Tension-compression spring design method based on improved particle swarm optimization algorithm

Abstract

The invention is suitable for the technical field of structure optimization design, and provides a tension-compression spring design method based on an improved particle swarm optimization algorithm. According to the improved particle swarm optimization (SuperMOP) algorithm provided by the invention, an elite archive strategy and an advantage combination learning strategy are fused, and a mutation operator based on an individual level is introduced. Different roles are allocated to particles according to the performance of the particles, so that a learning strategy is dynamically updated. The improved particle swarm optimization algorithm is compared with several most advanced algorithms, and the result shows that the SuperMOP algorithm effectively balances the exploration and utilization process, improves the population diversity and enhances the diversity of learning samples. According to the method, the improved particle swarm optimization algorithm is applied to the design problem of the tension-compression spring, a new mode with strong global search capability is provided for the design problem of the tension-compression spring, and the method is particularly suitable for solving the optimization problem of complex and changeable constraint engineering.

Description

Technical Field

[0001] The invention belongs to the technical field of structural optimization design, and in particular relates to a tension and compression spring design method based on an improved particle swarm optimization algorithm. Background Art

[0002] Particle Swarm Optimization (PSO) is a metaheuristic algorithm widely used to solve complex optimization problems such as multimodal, non-convex and non-differentiable. The algorithm simulates the foraging behavior of a flock of birds or a school of fish, so that particles continuously update their positions and velocities in the search space to find the global optimal solution. In PSO, the position of each particle is updated based on its velocity, personal best position and the global best position in the current iteration.

[0003] The tension and compression spring design problem is a multimodal constrained engineering design problem that is crucial in the optimization of engineering machinery equipment, such as air conditioning compressors and industrial internal combustion engines. Its quality is directly related to equipment performance, and optimization can improve efficiency, energy utilization and reduce costs. In structural optimization design, the problem aims to determine the optimal geometric parameters of the spring to meet specific tension and compression requirements by defining objective functions, constraints and applying optimization algorithms. The tension and compression spring design problem seems simple, but it is difficult to solve due to complex nonlinear constraints. Therefore, the tension and compression spring design problem is widely used to verify and test the performance of various swarm intelligence algorithms.

[0004] The performance of the PSO algorithm depends largely on the balance between exploration (global search capability) and utilization (local search accuracy). Excessive emphasis on exploration will lead to inefficiency, while excessive utilization may lose diversity and converge to suboptimal solutions prematurely, so it is necessary to effectively balance these two parts. Like other metaheuristic algorithms, the PSO algorithm also has an inherent conflict between exploration and utilization capabilities, and relies on a single search strategy, resulting in loss of diversity and slow convergence. In addition, the algorithm is also very sensitive to rotational transformations, which further limits its scope of application. A major challenge of the particle swarm optimization algorithm is the lack of communication between particles, which can cause the algorithm to converge prematurely and fall into local optimality. Therefore, although the PSO algorithm can find better solutions in some cases, its effect in practical applications is often limited due to the above-mentioned defects. For this reason, the present invention proposes a tension and compression spring design method based on an improved particle swarm optimization (SuperMOP) algorithm. Summary of the invention

[0005] The purpose of the present invention is to provide a tension and compression spring design method based on an improved particle swarm optimization algorithm, aiming to solve the problems raised in the above background technology.

[0006] To achieve the above object, the present invention provides the following technical solutions:

[0007] The tension and compression spring design method based on the improved particle swarm optimization algorithm includes the following steps:

[0008] Step 1: Establish a mathematical model for the tension and compression spring design problem with weight as the fitness function, and encode the variables in the tension and compression spring design as the individual position vectors of the particles;

[0009] Step 2: Based on the elite profile strategy and the advantage combination learning strategy, an improved particle swarm optimization algorithm is constructed. The model executes the improved particle swarm optimization algorithm to calculate the input fitness function to find the optimal value;

[0010] Step 3: After the model is calculated, the current best variable and the best solution are output;

[0011] The improved particle swarm optimization algorithm includes:

[0012] At the beginning of each cycle, the population is divided into excellent subgroups X according to the fitness ranking of the particles. t o and the general subgroup X t c ; define X t c The speed is X t o The best historical solution is X t c The best historical solution is

[0013] The particle size in the population is N, and the particle size in the excellent subgroup and the ordinary subgroup is N / 2; according to the elite archive strategy, the excellent particles in the excellent subgroup are divided into A t , B t , C t Three files, File A t The maximum length is N / 2, file A t Expressed as and in represents the individual best solution of the i-th particle in the excellent subgroup in the t-th iteration; File B t and File C t The maximum length of file B is N. t Expressed as File C t Expressed as Excellent particles directly enter the next iteration, ordinary particles in the ordinary subgroup are updated according to the six learning strategies, and learning particles in the six learning strategies are generated according to the advantage combination learning strategy; conditional judgment is performed on particles that have completed the update, and particles that meet the conditions are mutated.

[0014] Furthermore, the mathematical model includes three design variables, and a fitness function is defined based on the variables, as follows:

[0015] The three design variables include: spring coil diameter, denoted as x1 or d; spring coil diameter, denoted as x2 or D; number of coils denoted as x3 or P;

[0016] Fitness function:

[0017]

[0018] Constraints: Consider the constraints of minimum deflection, vibration frequency, and shear stress, including one linear constraint and three nonlinear constraints:

[0019]

[0020] Boundary constraints:

[0021] 0.05≤x1≤2.00,0.25≤x2≤1.30,2.00≤x3≤15.00;

[0022] Among them, x1 represents the spring coil diameter, x2 represents the spring coil diameter, x3 represents the number of winding coils; g1(x), g2(x), g3(x), and g4(x) represent four different constraint functions.

[0023] Furthermore, the process of updating the common particles in the common subgroup according to the six learning strategies is as follows:

[0024] Define the three learning particles generated from the three archives as and The particles to be updated are particle The fitness of The current global best particle is Gbest i , the average fitness of all particles in the current common subgroup is meanct;

[0025] when When the particle The fitness of is better than the average fitness of all particles in the common subgroup; randomly selected and One of them and Gbest i The particle update process for the guide is:

[0026] Formula 1:

[0027] when When , the average fitness of all particles in the common subgroup is better than that of particles The fitness of and The update process of the two guided particles in is:

[0028] Formula 2:

[0029] in represents the velocity of the ith particle in the common subgroup in the t+1th iteration; ω is the inertia weight, which is a random number in [0,1]; R1 and R2 are random numbers in [0,1]; V t c,i represents the velocity of the i-th particle in the ordinary subgroup in the t-th iteration; X t c,i represents the position of the i-th particle in the common subgroup in the t-th iteration.

[0030] Furthermore, the calculation formula of the fitness average meanct of all particles in the current common subgroup is as follows:

[0031] Formula 3:

[0032] Where N represents the population size; represents the position of the i-th particle in the common subgroup; Represents particles The fitness of .

[0033] Furthermore, in the step of performing conditional judgment on the particles that have completed the update and performing mutation on the particles that meet the conditions, the mutation operation of the particle i that is triggered needs to meet the following conditions:

[0034] Condition 1: The historical optimal solution of particle i has stagnated for k generations, as described below:

[0035] Formula 4: ST i >k;

[0036] Among them, ST i is the current stagnation iteration number of particle i;

[0037] Condition 2: The average distance between the current position of particle i and its most recent n positions is less than the threshold λ, which is described as follows:

[0038] Formula 5:

[0039] in is the position of particle i in generation t; represents the position of particle i in the tjth generation; j represents a positive integer between [1, n];

[0040] Condition 3: The distance between the current position of particle i and its previous nth position is less than the threshold λ, which is described as follows:

[0041] Formula 6:

[0042] When the three conditions are met, the Gaussian mutation operator or the Cauchy mutation operator is executed alternately on particle i until the fitness of the particle after mutation is better than the original value.

[0043] Further, the Gaussian mutation operator or Cauchy mutation operator is executed on the global optimal solution using the following formula, that is, Gaussian mutation or Cauchy mutation is performed:

[0044] Formula 7:

[0045] GBest newpos Indicates the updated position of the global best particle; GBest i Represents the current global best particle; Gaussion(0,1) represents the random number generated by the normal Gaussian distribution function; Cauthy(0,1) represents the random number generated by the standard Cauchy distribution function.

[0046] Compared with the prior art, the present invention has the following beneficial effects:

[0047] The SuperMOP algorithm proposed in this paper combines the elite archive strategy and the dominant combination learning strategy, and introduces a mutation operator based on the individual level. It assigns different roles to particles according to their performance, thereby dynamically updating the learning strategy. The results show that the SuperMOP algorithm effectively balances the exploration and utilization process, improves the population diversity, and enhances the diversity of learning samples. It provides a new model with powerful global search capabilities for the design problem of tension and compression springs, which is particularly suitable for solving complex and variable constraint engineering optimization problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 Flow chart of the method of the present invention.

[0049] Figure 2 The overall workflow of the SuperMOP framework.

[0050] Figure 3 Schematic diagram of tension and compression spring.

[0051] Figure 4is the number of functions for which the algorithm achieves optimal performance in three dimensions.

[0052] Figure 5 Ablation experiment results of SuperMOP algorithm with mutation operator. DETAILED DESCRIPTION

[0053] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0054] The specific implementation of the present invention is described in detail below in conjunction with specific embodiments.

[0055] One embodiment of the present invention provides a tension and compression spring design method based on an improved particle swarm optimization algorithm, the flow chart is as follows: Figure 1 As shown, the following steps are included:

[0056] Step 1: Establish a mathematical model for the tension and compression spring design problem with weight as the fitness function, and encode the variables in the tension and compression spring design as the individual position vectors of the particles;

[0057] Step 2: Based on the elite profile strategy and the advantage combination learning strategy, an improved particle swarm optimization algorithm is constructed. The model executes the improved particle swarm optimization algorithm to calculate the input fitness function to find the optimal value;

[0058] Step 3: After the model is calculated, the current best variable and the best solution are output;

[0059] like Figure 2 As shown, at the beginning of the Tth cycle, the elite archive strategy, the dominant combination learning strategy and the mutation operator are implemented for the N particles, and the updated particles are recombined into a population containing N particles and enter the T+1th cycle.

[0060] The mathematical model contains three design variables, and defines a fitness function based on the variables, as follows:

[0061] The three design variables include (see Figure 3 ): spring coil diameter, expressed as x1 or d; spring coil diameter, expressed as x2 or D; number of coils, expressed as x3 or P;

[0062] Fitness function:

[0063]

[0064] Constraints: Consider the constraints of minimum deflection, vibration frequency, and shear stress, including one linear constraint and three nonlinear constraints:

[0065]

[0066] Boundary constraints:

[0067] 0.05≤x1≤2.00,0.25≤x2≤1.30,2.00≤x3≤15.00;

[0068] Among them, x1 represents the spring coil diameter, x2 represents the spring coil diameter, x3 represents the number of winding coils; g1(x), g2(x), g3(x), and g4(x) represent four different constraint functions.

[0069] The improved particle swarm optimization algorithm includes:

[0070] (1) Elite Archive Strategy: The elite archive strategy has a dual purpose: improving the convergence of the algorithm and maintaining population diversity. By learning from individuals in the archive, particles can inherit and exchange valuable information and guide them to find promising solutions, preventing particle groups from gathering in local areas. This in turn expands the global search space and balances exploration and exploitation within the algorithm.

[0071] At the beginning of each cycle, the population is divided into excellent subgroups X according to the fitness ranking of the particles. t o and the general subgroup X t c ;definition:

[0072] For X t c The speed of the ordinary subgroup X t c Contains N / 2 particles, among which represents a common subgroup X t c The speed of each particle in

[0073] For X t o The best historical solution, excellent subgroup X t o Contains N / 2 particles, among which Denotes an excellent subgroup X t o The individual best solution for each particle in ;

[0074] For X t c The best historical solution is represents the common subgroup X tc The individual best solution for each particle in .

[0075] The particle size in the population is N, and the particle size in the excellent subgroup and the ordinary subgroup is N / 2; according to the elite archive strategy, the excellent particles in the excellent subgroup are divided into A t , B t , C t Three archives; excellent particles directly enter the next iteration, ordinary particles in the ordinary subgroup are updated according to the six learning strategies, and learning particles in the six learning strategies are generated according to the advantage combination learning strategy; conditional judgment is performed on particles that have completed the update, and particles that meet the conditions are mutated.

[0076] a. File A t By X t o The particle composition of A t The maximum length of A is N / 2, which is used to save the historical best solutions of all excellent particles. t It can be expressed as and in represents the individual best solution of the i-th particle in the excellent subgroup in the t-th iteration; Indicates file A t The archive information of each particle in the tth iteration, Indicates file A t Each piece of information in is composed of the individual best solutions in the tth iteration. Obviously, if the excellent subgroup and the ordinary subgroup are determined, then A is also determined. t , and will not change within a loop.

[0077] b. File B t To save X t There are some promising best historical solutions for all particles in , whose maximum length is the same as the population size, N. in Indicates file B t In one iteration, as the particle speed and position are updated, B t The individuals in are changing dynamically. If x t c,i Can find the best solution than the history A better solution, B t Update; where X t represents the population in the tth iteration, x t c,i represents the common subgroup X t c is the i-th particle in the t-th iteration.

[0078] c. File C t Used to save some promising global optimal solutions, whose maximum length is the same as the population size, which is N. in Indicates file C t After completing the tth cycle, the current global optimal solution is obtained, i.e. Gbest t Obviously, C t The individuals in a cycle are also dynamic.

[0079] (2) Dominant combination learning strategy: In the dominant combination learning strategy, each particle in the common subgroup with poor performance needs to learn from the learning particles generated from three archives.

[0080] The generation process of learning particles: Each file randomly selects particles from various dimensions to form learning particles. Assume that for a certain file, the learning particle learn x The generation process is that for each dimension j, we randomly select a particle from the archive Then the learning particle learn x The jth dimension is set as particle of The jth dimension of Represents particles The best individual solution.

[0081] Assume that the three learning particles generated from the three archives are and The particles to be updated are particle The fitness of The current global best particle is Gbest i , the average fitness of all particles in the current common subgroup is meanct;

[0082] Two archives are randomly selected to design six learning strategies (see Equation 1 and Equation 2 for details). The two selected archives will combine the advantages of all the particles in them to form two advantage combination learning particles. The three archives all store information about promising particles in the current population, most of which are better than particles in the ordinary subgroup. The two advantage combination learning particles generated by the external archives obtain advantage combinations from their respective corresponding archives, thereby achieving excellent guidance for particles in the ordinary subgroup. Based on the ordinary particles Fitness Whether it is lower than the average fitness of all particles in the current common subgroup, meanct selects the learning strategy;

[0083] when When the particle The fitness of is better than the average fitness of all particles in the common subgroup; according to this situation, randomly select and One of them and Gbest i The particle update process for the guide is:

[0084] Formula 1:

[0085] when When , the average fitness of all particles in the common subgroup is better than that of particles The fitness of and The update process of the two guided particles in is:

[0086] Formula 2:

[0087] in represents the velocity of the ith particle in the common subgroup in the t+1th iteration; ω is the inertia weight, which is a random number in [0,1]; R1 and R2 are random numbers in [0,1]; V t c,i represents the velocity of the i-th particle in the ordinary subgroup in the t-th iteration; X t c,i represents the position of the i-th particle in the common subgroup in the t-th iteration.

[0088] The calculation formula of the fitness average meanct of all particles in the above current common subgroup is as follows:

[0089] Formula 3:

[0090] Where N represents the population size; represents the position of the i-th particle in the common subgroup; Represents particles The fitness of .

[0091] (3) Individual-level mutation operator: The SuperMOP algorithm introduces an individual-level mutation operator, which combines the position distribution of each particle with its proximal fitness change.

[0092] The peak value of the Cauchy distribution function at the origin is small but the distribution at both ends is long. Using Cauchy mutation can generate larger disturbances near the current global optimal solution. Gaussian regional disturbance (i.e. Gaussian mutation) helps the algorithm to approach the global optimal solution area in the later stage, improves the local search ability of the algorithm, and enhances the ability of the algorithm to jump out of local extreme values.

[0093] The following conditions must be met to trigger the mutation operation of particle i:

[0094] Condition 1: The historical optimal solution of particle i has stagnated for k generations, as described below:

[0095] Formula 4: ST i >k;

[0096] Among them, ST i is the current stagnation iteration number of particle i;

[0097] Condition 2: The average distance between the current position of particle i and its most recent n positions is less than the threshold λ, which is described as follows:

[0098] Formula 5:

[0099] in is the position of particle i in generation t; represents the position of particle i in the tjth generation; j represents a positive integer between [1, n];

[0100] Condition 3: The distance between the current position of particle i and its previous nth position is less than the threshold λ, which is described as follows:

[0101] Formula 6:

[0102] When the three conditions are met, the Gaussian mutation operator or the Cauchy mutation operator is executed alternately on particle i until the fitness of the particle after mutation is better than the original value. The Gaussian mutation operator or the Cauchy mutation operator is executed on the global optimal solution using the following formula, that is, Gaussian mutation or Cauchy mutation:

[0103] Formula 7:

[0104] GBest newpos Indicates the updated position of the global best particle; GBest i Represents the current global best particle; Gaussion(0,1) represents the random number generated by the normal Gaussian distribution function; Cauthy(0,1) represents the random number generated by the standard Cauchy distribution function.

[0105] Embodiment 1, optimal solution to the tension and compression spring design problem;

[0106] The tension and compression spring design problem has three design variables, including the spring coil diameter x1, the spring coil diameter x2, and the number of coils x3. In each experiment, the maximum function evaluation times (MaxFEs) of each benchmark function is set to 150,000. To consider the randomness of the algorithm, each algorithm is executed independently 30 times. In order to evaluate the effectiveness of the SuperMOP algorithm proposed in the present invention, the SuperMOP algorithm is compared with PCFMO, HPSO, GSA, SCSO, GA, and GWO. Table 1 summarizes the best decision variables and optimal solutions obtained by the seven algorithms when solving the tension and compression spring problem.

[0107] Table 1 Best weight results of SuperMOP algorithm and other algorithms

[0108]

[0109] It can be seen from Table 1 that the best weight obtained by the present invention is 0.012665. Compared with PCFMO, HPSO, GSA, SCSO, GA, and GWO, the optimization result of the SuperMOP algorithm of the present invention is the best among all algorithms.

[0110] Example 2: Numerical verification using the unconstrained CEC2013 test suite;

[0111] The CEC2013 test suite has a total of 28 functions, of which more than 82% of the test functions are multimodal functions. We considered three different dimensions (30D, 50D, and 100D) to evaluate the global search capability of SuperMOP on unconstrained function problems. The maximum number of function evaluations (MaxFEs) for each benchmark function in each experiment is set to 5000D, where D represents the dimension. To consider the randomness of the algorithm, each benchmark function is executed 30 times independently.

[0112] In order to evaluate the effectiveness of the SuperMOP algorithm proposed in this paper, the SuperMOP algorithm is compared with eight state-of-the-art PSO variants: HGSPSO, EPSO, QPSO, AWPSO, EAPSO, CHCLPSO-I, CHCLPSO-II, and CHCLPSO-III. The time span of these algorithms is from 2004 to 2023, making the experimental analysis more comprehensive and convincing. Figure 4 The number of functions where the SuperMOP algorithm and the other eight algorithms achieve the best performance in three dimensions is shown. Figure 4 It can be seen that the number of functions for which the SuperMOP algorithm achieves the best performance is the highest in all three dimensions.

[0113] Example 3: Verify the statistical significance of the comparative experimental results;

[0114] The Wilcoxon rank sum test was used to determine whether there was a statistically significant difference between the results of SuperMOP and the comparison algorithm. Table 2 shows the comparison results, where + / = / - respectively indicates that the SuperMOP algorithm performs better, similar, and worse than a certain algorithm. The average value of 30 runs was calculated as the indicator MDT, and a smaller MDT value indicates that the algorithm performs better.

[0115] Table 2 Wilcoxon rank sum test results of SuperMOP algorithm and other algorithms

[0116]

[0117] From the data in Table 2, we can see that the SuperMOP algorithm significantly outperforms similar algorithms in three dimensions. At 30D, for all similar algorithms, SuperMOP obtained at least 13 '+' and at most 23 '+', which means that SuperMOP achieved significantly better results in at least 13 test problems. In addition, in three dimensions, compared with the other 8 algorithms, SuperMOP obtained more "+" than "-", indicating that the SuperMOP algorithm is statistically significantly better than other algorithms.

[0118] The average ranking of all algorithms in 28 functions is calculated based on Friedman-test. The results are shown in Table 3:

[0119] Table 3 Friedman-test results of SuperMOP algorithm and other algorithms in three dimensions

[0120]

[0121] From the data in Table 3, we can see that the SuperMOP algorithm has the best overall performance on 30D, 50D, and 100D. It also achieved the best ranking in the single dimension test.

[0122] Example 4: Verify the impact of mutation operators on algorithm performance;

[0123] The ablation experiment is performed on the CEC2013 test suite to compare the SuperMOP algorithm with and without mutation operator (NoMutationOperator). MDT ratio is the MDT ratio of the SuperMOP algorithm with and without mutation operator. Figure 5The comparison results of 30D are shown. The results show that the SuperMOP algorithm with mutation operator always achieves the best results. Among the 28 functions, the SuperMOP algorithm with mutation operator achieves the best average on 26 functions, while the SuperMOP algorithm without mutation operator only achieves the best results on 3 functions (F5, F13 and F25).

[0124] The above are only preferred embodiments of the present invention. It should be pointed out that, for those skilled in the art, several modifications and improvements can be made without departing from the concept of the present invention. These should also be regarded as the protection scope of the present invention. These will not affect the effect of the implementation of the present invention and the practicality of the patent.

Claims

1. A tension and compression spring design method based on an improved particle swarm optimization algorithm is characterized in that: The following steps are involved: Step 1: Establish a mathematical model for the tension and compression spring design problem with weight as the fitness function, and encode the variables in the tension and compression spring design as the individual position vectors of the particles; Step 2: Based on the elite profile strategy and the advantage combination learning strategy, an improved particle swarm optimization algorithm is constructed. The model executes the improved particle swarm optimization algorithm to calculate the input fitness function to find the optimal value; Step 3: After the model is calculated, the current best variable and the best solution are output; The improved particle swarm optimization algorithm includes: At the beginning of each cycle, the population is divided into excellent subgroups X according to the fitness ranking of the particles. t o and the general subgroup X t c ; define X t c The speed is X t o The best historical solution is X t c The best historical solution is The particle size in the population is N, and the particle size in the excellent subgroup and the ordinary subgroup is N / 2; according to the elite archive strategy, the excellent particles in the excellent subgroup are divided into A t , B t , C t Three files, File A t The maximum length is N / 2, file A t Expressed as and in represents the individual best solution of the i-th particle in the excellent subgroup in the t-th iteration; File B t and File C t The maximum length of file B is N. t Expressed as File C t Expressed as Excellent particles directly enter the next iteration, ordinary particles in the ordinary subgroup are updated according to the six learning strategies, and learning particles in the six learning strategies are generated according to the advantage combination learning strategy; conditional judgment is performed on particles that have completed the update, and particles that meet the conditions are mutated.

2. The method according to claim 1, characterized in that The mathematical model contains three design variables, and defines a fitness function based on the variables, as follows: The three design variables include: spring coil diameter, denoted as x1 or d; spring coil diameter, denoted as x2 or D; number of coils denoted as x3 or P; Fitness function: Constraints: Consider the constraints of minimum deflection, vibration frequency, and shear stress, including one linear constraint and three nonlinear constraints: Boundary constraints: 0.05≤x1≤2.00,0.25≤x2≤1.30,2.00≤x3≤15.00; Among them, x1 represents the spring coil diameter, x2 represents the spring coil diameter, x3 represents the number of winding coils; g1(x), g2(x), g3(x), and g4(x) represent four different constraint functions.

3. The method according to claim 1, characterized in that The process of updating the ordinary particles in the ordinary subgroup according to the six learning strategies is as follows: Define the three learning particles generated from the three archives as and The particles to be updated are particle The fitness of The current global best particle is Gbest i , the average fitness of all particles in the current common subgroup is meanct; when When the particle The fitness of is better than the average fitness of all particles in the common subgroup; randomly selected and One of them and Gbest i The particle update process for the guide is: Formula 1: when When , the average fitness of all particles in the common subgroup is better than that of particles The fitness of and The update process of the two guided particles in is: Formula 2: in represents the velocity of the ith particle in the common subgroup in the t+1th iteration; ω is the inertia weight, which is a random number in [0,1]; R1 and R2 are random numbers in [0,1]; V t c,i represents the velocity of the i-th particle in the ordinary subgroup in the t-th iteration; X t c,i represents the position of the i-th particle in the common subgroup in the t-th iteration.

4. The method according to claim 3, characterized in that The calculation formula of the fitness average value meanct of all particles in the current common subgroup is as follows: Formula 3: Where N represents the population size; represents the position of the i-th particle in the common subgroup; Represents particles The fitness of .

5. The method according to claim 1, characterized in that In the step of performing conditional judgment on the particles that have completed the update and performing mutation on the particles that meet the conditions, the mutation operation of the particle i that is triggered must meet the following conditions: Condition 1: The historical optimal solution of particle i has stagnated for k generations, as described below: Formula 4: ST i >k; Among them, ST i is the current stagnation iteration number of particle i; Condition 2: The average distance between the current position of particle i and its most recent n positions is less than the threshold λ, which is described as follows: Formula 5: in is the position of particle i in generation t; represents the position of particle i in the tjth generation; j represents a positive integer between [1, n]; Condition 3: The distance between the current position of particle i and its previous nth position is less than the threshold λ, which is described as follows: Formula 6: When the three conditions are met, the Gaussian mutation operator or the Cauchy mutation operator is executed alternately on particle i until the fitness of the particle after mutation is better than the original value.

6. The method according to claim 5, characterized in that The Gaussian mutation operator or Cauchy mutation operator is executed on the global best particle using the following formula, that is, Gaussian mutation or Cauchy mutation is performed: Formula 7: GBest newpos Indicates the updated position of the global best particle; GBest i Represents the current global best particle; Gaussion(0,1) represents the random number generated by the normal Gaussian distribution function; Cauthy(0,1) represents the random number generated by the standard Cauchy distribution function.

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