A multi-objective intelligent decision method for cable force of completed cable-stayed bridge based on balance degree

Abstract

This invention belongs to the field of bridge engineering technology, and specifically relates to a multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on equilibrium degree. This invention includes a method for constructing a small-rate-increase influence matrix for cable-stayed bridges and a multi-objective intelligent decision-making algorithm for cable forces based on a high-low equilibrium degree strategy. The method for constructing the small-rate-increase influence matrix adjusts the cable forces at the same small rate of increase based on the initial cable forces, and normalizes the resulting effect increments to form a small-rate-increase influence matrix, resulting in a smaller linear approximation error when calculating the structural response of cable-stayed bridges. The multi-objective intelligent decision-making algorithm for cable forces based on a high-low equilibrium degree strategy is a strategy of multi-objective particle swarm optimization that simultaneously retains high-equilibrium-degree and low-equilibrium-degree solutions during the iterative process to guide the optimization direction. It can efficiently and reliably select Pareto non-dominated solutions that achieve the preset standard of optimization degree on multiple objectives, thus improving decision-making efficiency.

Description

Technical Field

[0001] This invention belongs to the field of bridge engineering technology, and specifically relates to a multi-objective intelligent decision-making method for cable forces in cable-stayed bridges based on equilibrium degree. Background Technology

[0002] Cable-stayed bridges are highly competitive cable-supported structures among long-span bridges with spans of up to 1,000 meters. Because the stress states of the main girder and towers differ depending on the cable tension of the stay cables, selecting appropriate cable tensions to achieve rational and even optimized stress distribution in the cable-stayed bridge design process is a crucial task.

[0003] Methods for optimizing cable forces in cable-stayed bridges include traditional empirical methods for determining cable forces, classical methods for optimizing cable forces, and modern intelligent methods for optimizing cable forces. Traditional empirical methods for determining cable forces include the zero-displacement method, the rigid-supported continuous beam method, and the moment distribution method. Classical methods for optimizing cable forces include the minimum bending energy method and the influence matrix method. Modern intelligent methods for optimizing cable forces include genetic algorithms, particle swarm optimization, and simulated annealing algorithms. Cable force optimization in cable-stayed bridges can be divided into single-objective optimization and multi-objective optimization based on the number of objective functions. In single-objective cable force optimization, both traditional empirical and classical methods are widely used. With the development of technology, cable-stayed bridge engineering has been presented with multiple objective requirements, including alignment, stress distribution, and economic efficiency. Therefore, multi-objective optimization of cable forces in cable-stayed bridges has begun. Unlike single-objective optimization, where the optimization result is the decision result, multi-objective optimization actually includes both optimization and decision-making processes.

[0004] Traditional multi-objective optimization methods generally follow a decision-then-optimize approach. This involves first determining the importance of multiple objectives, then transforming the multi-objective optimization problem into a single-objective optimization problem. Transformation methods include weighted methods, constraint methods, and linear programming. The drawbacks of this approach are that each optimization iteration yields only one solution; obtaining solutions with different weights requires multiple computations, resulting in extremely low efficiency. Furthermore, when multiple objectives have inconsistent dimensions or large differences in magnitude, manually setting weights becomes highly subjective and even unreasonable. Finally, determining weights beforehand prevents the exploration of all optimal solutions before making a final decision.

[0005] Multi-objective intelligent algorithms such as MOGA, NSGA-II / III, and MOPSO, developed based on genetic algorithms and particle swarm optimization, employ a process of intelligent optimization followed by human decision-making. The algorithms do not yield a single optimal solution, but rather a solution set containing multiple Pareto optimal solutions. During multi-objective intelligent optimization, the lack of guidance for decision direction means that each iteration considers all possible decisions, reducing computational efficiency and optimization accuracy. Furthermore, efficiently determining one or more satisfactory solutions from a Pareto optimal solution set still requires embedding a decision-making method at the end of the algorithm, such as the equilibrium method, hierarchical method, objective weighting method, or game-theoretic fairness method.

[0006] To further improve the computational efficiency and optimization accuracy of multi-objective decision-making methods, it is necessary to embed the decision-making methods into multi-objective optimization iterative operations. However, the choice of which decision-making method to adopt and how to combine it with optimization iterative operations have become new problems to be solved.

[0007] Meanwhile, when applying multi-objective intelligent optimization methods to the cable force optimization problem of cable-stayed bridges, each iteration requires calling the finite element model, influence matrix, or intelligent surrogate model of the cable-stayed bridge to calculate the structural response. Calling the large finite element model results in significantly low iteration efficiency. Furthermore, establishing an intelligent surrogate model for cable-stayed bridges requires a large amount of training data, making the preliminary work quite arduous. For typical cable force optimization of cable-stayed bridges, calling the influence matrix is ​​a very convenient operation; however, the influence matrix is ​​derived based on the assumptions of linear elasticity and small deformation. Therefore, reducing the linear approximation error of the influence matrix is ​​a key issue.

[0008] In conclusion, it is necessary to improve the method for constructing the linear response influence matrix of cable-stayed bridges and develop a multi-objective intelligent decision-making algorithm that considers the decision-making direction, so as to improve work efficiency and ensure the reliability of results. Summary of the Invention

[0009] The problem to be solved by the present invention is to overcome the shortcomings of the prior art and provide a multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on the degree of equilibrium.

[0010] This invention is achieved through the following technical solution: A multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on equilibrium degree includes the following steps: Step S1: Establish a finite element model of the cable-stayed bridge; Step S2: Preliminary optimization of cable tension; Step S3: Initial uniform cable force; Step S4: Determine the optimization objective, select the small increment value, and calculate the small increment influence matrix of the cable force corresponding to the effect. Step S5: Select the objective function of the optimization objective, define the constraints, set the initial value column vector of cable force, and establish a multi-objective optimization mathematical model of cable force of cable-stayed bridge based on the small increase rate influence matrix; Step S6: Initialize particle position and velocity using the initial force column vector, calculate particle fitness based on the objective function, and create an external file and divide the mesh according to the fitness and non-dominance principles; Step S7: Determine the optimal position of each individual particle; Step S8: Calculate the mesh quality for the external archive, execute the roulette wheel strategy, and determine the global optimal position of the particles; Step S9: Update the particle velocity and position according to the particle swarm algorithm iterative formula; Step S10: Recalculate the particle fitness based on the objective function, and update the optimal position of each particle according to the dominance relationship; Step S11: Update the external file according to the non-dominated sorting strategy, and then further update the external file according to the high and low balance strategy; if the number of non-dominated particles exceeds the size of the external file, delete the excess particles according to the crowding strategy; the high and low balance strategy includes setting a high balance standard and a low balance standard, calculating the balance of the Pareto non-dominated solution, and the balance between the high balance standard and the low balance standard is the medium balance solution. Then, in each iteration step, discard the medium balance solutions in the Pareto non-dominated solution set, and retain the high balance solutions and low balance solutions in the Pareto non-dominated solution set to update and maintain the external file of the algorithm. Step S12: Determine whether the termination condition is met. If it is met, output the external file particle as the optimal solution set. If it is not met, return to step S8 for iteration.

[0011] As a preferred option: In step S2, the cable force is initially optimized using the unconstrained minimum bending energy method.

[0012] In step S3, the cable force is initially uniformly applied according to the principle that the longer the cable has a larger force and the shorter the cable has a smaller force.

[0013] In step S4, the relative error between the effect vector calculated when the increment rate is different and the effect vector directly calculated by the finite element model of the cable-stayed bridge is compared. The increment rate value with the smallest relative error is selected as the small increment rate value. The increment of the effect vector is divided by the increment of the cable force and normalized to obtain the small increment rate influence vector. The small increment rate influence vectors caused by all cables are arranged to form the small increment rate influence matrix.

[0014] In step S4, the effect vector includes bending moment and displacement.

[0015] In step S5, the bending strain energy U is minimized. M The objective functions are to minimize the resultant displacement D.

[0016] In step S7, the initial position value of the first selected particle is the optimal position of the particle.

[0017] In step S9, the particle velocity and position are updated according to the standard particle swarm algorithm iterative formula.

[0018] In step S11, the high balance standard is set to 0.9 / m ~ 0.99 / m, and the low balance standard is set to 0.1 / m ~ 0.01 / m, where m is the number of optimization targets.

[0019] In step S11, calculating the balance of Pareto nondominated solutions includes: normalizing fitness using the max-min normalization method; and calculating the balance of fitness, where the balance is the ratio of the minimum normalized fitness to the sum of the normalized fitness.

[0020] Compared with the prior art, the beneficial effects of the present invention are as follows: The small increment rate influence matrix method in this invention has a smaller linear approximation error when calculating the response of cable-stayed bridge structures. At the same time, this invention sorts the optimal solutions in the Pareto optimal solution set according to their balance degree. In each iteration, the Pareto optimal solutions with the highest and lowest balance degrees are selected and entered into the external archive of the multi-objective particle swarm optimization algorithm. Then, the global optimal particle is determined from it to guide the operation towards the decision direction and ensure the optimization accuracy. Finally, the optimal solution with the balance degree that meets the decision criteria is directly given, which improves the decision efficiency. Attached Figure Description

[0021] Figure 1 This is a flowchart of the present invention; Figure 2 This is a schematic diagram illustrating the linear characteristics of the small increase rate influence matrix of the cable-stayed bridge according to the present invention; Figure 3 This is a schematic diagram illustrating the retained and discarded solutions for the high and low balance strategy of this invention; Figure 4 This is a schematic diagram of the sparse Pareto front of the present invention. Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are some embodiments of the present invention, but not all embodiments.

[0023] This invention provides a multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on equilibrium degree, including a method for constructing a small-rate-increase influence matrix for cable-stayed bridges and a multi-objective intelligent decision-making algorithm for cable forces based on a high-low equilibrium degree strategy. The method for constructing the small-rate-increase influence matrix for cable-stayed bridges is a modeling method that expresses the explicit relationship between the increase in cable force and its effect. This method adjusts the cable force at the same small rate of increase based on the initial cable force, and normalizes the resulting increase in effect to form the small-rate-increase influence matrix. The multi-objective intelligent decision-making algorithm for cable forces based on a high-low equilibrium degree strategy is an improved multi-objective particle swarm optimization algorithm that incorporates a high-low equilibrium degree strategy. The equilibrium degree is the ratio of the minimum multi-objective fitness of a single Pareto non-dominated solution to the sum of the fitnesss. The fitness should be normalized during calculation to measure the optimization degree of a single Pareto non-dominated solution across multiple objectives. The high-low equilibrium degree strategy is a strategy in which the multi-objective particle swarm optimization algorithm simultaneously retains solutions with high and low equilibrium degrees during the iterative process to guide the optimization direction. The multi-objective intelligent decision-making algorithm for cable-stayed bridge cable forces based on the high-low balance strategy of the present invention can efficiently and reliably select Pareto non-dominated solutions that achieve the preset standard in terms of optimization degree on multiple objectives.

[0024] This embodiment is a single-tower, single-cable-stayed bridge with a span of (96+60+30) m, featuring a tower-pier-beam rigid connection system, with sliding bearings at the transition piers and auxiliary piers. For this embodiment, the multi-objective intelligent decision-making method for cable forces in cable-stayed bridges based on equilibrium includes the following steps: Step S1: Establish the finite element model of the cable-stayed bridge A finite element model of the cable-stayed bridge was established. In this embodiment, the finite element model is divided into 28 cable-stayed elements, 30 main beam elements, 16 bridge tower elements, 31 main beam nodes, and 16 bridge tower nodes.

[0025] Step S2: Preliminary optimization of cable tension Based on the finite element model of a cable-stayed bridge, the unconstrained minimum bending energy method is used for preliminary optimization of cable forces.

[0026] Step S3: Initial Uniform Cable Force The cable forces in the finite element model of the cable-stayed bridge are initially uniformd according to the principle of "longer cables have greater cable forces, and shorter cables have smaller cable forces." This embodiment comprises... Genlaso, the The initial uniform force of the cable is , It is an integer. .

[0027] Step S4: Determine the optimization objective and calculate the small-rate effect matrix of the cable force corresponding to the effect. Based on the finite element model of the cable-stayed bridge after initial uniform cable force, the corresponding initial cable force can be obtained. and initial action effect vector By changing the cable force according to a fixed increment rate, the cable force increment and the effect vector increment are obtained. Each cable force increment is set as... The increments of the cable forces are not equal, but the ratio of each cable force increment to its initial value is a small fixed value. That is, a fixed small growth rate. Values ​​can be selected through parameter analysis, for example... Values ​​of 1%, 2%, ... 10% were used to extract values ​​from the finite element model of the cable-stayed bridge after initial uniform cable force. The cable force and effect vector for different values ​​are selected. The cable force and effect vector corresponding to a certain value, based on the initial cable force. and initial action effect vector Linear interpolation yields For other values, the effect vector is then compared. The relative errors between the calculated effect vector and the effect vector directly calculated from the finite element model of the cable-stayed bridge for different values ​​were selected, with the smallest relative error being chosen. The value is used as a small increment, and then the increment of the effect vector is calculated. Divided by the increase in cable force After normalization, the small increase rate influence vector is obtained. ; Root cable caused vectors The arrangement forms a small increase rate influence matrix .

[0028] The effect vector in this embodiment includes bending moment and displacement. Based on the parameters required for the optimization objective, this embodiment specifically selects the bending moment of 54 sections, including the support point, cable-beam connection point, cable-tower connection point, beam-tower-pier connection point, pier bottom, midpoint of the cable-free zone of the beam, and midpoint of the cable-free zone of the tower, and the displacement of 42 nodes, including the cable-beam connection point and cable-tower connection point.

[0029] Step S5: Establish a multi-objective optimization mathematical model for cable forces in the completed cable-stayed bridge. By selecting the objective function for optimization, defining constraints, setting the initial values ​​of the cable force column vector, and establishing a multi-objective optimization mathematical model for the cable force of a cable-stayed bridge based on the small-rate-increase influence matrix, a multi-objective optimization model for the cable force of a cable-stayed bridge is created.

[0030] This embodiment selects the option of minimizing bending strain energy U. M A dual objective function is constructed by minimizing the resultant displacement D. The two objective functions are nonlinear functions controlling the bending moment at the cross section and the displacement at the nodes, respectively. Specifically, minimizing the bending strain energy U... MThe control section bending moment is a quadratic function, calculated using a small-incremental-rate influence matrix based on the cable force variables. The minimized resultant displacement D is a nonlinear function of the control node displacement, i.e., the arithmetic square root of the sum of squares of the control node displacements, calculated using a small-incremental-rate influence matrix based on the cable force variables.

[0031] The constraints include: maximum and minimum bending moments of the main girder are ±35000kNm; maximum and minimum bending moments of the bridge tower are ±25000kNm; maximum and minimum displacements of the main girder are ±20mm; maximum displacement of the main tower is 0mm and minimum displacement is -20mm; maximum cable force is 13000kN and minimum cable force is 6000kN; minimum support reaction force is 1000kN.

[0032] The initial column vector of cable force is t0. T =[11027.0;10880.2;10620.4;10258.9;9958.2;9735.0;9271.5;8788.4;8374.3;8044.4;7623.3;7110.8;6581.9;6068.8;6054.4;6574.8;7051.4;7535.7;7988.8;8184.4;8361.5;9422.0;11468.1;11601.3;11732.6;11835.5;11938.4;12133.4].

[0033] Step S6: Create external files and divide the grid Particle positions and velocities are initialized using a column vector of initial force values. Particle fitness is calculated based on the objective function. An external archive is created and a mesh is generated based on fitness and non-dominance principles. In this calculation, the particle population size is set to 200, the external archive capacity is 20, and the column vector of initial force values ​​t0 is used. T The initial particle position and velocity are randomly varied within a range of ±600kN.

[0034] Step S7: Determine the optimal position of each individual particle. To determine the optimal position of an individual particle, the initial position value of the individual particle can be selected for the first selection of the optimal position.

[0035] Step S8: Determine the global optimal position of the particle The external archive is used to calculate the grid quality, and a roulette wheel strategy is executed to determine the global optimal position of the particles.

[0036] Step S9: Update particle velocity and position The particle velocity and position are updated according to the standard particle swarm optimization (PSO) iterative formula. The standard PSO iterative formula includes velocity update formulas and position update formulas, which are standard update rules and common knowledge to those skilled in the art, and will not be elaborated here.

[0037] Step S10: Update the optimal position of individual particles Recalculate the particle fitness based on the objective function, and update the optimal position of each particle according to the dominance relationship.

[0038] Step S11: Update the external file according to the non-dominated sorting strategy, and then further update the external file according to the high-low balance strategy; if the number of non-dominated particles exceeds the size of the external file, delete the excess particles according to the crowding strategy.

[0039] The high-low balance strategy is an improved strategy for updating and maintaining the external archive of the multi-objective particle swarm optimization algorithm. In each iteration step, solutions with medium balance in the Pareto non-dominated solution set are discarded, while high- and low-balance solutions in the Pareto non-dominated solution set are retained to update and maintain the algorithm's external archive. The high- and low-balance standards are pre-set. The high-balance standard can take values ​​between 0.9 / m and 0.99 / m, and the low-balance standard can take values ​​between 0.1 / m and 0.01 / m, where m is the number of optimization objectives. In this embodiment, there are two optimization objectives, so m=2. The high-balance standard is set to 0.95 / m, i.e., 0.95 / 2=0.425; the low-balance standard is set to 0.05 / m, i.e., 0.05 / 2=0.025. The selection criteria for high and low equilibrium solutions are as follows: solutions with an equilibrium degree greater than the high equilibrium degree standard are considered high equilibrium solutions, solutions with an equilibrium degree less than the low equilibrium degree standard are considered low equilibrium solutions, and solutions with an equilibrium degree between the high and low equilibrium degree standards are considered medium equilibrium solutions. Retaining high equilibrium solutions can guide optimization towards the direction of decision requirements; retaining low equilibrium solutions helps optimization explore towards the single-objective extremum, thereby improving optimization accuracy.

[0040] When implementing the high-low balance strategy, first calculate the balance of the Pareto non-dominated solutions, and then retain high-balance and low-balance solutions according to the set high-balance and low-balance standards, while discarding medium-balance solutions.

[0041] The equilibrium calculation of Pareto non-dominated solutions involves two steps: first, calculating the normalized fitness, and then calculating the equilibrium of multi-objective fitness. The calculation steps and formulas are as follows: (1) Calculate the normalized fitness. Fitness We directly take the objective function value and use the min-max normalization method to normalize the fitness, i.e., the fitness is... At that time, normalized fitness for: (1); in This represents the minimum fitness value. This represents the maximum fitness value.

[0042] (2) Calculate the balance of multi-objective fitness. The balance is defined as the ratio of the minimum normalized fitness to the sum of normalized fitness, i.e.: (2).

[0043] Step S12: Determine whether the termination condition is met. Determine if the termination condition is met. If it is, output the external file particle as the optimal solution set. If not, return to step S8 for iteration.

[0044] In this calculation, the iterations terminated after 1000 steps. The final Pareto optimal solution for the cable force obtained in this embodiment is: t p T =[10514.8;10514.6;10514.8;10514.8;10335.6;10335.0;9817.6;9388.4;8948.4;8644.0;8061.6;7710.8;6527.2;6527.2;6654.4;7127.9;7168.6;7312.5;7388.8;7706.0;8961.5;9684.6;11435.2;11435.2;11864.9;11958.0;12371.5;12427.6].

[0045] The sparse Pareto front and the two objective values ​​corresponding to the high-equilibrium and low-equilibrium solutions are attached. Figure 4 As shown.

[0046] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; those skilled in the art should understand that modifications can still be made to the technical solutions described in the above embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the various embodiments of the present invention.

Claims

1. A multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on equilibrium degree, characterized in that: Includes the following steps: Step S1: Establish a finite element model of the cable-stayed bridge; Step S2: Preliminary optimization of cable tension; Step S3: Initial uniform cable force; Step S4: Determine the optimization objective, select the small increment value, and calculate the small increment influence matrix of the cable force corresponding to the effect. Step S5: Select the objective function of the optimization objective, define the constraints, set the initial value column vector of the cable force, and establish a multi-objective optimization mathematical model of the cable force of the cable-stayed bridge based on the small increase rate influence matrix; Step S6: Initialize particle position and velocity using the initial force column vector, calculate particle fitness based on the objective function, and create an external file and divide the mesh according to the fitness and non-dominance principles; Step S7: Determine the optimal position of each individual particle; Step S8: Calculate the mesh quality for the external archive, execute the roulette wheel strategy, and determine the global optimal position of the particles; Step S9: Update the particle velocity and position according to the particle swarm algorithm iterative formula; Step S10: Recalculate the particle fitness based on the objective function, and update the optimal position of each particle according to the dominance relationship; Step S11: Update the external file according to the non-dominated sorting strategy, and then further update the external file according to the high and low balance strategy; if the number of non-dominated particles exceeds the size of the external file, delete the excess particles according to the crowding strategy; the high and low balance strategy includes setting a high balance standard and a low balance standard, calculating the balance of the Pareto non-dominated solution, and the balance between the high balance standard and the low balance standard is the medium balance solution. Then, in each iteration step, discard the medium balance solutions in the Pareto non-dominated solution set, and retain the high balance solutions and low balance solutions in the Pareto non-dominated solution set to update and maintain the external file of the algorithm. Step S12: Determine whether the termination condition is met. If it is met, output the external file particle as the optimal solution set. If it is not met, return to step S8 for iteration.

2. The multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on equilibrium degree as described in claim 1, characterized in that: In step S2, the cable force is initially optimized using the unconstrained minimum bending energy method.

3. The multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on equilibrium degree as described in claim 1, characterized in that: In step S3, the cable force is initially uniformly applied according to the principle that the longer the cable has a larger force and the shorter the cable has a smaller force.

4. The multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on equilibrium degree as described in claim 1, characterized in that: In step S4, the relative error between the effect vector calculated when the increment rate is different and the effect vector directly calculated by the finite element model of the cable-stayed bridge is compared. The increment rate value with the smallest relative error is selected as the small increment rate value. The increment of the effect vector is divided by the increment of the cable force and normalized to obtain the small increment rate influence vector. The small increment rate influence vectors caused by all cables are arranged to form the small increment rate influence matrix.

5. The multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on equilibrium degree as described in claim 1, characterized in that: In step S4, the effect vector includes bending moment and displacement.

6. The multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on equilibrium degree as described in claim 1, characterized in that: In step S5, the bending strain energy U is minimized. M The objective functions are to minimize the resultant displacement D.

7. The multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on equilibrium degree as described in claim 1, characterized in that: In step S7, the initial position value of the first selected particle is the optimal position of the particle.

8. The multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on equilibrium degree as described in claim 1, characterized in that: In step S9, the particle velocity and position are updated according to the standard particle swarm algorithm iterative formula.

9. The multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on equilibrium degree as described in claim 1, characterized in that: In step S11, the high balance standard is set to 0.9 / m ~ 0.99 / m, and the low balance standard is set to 0.1 / m ~ 0.01 / m, where m is the number of optimization targets.

10. The multi-objective intelligent decision-making method for cable-stayed bridge cable forces based on equilibrium degree as described in claim 1, characterized in that: In step S11, calculating the balance of Pareto nondominated solutions includes: normalizing fitness using the max-min normalization method; and calculating the balance of fitness, where the balance is the ratio of the minimum normalized fitness to the sum of the normalized fitness.

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