as follows: from top to bottom, it consists of a sealing layer (2 mm), an impermeable layer (10 cm), and a trapezoidal thickened area (top width). b ,thickness h Slope i The structure consists of a leveling cementitious layer (10 cm), a crushed stone drainage cushion layer (60 cm), and the bottom layer is bedrock and backfill. The asphalt concrete panel excavation-fill interface structure involves four key parameters, therefore, [the following is selected]. b, h, i Four parameters are used as structural optimization design variables for the cut-fill interface area of ​​asphalt concrete panels. For the optimized deformation geometry parameters, the panel thickening width is determined through relevant engineering design specifications and engineering experience. b and panel thickness h The range of values ​​is determined by statistically analyzing the excavation slope at the cut-fill boundary of existing pumped storage power stations both domestically and internationally, and by appropriately expanding the excavation slope ratio to determine the appropriate value. i Based on the "Design Code for Rolled Earth-Rock Dams", the range of compaction density values ​​is obtained, and the range of optimized parameter values ​​is shown in Table 1. Table 1. Range of Optimization Parameter Values

[0045] The maximum compressive strain, maximum tensile strain, and maximum settlement gradient were selected as stress-deformation constraints. Based on extensive test results and engineering design experience of asphalt concrete materials, relevant specifications stipulate that the maximum compressive strain of asphalt concrete panels should be <0.08%, the maximum tensile strain should be <0.1%, and the maximum settlement gradient should be <0.1%.

[0046] In step 1, based on the structure of the cut-fill interface zone of the asphalt concrete panel, the geometric mapping relationship between the volume of the thickened area of ​​the panel and the structural optimization parameters is analyzed to construct an explicit expression between the volume of the thickened area of ​​the panel and the structural parameters. The specific expression is as follows: (1) In the formula: V Optimize the volume of the asphalt concrete panel; L The length of the asphalt concrete panel at the bottom of the reservoir around the cut-and-fill junction; b Increase the width of the panel; h To increase the thickness of the panel, i The slope for excavation at the junction of cut and fill.

[0047] The main cause of the asphalt concrete panel failure at the cut-fill interface in Step 1 was excessive tensile strain. Tensile strain is a comprehensive indicator suitable for evaluating the structural safety of asphalt concrete panels at the cut-fill interface. Therefore, the maximum tensile strain of the panel can be used as the optimization target for panel structural safety. Using response surface methodology, an explicit correlation between the tensile strain safety index and the optimization parameters is constructed: (2) In the formula: E For tensile strain safety indicators; b Increase the width of the panel; h Increase the thickness of the panel; i The excavation slope ratio for the cut-fill boundary area; This represents the compaction density of the backfill area.

[0048] Step 2: Generate parameter combinations for optimization parameters based on the Box-Behnken experimental design method; Step 2 is implemented in the following steps: The panel thickening width, panel thickening thickness, excavation slope ratio at the cut-fill junction, and backfill compaction density were selected as optimization parameters. Based on the Box-Behnken experimental design method, 29 sets of parameter combinations for the four optimization design variables were generated, as shown in Table 2. Table 2 Results of structural optimization tests

[0049] Step 3: Establish a finite element numerical model, obtain the response values ​​under each combination through finite element numerical simulation, and construct a parameterized prediction model for safety indicators; Step 4: Solve the model using the CA-MOEA algorithm to obtain the Pareto optimal solution set representing the optimal trade-off relationship; Step 4 is implemented in the following steps: Based on the defined objective function, the CA-MOEA algorithm is used for multi-objective optimization. First, the feasible solution space of the multi-objective optimization mathematical model is constructed according to the mapping relationship of panel stress-deformation constraint indices. Then, within the defined feasible solution space, the algorithm strictly follows... Figure 4 The CA-MOEA algorithm is used to solve the model. The CA-MOEA algorithm adopts a bottom-up hierarchical clustering strategy. First, each data point is treated as an independent cluster. Then, two clusters are merged according to the Ward link criterion until all clusters are merged into one. The Ward link criterion minimizes the sum of squared errors within the same group while maximizing the errors between different groups. The sum of squared errors is equivalent to the distance metric calculated by equation (4). d ( r，s ): (4) In the formula: and For class r and s The center; for and The Euclidean distance between them; and For class r and s The number of elements in; Taking the bi-objective optimization problem shown in Figure 4 as an example, assuming there are 6 candidate solutions A, B, C, D, E, and F in the objective space, in the initial stage, each individual is regarded as an independent cluster, and the coordinates of its cluster center directly correspond to the coordinates of the individual in the objective space, thus forming 6 initial clusters. Subsequently, based on the distance metric defined by formula (4) d ( r，s ), calculate the sum of squared errors between every two clusters, and obtain the sum of squared errors between 15 cluster pairs, assuming d If (A, B) is the smallest, then A and B are merged into a new cluster. H In, and calculate using the following formula H Center: (5) In the formula: Representing cluster centers a In the m Coordinate values ​​in each target dimension For class a The Middle j The first individual m The component values ​​of each target space. n This represents the total number of individuals in the current class a. The specific operation process of the CA-MOEA algorithm is as follows: First, based on the specific parameters of the multi-objective problem to be solved, a random number of parameters of size is generated. N The initial parent population P The number of decision variables is d The number of targets is set to M Constraint B is the upper and lower bounds of the decision variable. Subsequently, simulated binary crossover (SBX) (Deb et al., 2014) is applied (crossover probability...). =1.0) and polynomial mutation (probability of mutation) =1 / d) Generate a child population Q of size N. Based on this, introduce an elite retention mechanism to merge the parent population P and the child population Q into a population of size 2N. Subsequently, selection was made from this mixed population. NThe best-performing individuals constitute the next generation of the parent population. P ; When performing environmental selection, the fast non-dominated ordination method is first used to assign individuals in the population that are not dominated by any other individual to the first non-dominated frontier. Next, individuals that are not dominated by other individuals are selected from the remaining population and placed on the non-dominated frontier. And so on, 2 from the merged population N Individuals are divided into L Non-dominated frontier , , ..., From the front arrive The process of gradually selecting solutions continues until N solutions are chosen. The process of generating offspring and selecting environments is repeated until the preset termination condition is met. Finally, the individuals selected from the last generation of the population will be the final set of solutions to the problem. To ensure efficient algorithm operation, initialization parameters for the CA-MOEA algorithm were set based on algorithm principles and empirical methods. The attribute values ​​of the generated population particles during the initialization phase were determined. The key parameter settings for the CA-MOEA algorithm are shown in Table 3. Table 3 CA-MOEA Algorithm Parameters

[0050] After iterative optimization using the CA-MOEA algorithm, 50 sets of candidate optimization parameters were obtained. The correspondence between the panel thickening volume and the tensile strain value of the safety index is as follows: Figure 5 As shown in the figure, the distribution of points on the Pareto front is relatively uniform, indicating good convergence of the algorithm and its ability to fully explore the feasible solution space and find representative non-dominated solutions. Simultaneously, it reveals a contradictory relationship between the increase in panel volume and the panel safety index; the two are unlikely to simultaneously reach their optimal state in the same optimization scheme. When the tensile strain value is at a low level, the panel thickening volume tends to be in a higher range. Conversely, when the panel thickening volume is at a low value, the tensile strain value will be relatively high.

[0051] Step 5: Select the final solution from the Pareto optimal solution set based on actual engineering preferences.

[0052] Example 6 This invention uses the design of the asphalt concrete panel structure at the bottom of the Zhen'an pumped storage power station as an engineering example. Firstly, it focuses on increasing the width of the panel. b Thickened panel h Bedrock excavation slope ratio at the cut-fill boundary i and backfill compaction density ρ To optimize the structural parameters of the cut-fill interface zone of asphalt concrete panels, a multi-objective optimization mathematical model for the cut-fill interface zone of asphalt concrete panels is established, with the maximum settlement deformation gradient and maximum tensile / compressive strain as constraints, the maximum tensile strain of the panel as the optimization safety objective, and the panel volume as the economic objective. A four-factor, three-level combination system is constructed using the Box-Behnken Design (BBD) experimental design method. Response surface methodology (RSM) is used to perform multivariate nonlinear regression fitting on the experimental data to establish a predictive model between the constraints, safety objectives, and optimization parameters. Based on this, the CA-MOEA algorithm is used to solve the multi-objective optimization of the structural parameters of the cut-fill interface zone of asphalt concrete panels.

[0053] This invention presents a multi-objective optimization method for asphalt concrete panel structures based on the CA-MOEA algorithm, combined with... Figure 1 The specific steps are as follows: Step 1: Define the optimization problem, and clarify the optimization objective, optimization parameters, and constraints; Step 1 is implemented in the following steps: To address the need for multi-objective optimization of the economy and safety of asphalt concrete panel cut-fill interface structures, the system defines the set of design variables. b, h, i and constraint space, b Indicates the increased width of the panel. h This indicates the thickness of the panel. i Indicates the excavation slope. To represent the compaction density, we establish an economic objective function based on the increased panel thickness and a safety performance objective function based on the tensile strain index. Figure 2 The asphalt concrete panel structure at the cut-fill boundary is as follows: from top to bottom, it consists of a sealing layer (2 mm), an impermeable layer (10 cm), and a trapezoidal thickened area (top width). b ,thickness h Slope i The structure consists of a leveling cementitious layer (10 cm), a crushed stone drainage cushion layer (60 cm), and the bottom layer is bedrock and backfill. The asphalt concrete panel excavation-fill interface structure involves four key parameters, therefore, [the following is selected]. b, h, i Four parameters are used as structural optimization design variables for the cut-fill interface area of ​​asphalt concrete panels. For the optimized deformation geometry parameters, the panel thickening width is determined through relevant engineering design specifications and engineering experience. b and panel thickness h The range of values ​​is determined by statistically analyzing the excavation slope at the cut-fill boundary of existing pumped storage power stations both domestically and internationally, and by appropriately expanding the excavation slope ratio to determine the appropriate value. iBased on the "Design Code for Rolled Earth-Rock Dams", the range of compaction density values ​​is obtained, and the range of optimized parameter values ​​is shown in Table 1. Table 1. Range of Optimization Parameter Values

[0054] The maximum compressive strain, maximum tensile strain, and maximum settlement gradient were selected as stress-deformation constraints. Based on extensive test results and engineering design experience of asphalt concrete materials, relevant specifications stipulate that the maximum compressive strain of asphalt concrete panels should be <0.08%, the maximum tensile strain should be <0.1%, and the maximum settlement gradient should be <0.1%.

[0055] In step 1, based on the structure of the cut-fill interface zone of the asphalt concrete panel, the geometric mapping relationship between the volume of the thickened area of ​​the panel and the structural optimization parameters is analyzed to construct an explicit expression between the volume of the thickened area of ​​the panel and the structural parameters. The specific expression is as follows: (1) In the formula: V Optimize the volume of the asphalt concrete panel; L The length of the asphalt concrete panel at the bottom of the reservoir around the cut-and-fill junction; b Increase the width of the panel; h To increase the thickness of the panel, i The slope for excavation at the junction of cut and fill.

[0056] The main cause of the asphalt concrete panel failure at the cut-fill interface in Step 1 was excessive tensile strain. Tensile strain is a comprehensive indicator suitable for evaluating the structural safety of asphalt concrete panels at the cut-fill interface. Therefore, the maximum tensile strain of the panel can be used as the optimization target for panel structural safety. Using response surface methodology, an explicit correlation between the tensile strain safety index and the optimization parameters is constructed: (2) In the formula: E For tensile strain safety indicators; b Increase the width of the panel; h Increase the thickness of the panel; i The excavation slope ratio for the cut-fill boundary area; This represents the compaction density of the backfill area.

[0057] Step 2: Generate parameter combinations for optimization parameters based on the Box-Behnken experimental design method; Step 2 is implemented in the following steps: The panel thickening width, panel thickening thickness, excavation slope ratio at the cut-fill junction, and backfill compaction density were selected as optimization parameters. Based on the Box-Behnken experimental design method, 29 sets of parameter combinations for the four optimization design variables were generated, as shown in Table 2. Table 2 Results of structural optimization tests

[0058] Step 3: Establish a finite element numerical model, obtain the response values ​​under each combination through finite element numerical simulation, and construct a parameterized prediction model for safety indicators; Step 3 is implemented in the following steps: A refined finite element numerical model of the upper reservoir basin was established based on the engineering data of the Zhen'an pumped storage power station, as follows: Figure 3 As shown in the diagram. The calculation model defines the positive x-axis as the direction downstream of the water flow, the positive y-axis as the direction perpendicular to the water flow pointing to the left bank, and the positive z-axis as the vertical direction pointing upwards. To ensure that the model's calculation domain covers the main structures of the dam and its surrounding soil and rock mass, and to reduce the influence of the boundary on the calculation results of the reservoir basin cut-fill interface area, the model's calculation range extends upwards 170-178 m along the water flow direction from the upper boundary of the reservoir basin, downwards 123-131 m from the downstream dam slope toe, and along the axial direction, extends 226-234 m at the left and right bank boundaries of the reservoir basin, respectively. Along the depth direction, the bottom of the dam body extends downwards 196-204 m. The model was meshed using tetrahedral four-node C3D4 elements, containing a total of 489,358 nodes and 123,842 elements. Fixed constraints were applied to the bottom boundary of the model, and corresponding normal constraints were applied to the four sides of the model. Gravity loads were applied to the overall model to simulate the effect of the gravity field. The upstream reservoir water pressure was simulated as the surface force on the panel, and applied along the gradient of the panel surface according to the hydrostatic pressure formula to simulate the reservoir water pressure borne by the seepage barrier. The dam filling process was simulated in 22 analysis steps, and the water storage process was simulated in 5 analysis steps, for a total of 27 analysis steps. The maximum settlement deformation gradient, maximum compressive strain, and maximum tensile strain response values ​​of the panel under each combination were obtained through finite element numerical simulation. A second-order polynomial prediction model between the maximum compressive strain, maximum tensile strain, maximum settlement deformation gradient, and optimized parameters was established based on the response surface methodology. After variance analysis and model credibility verification, a parameterized prediction model for the safety index was constructed. The objective function for the panel tensile stress safety index is shown in the following equation: (3) In the formula: E For tensile strain safety indicators; b Increase the width of the panel; h Increase the thickness of the panel; i The excavation slope ratio for the cut-fill boundary area; This represents the compaction density of the backfill area.

[0059] Step 4: Solve the model using the CA-MOEA algorithm to obtain the Pareto optimal solution set representing the optimal trade-off relationship; Step 4 is implemented in the following steps: Based on the defined objective function, the CA-MOEA algorithm is used for multi-objective optimization. First, the feasible solution space of the multi-objective optimization mathematical model is constructed according to the mapping relationship of panel stress-deformation constraint indices. Then, within the defined feasible solution space, the algorithm strictly follows... Figure 4 The CA-MOEA algorithm is used to solve the model. The CA-MOEA algorithm adopts a bottom-up hierarchical clustering strategy. First, each data point is treated as an independent cluster. Then, two clusters are merged according to the Ward link criterion until all clusters are merged into one. The Ward link criterion minimizes the sum of squared errors within the same group while maximizing the errors between different groups. The sum of squared errors is equivalent to the distance metric calculated by equation (4). d ( r，s ): (4) In the formula: and For class r and s The center; for and The Euclidean distance between them; and For class r and s The number of elements in; Taking the bi-objective optimization problem shown in Figure 4 as an example, assuming there are 6 candidate solutions A, B, C, D, E, and F in the objective space, in the initial stage, each individual is regarded as an independent cluster, and the coordinates of its cluster center directly correspond to the coordinates of the individual in the objective space, thus forming 6 initial clusters. Subsequently, based on the distance metric defined by formula (4) d ( r，s ), calculate the sum of squared errors between every two clusters, and obtain the sum of squared errors between 15 cluster pairs, assuming d If (A, B) is the smallest, then A and B are merged into a new cluster. H In, and calculate using the following formula H Center: （5） In the formula: Representing cluster centers a In the m Coordinate values ​​in each target dimension For class a The Middle j The first individual m The component values ​​of each target space. n This represents the total number of individuals in the current class a. The specific operation process of the CA-MOEA algorithm is as follows: First, based on the specific parameters of the multi-objective problem to be solved, a random number of parameters of size is generated. N The initial parent population P The number of decision variables is d The number of targets is set to M Constraint B is the upper and lower bounds of the decision variable. Subsequently, simulated binary crossover (SBX) (Deb et al., 2014) is applied (crossover probability...). =1.0) and polynomial mutation (probability of mutation) =1 / d) The generation scale is also... N Based on the offspring population Q, an elite preservation mechanism is introduced to merge the parent population P and the offspring population Q into a population of size 2N. Then, N individuals with the best performance are selected from the mixed population to form the next generation parent population P; When performing environmental selection, the fast non-dominated ordination method is first used to assign individuals in the population that are not dominated by any other individual to the first non-dominated frontier. Next, individuals that are not dominated by other individuals are selected from the remaining population and placed on the non-dominated frontier. And so on, 2 from the merged population N Individuals are divided into L Non-dominated frontier , , ..., From the front arrive The process of gradually selecting solutions continues until N solutions are chosen. The process of generating offspring and selecting environments is repeated until the preset termination condition is met. Finally, the individuals selected from the last generation of the population will be the final set of solutions to the problem. To ensure efficient algorithm operation, initialization parameters for the CA-MOEA algorithm were set based on algorithm principles and empirical methods. The attribute values ​​of the generated population particles during the initialization phase were determined. The key parameter settings for the CA-MOEA algorithm are shown in Table 3. Table 3 CA-MOEA Algorithm Parameters

[0060] After iterative optimization using the CA-MOEA algorithm, 50 sets of candidate optimization parameters were obtained. The correspondence between the panel thickening volume and the tensile strain value of the safety index is as follows: Figure 5As shown in the figure, the distribution of points on the Pareto front is relatively uniform, indicating good convergence of the algorithm and its ability to fully explore the feasible solution space and find representative non-dominated solutions. Simultaneously, it reveals a contradictory relationship between the increase in panel volume and the panel safety index; the two are unlikely to simultaneously reach their optimal state in the same optimization scheme. When the tensile strain value is at a low level, the panel thickening volume tends to be in a higher range. Conversely, when the panel thickening volume is at a low value, the tensile strain value will be relatively high.

[0061] Step 5: Select the final solution from the Pareto optimal solution set based on actual engineering preferences.

[0062] Step 5 is implemented in the following steps: Given that the optimal solution obtained by the multi-objective optimization algorithm is not unique, it is necessary to closely combine it with the actual engineering needs and select the best-fitting solution from the Pareto front as the optimal panel structure scheme. The optimal structure configuration scheme based on differentiated decision preferences is extracted from the Pareto non-dominated solution set using the linear weighted sum method, which includes the following two steps: First, eliminate the incomparability caused by the difference in dimensions of different objective functions by transforming the panel thickness volume value and the panel safety index tensile strain into a single objective function form that can be uniformly compared. The specific normalization process is shown in the following equation: (6) Secondly, using the linear weighting method, based on the importance of each objective in the project, corresponding weight coefficients are assigned to the panel thickness volume value and the panel safety index tensile strain. By linearly combining the normalized values, structural optimization schemes with different tendencies are obtained. The calculation formula of the linear weighting method is shown in the following formula: (7) In the formula: The value is a normalized value for the target value; , , These are the k-th target value, the minimum value, and the maximum value, respectively. Weighting coefficients for the thickness volume value of the panel; The weighting coefficient for the tensile strain value of the panel safety index; With a balanced and coordinated configuration W=(0.5,0.5), following the steps above, a balanced and coordinated panel structure optimization scheme is obtained. The structural optimization parameters of the balanced and coordinated scheme correspond to the 47th Pareto solution (10.96, 0.61, 1.99, 2.25), exhibiting a significant balance between safety and economy.

Claims

1. A method for multi-objective optimization of asphalt concrete panel structure based on CA-MOEA algorithm, characterized in that, The step 1 is implemented according to the following steps: b, h, i, b, h, i, Table 1 Value range of optimization parameters The maximum compressive strain, the maximum tensile strain and the maximum settlement deformation gradient are selected as the stress and deformation constraints, according to a large number of test results and engineering design experience of asphalt concrete materials, and relevant specifications stipulate that the maximum compressive strain of asphalt concrete panel is <0.08%, the maximum tensile strain is <0.1%, and the maximum deformation settlement gradient is <0.1%. In the step 1, according to the structure of the excavation-filling junction area of the asphalt concrete panel, the geometric mapping relationship between the volume of the panel thickening area and the structure optimization parameters is solved, and the explicit expression between the volume of the panel thickening area and the structure parameters is constructed, and the specific expression form is as follows:

2. The method for multi-objective optimization of asphalt concrete panel structure based on CA-MOEA algorithm according to claim 1, characterized in that, In the step 1, the maximum tensile strain of the panel is taken as the safety optimization target of the panel structure, and the explicit correlation between the tensile strain safety index and the optimization parameters is constructed by the response surface method: Aiming at the demand of multi-objective optimization of economy and safety of the structure of asphalt concrete panel in cut-fill junction area, the design variable set and the constraint condition space are defined The step 2 is implemented according to the following steps: b represents the panel thickening width, h represents the panel thickening thickness, i represents the excavation slope, represents the rolling density, the economic objective function based on the panel thickening volume and the safety performance objective function of the tensile strain index are respectively established;​ The asphalt concrete panel cut-and-fill junction structure specifically comprises, from top to bottom, a sealing layer, an anti-seepage layer, a trapezoidal thickening area, a leveling cementing layer, and a gravel drainage cushion layer, and the bottom layer is bedrock and a backfill area. The asphalt concrete panel cut-and-fill junction structure involves four key parameters, and therefore, the four parameters are selected as the optimization design variables of the asphalt concrete panel cut-and-fill junction structure. The panel thickening width, the panel thickening thickness, the excavation slope ratio of the excavation-filling junction area and the rolling density of the backfill body are selected as the optimization parameters: The four parameters are used as the optimization design variables of the asphalt concrete panel cut-and-fill junction structure, and the deformation geometry is optimized. The value ranges of the panel thickening width b and the panel thickening thickness h are determined according to relevant engineering design specifications and engineering experience. The excavation slope of the cut-and-fill junction area of the built pumped storage power station at home and abroad is counted, the excavation slope ratio is appropriately expanded, and the range of i is determined. According to the Design Specification for Roller Compacted Earth and Rockfill Dams, the value range of the rolling density is obtained, and the optimization parameter value range is shown in Table 1. Table 2 Results of structure optimization test Based on the Box-Behnken experimental design method, 29 groups of parameter combinations about the four optimization design variables are generated as shown in Table 2.

3. The method for multi-objective optimization of asphalt concrete panel structure based on CA-MOEA algorithm according to claim 2, characterized in that, The step 3 is implemented according to the following steps: （1） wherein: V is the volume of the optimized section of the asphalt concrete panel; L is the length of the banked asphalt concrete panel around the cut-fill interface; b is the width of the panel thickening; h is the thickness of the panel thickening, i is the slope of the cut-fill interface.

4. The method for multi-objective optimization of asphalt concrete panel structure based on CA-MOEA algorithm according to claim 3, characterized in that, ​ （2） In the formula: E is a safety index of tensile strain; b is the thickening width of the panel; h is the thickening thickness of the panel; i is the excavation slope ratio of the excavation-intersection area; is the compaction density of the backfill area.

5. The method for multi-objective optimization of asphalt concrete panel structure based on CA-MOEA algorithm according to claim 4, characterized in that, ​ ​ ​ ​ 6. The method for multi-objective optimization of asphalt concrete panel structure based on CA-MOEA algorithm according to claim 5, characterized in that, ​ The fine finite element numerical model of the upper reservoir basin is established. The model defines the downstream direction as the positive direction of the x-axis, the left bank direction as the positive direction of the y-axis, and the vertical direction as the positive direction of the z-axis. To ensure that the model calculation domain covers the main structures of the dam and its surrounding rock-soil body and to reduce the influence of the boundary on the calculation results of the excavation-filling junction area of the reservoir basin, the model calculation range extends 170-178 m upward along the water flow direction from the upper boundary of the reservoir basin, 123-131 m downward from the toe of the downstream dam slope, 226-234 m along the axis direction from the left and right bank boundaries, and 196-204 m downward from the bottom of the dam. Tetrahedral four-node element C3D4 is used for meshing different parts of the model, which contains 489358 nodes and 123842 elements. Fixed constraints are applied to the bottom boundary of the model, and corresponding normal constraints are applied to the four sides of the model. Gravity loads are applied to the overall model to simulate the gravity field. The water pressure of the upper reservoir is simulated as a surface force on the face plate, which is applied along the surface gradient of the face plate according to the hydrostatic pressure formula to simulate the water pressure on the impervious body. The dam filling process is simulated by 22 analysis steps, and the reservoir filling process is simulated by 5 analysis steps, totaling 27 analysis steps. The maximum settlement deformation gradient, maximum compressive strain, and maximum tensile strain response values of each combination of the face plate are obtained through finite element numerical simulation. Based on the response surface method, a second-order polynomial prediction model is established between the maximum compressive strain, the maximum tensile strain, the maximum settlement deformation gradient, and the optimization parameters. After variance analysis and model credibility verification, the parameterized prediction model of the safety index is constructed, and the target function of the face plate tensile stress safety index is as follows: （3） In the formula: E is a safety index of tensile strain; b is the thickening width of the panel; h is the thickening thickness of the panel; i is the excavation slope ratio of the excavation-intersection area; is the compaction density of the backfill area.

7. The method for multi-objective optimization of asphalt concrete panel structure based on CA-MOEA algorithm according to claim 6, characterized in that, Step 4 is specifically implemented according to the following steps: On the basis of determining the objective function, the CA-MOEA algorithm is used for multi-objective optimization solution. First, according to the mapping relationship of the panel stress deformation constraint index, the feasible solution space of the multi-objective optimization mathematical model is constructed, and then the CA-MOEA algorithm is used to solve the model in the determined feasible solution space. The CA-MOEA algorithm adopts a bottom-up hierarchical clustering strategy. First, each data point is taken as an independent cluster. Then, according to the Ward linkage criterion, two clusters are merged until all clusters are merged into one. The Ward linkage criterion minimizes the sum of squared errors within the same group while maximizing the error between different groups. The sum of squared errors is equivalent to the distance metric calculated by formula (4) d ( r，s ): (4) wherein: and are centers of classes r and s ; is and Euclidean distance between and are numbers of elements in classes r and s ; Assuming that there are 6 candidate solution individuals A, B, C, D, E, F in the target space, in the initial stage, each individual is regarded as an independent clustering cluster, and the cluster center coordinates directly correspond to the coordinate values of the individual in the target space, thereby forming 6 initial clusters, then, based on the distance metric defined by formula (4) d ( r，s ), the error sum of squares between each two clusters is calculated, and the error sum of squares between 15 cluster pairs is obtained, assuming that d(A, B) is the smallest, then A and B are merged into a new cluster H, and the center of H is calculated using the following formula: (5) wherein: denotes the cluster center a the coordinate value in the m target dimension, the component value of the a individual in the j target space, m the component value of the n individual in the current class a; The specific operation process of the CA-MOEA algorithm is as follows: First, based on the specific parameters of the multi-objective problem to be solved, a random number of parameters of size is generated. N The initial parent population P The number of decision variables is d The number of targets is set to M Constraint B, where constraint B is the upper and lower bounds of the decision variable, is then used to simulate binary crossover SBX and polynomial mutation, respectively, generating variables of similar scale. N Based on the offspring population Q, an elite preservation mechanism is introduced to merge the parent population P and the offspring population Q into a population of size 2N. Subsequently, selection was made from this mixed population. N The best-performing individuals constitute the next generation of the parent population. P ; In the process of environment selection, first, the fast non-dominated sorting method is used to divide the individuals in the population which are not dominated by any other individual into the first non-dominated front , then, the individuals which are not dominated by other individuals are screened from the remaining population and put into the non-dominated front , and so on, 2 N individuals in the combined population are divided into L non-dominated fronts , , …, ; solutions are gradually selected from the preceding to until N solutions are selected; the above process of generating offspring and environment selection is repeated until a preset termination condition is met, and finally, the individuals screened from the last generation population will be the final solution set of the problem; The initialization parameters of the CA-MOEA algorithm are set, and the attribute values of the particles generated in the initialization stage are determined. The key parameter settings of the CA-MOEA algorithm are shown in Table 3. Table 3 CA-MOEA algorithm parameters After iteration optimization by the CA-MOEA algorithm, 50 groups of candidate optimization parameters are obtained. When the tensile strain value is at a relatively low level, the face plate thickening volume is often in a relatively high range, and vice versa. When the face plate thickening volume is at a relatively low value, the tensile strain value is relatively high.

8. The method for multi-objective optimization of asphalt concrete panel structure based on CA-MOEA algorithm according to claim 7, characterized in that, Step 5 is specifically implemented according to the following steps: Due to the non-unique characteristics of the optimal solution obtained by the multi-objective optimization algorithm, it is necessary to closely combine with the actual engineering requirements and select the most suitable solution from the Pareto front as the optimal face plate structure scheme. The linear weighted sum method is used to extract the optimal structure configuration scheme with differentiated decision preferences in the Pareto non-inferior solution set, which includes the following two steps: First, eliminate the incomparability caused by the dimensional difference of different objective functions, and convert the face plate thickening volume and the face plate safety index tensile strain into a single objective function form that can be uniformly compared. The specific process of normalization is as follows: （6） Secondly, according to the importance of each target in the project, the linear weighting method is used to give the corresponding weight coefficient to the panel thickening volume value and the panel safety index tensile strain. Through linear combination of the normalized values, the structure optimization scheme of different inclination is obtained. The calculation formula of linear weighting sum method is shown in the following formula: （7） In the formula: is a target value normalized value; , , is the kth target value of the target value, minimum value, maximum value, respectively; is a weight coefficient of the panel thickening volume value; is a weight coefficient of the panel safety index tensile strain value; The balanced coordination type configuration W=(0.5, 0.5), according to the above steps, the balanced coordination type panel structure optimization scheme is obtained.

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