Spring optimization design method and system

Abstract

The invention discloses a spring optimization design method and system, relates to the technical field of machine learning, and aims to solve the problem that in the prior art, it is difficult to ensure that spring design obtains an optimal optimization result under various conditions through a single optimization algorithm. Therefore, the invention provides a mixed meta-heuristic algorithm which combines an improved black wing plenum algorithm with a doliolaria algorithm. Specifically, the method comprises the following steps: initializing a population by adopting a good point set method, and calculating fitness values of individuals; dividing the population into an elite sub-population and an exploration sub-population; focusing the elite sub-population near the current optimal solution, and performing local search to improve the precision of the solution; exploring the sub-population is devoted to maintain population diversity and execute global search to enhance the exploration ability of the algorithm. According to the method, the improved black-wing plinual algorithm and the improved doliolaria algorithm are alternately adopted to update individuals in the population, the convergence speed and the solving precision are improved, exploration and development in the searching process are balanced, and an efficient and reliable solution is provided for spring optimization design.

Description

Technical Field

[0001] This application relates to the field of machine learning technology, and in particular to a spring optimization design method and system. Background Technology

[0002] Springs, as indispensable components in the industrial field, are widely used in machinery, automobiles, electronics, and many other sectors. Their design quality directly affects the performance and reliability of equipment. In practical applications, springs must meet diverse requirements, such as load-bearing capacity, elastic modulus, and fatigue resistance, while also considering lightweight design to reduce equipment weight and improve energy efficiency. However, traditional spring design methods have many limitations, relying mainly on empirical formulas and manual calculations. This makes it difficult to accurately balance the relationship between various performance indicators and quality, especially when facing complex design requirements, making it difficult to find the optimal design solution.

[0003] With the rapid development of computer technology, optimization algorithms have been gradually introduced into the field of spring design, providing new ideas for solving this problem. Optimization algorithms can systematically search and optimize various parameters in spring design through mathematical modeling, aiming to find the optimal solution that meets performance requirements. Among many optimization algorithms, heuristic algorithms have attracted much attention due to their unique advantages. Based on their design principles and characteristics, heuristic algorithms are mainly divided into two categories: individual-based algorithms and population-based algorithms. Compared to algorithms based on a single individual, population-based algorithms exhibit a more powerful global search capability. These algorithms construct a set of candidate solutions (i.e., a population) and simulate the cooperation and competition mechanisms among individuals, enabling them to simultaneously explore different regions of the solution space, thus increasing the likelihood of finding the globally optimal solution. Even if some individual solutions perform poorly, the overall population may still contain excellent solutions, giving the algorithm strong robustness. Furthermore, the parallel execution of the population significantly improves computational efficiency.

[0004] In the optimization problem of spring design, metaheuristic algorithms provide an effective technical means to solve complex optimization problems by mathematically modeling evolutionary behavior. These algorithms can find near-optimal solutions within an acceptable computational cost. However, due to the random search characteristic of metaheuristic algorithms, the obtained solution cannot be guaranteed to be the global optimum; therefore, they are usually regarded as approximate methods for solving complex optimization problems. Nevertheless, these solutions are often close enough to the original optimum, and even completely equivalent to the optimum in some cases, and can therefore be applied as near-optimal solutions in practical designs. In the context of the spring design optimization problem, the performance differences of different metaheuristic algorithms mainly lie in the strategies adopted when searching and updating candidate solutions. The NFL (No Free Lunch) theorem further points out that the excellent performance of an algorithm on a specific optimization problem does not mean that it will maintain the same efficiency on other problems. This means that relying solely on a single algorithm cannot guarantee finding the best result for the spring design optimization problem in all cases. Therefore, in practical applications, it is often necessary to combine the advantages of multiple algorithms, or to improve and mix existing algorithms, to improve the optimization effect and meet the diverse needs of spring design.

[0005] When multiple algorithms are combined, the optimization problem becomes very challenging. Therefore, it is necessary to provide a technique that can solve the complex optimization problem in spring design. Summary of the Invention

[0006] This application provides a spring optimization design method and system to solve the problem that a single optimization algorithm in the prior art cannot guarantee the best results in all cases when optimizing spring design.

[0007] On one hand, embodiments of this application provide a spring optimization design method, including: Obtain the performance data of the spring and establish the existing objective function and constraints; The population is initialized using the optimal point set method to obtain an initial population, which contains multiple individuals, each representing a potential spring design scheme. The fitness value of each individual is evaluated according to the objective function, and the individuals are divided into elite subpopulation and exploratory subpopulation based on the fitness value; During the iteration process, the Black-winged Kite Algorithm and the Salicet Algorithm are used to update individuals alternately. When updating, the Black-winged Kite Algorithm uses an elite differential mutation strategy to enhance the diversity of the initial population; the Salicet Algorithm uses a sine and cosine exploration strategy to improve the global search capability. After each iteration, the best individual is selected based on its fitness value to form the basis of the next generation population; Determine if the termination condition is met. If it is, select the optimal individual as the final spring design. If not, continue iterating.

[0008] In one possible implementation, after determining the fitness value of each individual in the initial population, the individuals with the best fitness values ​​are divided into an elite subpopulation, and the individuals with the worst fitness values ​​are divided into an exploratory subpopulation.

[0009] In one possible implementation, the Blackwing Kite algorithm utilizes an elite differential mutation strategy to account for the impact of individuals with poor fitness, thereby reducing the risk of premature convergence.

[0010] In one possible implementation, the tunic algorithm uses sine and cosine functions and the optimal individual position to guide the follower position update.

[0011] In one possible implementation, when the number of iterations is odd, the initial population is updated using the Black-winged Kite Algorithm or the population updated by the Tulipe Algorithm; when the number of iterations is even, the population updated by the Black-winged Kite Algorithm is updated using the Tulipe Algorithm.

[0012] On the other hand, embodiments of this application also provide a spring optimization design system, including: The function creation module is used to obtain the performance data of the spring and establish the existing objective function and constraints. The initialization module is used to initialize the population using the optimal point set method to obtain an initial population. The initial population contains multiple individuals, and each individual represents a potential spring design scheme. The subpopulation partitioning module is used to evaluate the fitness value of each individual according to the objective function, and to divide the individuals into elite subpopulation and exploratory subpopulation based on the fitness value; The iterative module is used to alternately update individuals using the Black-winged Kite Algorithm and the Salicet Algorithm during the iteration process. During the update, the Black-winged Kite Algorithm uses an elite differential mutation strategy to enhance the diversity of the initial population; the Salicet Algorithm uses a sine and cosine exploration strategy to improve the global search capability; after each iteration, the optimal individual is selected based on the fitness value as the basis for the next generation population. The parameter output module is used to determine whether the termination condition is met. If it is met, the optimal individual is taken as the final spring design scheme; otherwise, the iteration continues.

[0013] The spring optimization design method and system disclosed in this application have the following advantages: (1) The Black-winged Kite algorithm and the Salicornia algorithm are combined to form a hybrid meta-heuristic algorithm, which improves the convergence speed and solution accuracy.

[0014] (2) The population is divided into an elite subpopulation and an exploratory subpopulation according to the fitness value. The elite subpopulation is close to the current optimal solution and is responsible for local search, while the exploratory subpopulation helps maintain population diversity and is responsible for global search, so as to ensure that exploration and development can be balanced during the search process.

[0015] (3) Based on the Black-winged Kite algorithm, the algorithm is further improved by introducing an elite differential mutation strategy to enhance local search capability, improve solution accuracy and accelerate convergence speed. The Tulip algorithm incorporates sine and cosine functions to optimize position update, thereby enhancing the search capability for the global optimal solution and finally improving convergence speed and solution accuracy.

[0016] (4) Initialization is performed using a set of optimal points to ensure a uniform population distribution. This effectively avoids the problem of uneven population distribution during random initialization, which can lead to reduced population diversity and lower quality, thus affecting the convergence speed of the algorithm. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 A flowchart illustrating a spring optimization design method provided in this application embodiment.

[0019] Figure 2 This diagram illustrates the ranking of the BKASSA algorithm and the comparison algorithm provided in this application embodiment on CEC2005.

[0020] Figure 3 This is a schematic diagram showing the ranking of the BKASSA algorithm and the comparison algorithm provided in the embodiments of this application at CEC2017.

[0021] Figure 4 A schematic diagram of the design principle of a tension / compression spring provided for an embodiment of this application.

[0022] Figure 5 These are the convergence curves of the BKASSA algorithm and the comparison algorithm provided in the embodiments of this application in the design of tension / compression springs. Detailed Implementation

[0023] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0024] Figure 1 A flowchart illustrating a spring optimization design method provided in this application embodiment. This application embodiment provides a spring optimization design method, including: S100: Obtain the performance data of the spring and establish the existing objective function and constraints.

[0025] For example, the core of the tension spring design optimization problem lies in minimizing material and manufacturing costs while meeting performance and safety constraints. Design variables include wire diameter. Spring outer diameter Valid number of laps The objective function is typically expressed as a combination of these variables, a mathematical expression reflecting total cost or performance. Constraints include maximum stress. Maximum compression or stretch Minimum safety factor and space size limitations The design variables must take values ​​within a specific range. The specific mathematical expression is as follows: variable:

[0026] Objective function: min

[0027] Constraints:

[0028]

[0029]

[0030]

[0031] Variable range:

[0032]

[0033]

[0034] in, x 1. x 2 and x3 represents d , D and N The parameters, This represents the parameter set, where min indicates taking the minimum value. This represents the objective function.

[0035] S110. The population is initialized using the optimal point set method to obtain an initial population. The initial population contains multiple individuals, each representing a potential spring design scheme. For example, in the embodiments of this application, the optimal point set method is used to initialize the initial population.

[0036] Specifically, assuming yes The unit cube in Euclidean space, if Then the optimal point set can be represented as:

[0037] Its deviation satisfy:

[0038] in, It is only with If a constant is related to any positive number, then it is called a constant. For the best collection, For best results, For the first s A good point. Represents taking the decimal part. To represent the number of points, take as follows:

[0039] in, It is to satisfy The smallest prime number. Mapping it to the search space:

[0040] in, and Individual number The upper and lower bounds of a dimension, For the search space.

[0041] S120 evaluates the fitness value of each individual based on the objective function, and divides the individuals into elite subpopulation and exploratory subpopulation based on the fitness value.

[0042] For example, after calculating the fitness value of each individual, all individuals can be sorted according to their fitness values. After sorting, the individuals with the best fitness values ​​(i.e., the highest number), such as half of the individuals, are selected as the elite subpopulation. They are close to the current optimal solution and are responsible for local search. Conversely, the individuals with the worst fitness values ​​(i.e., the lowest number), such as the other half of the individuals, constitute the exploratory subpopulation. They help maintain population diversity and are responsible for global search. This partitioning process is dynamic. After each iteration, the individuals are re-sorted and the subpopulations are re-partitioned based on the latest fitness values ​​to ensure that the algorithm can balance exploration and exploitation during the search process, improving convergence speed and solution accuracy.

[0043] In S130, during the iteration process, the Black-winged Kite Algorithm and the Salicet Algorithm are used to update individuals alternately. During the update, the Black-winged Kite Algorithm uses an elite differential mutation strategy to enhance the diversity of the initial population; the Salicet Algorithm uses a sine and cosine exploration strategy to improve the global search capability.

[0044] For example, in this application embodiment, the Black-winged Kite algorithm or the Sea Squirt algorithm is selected according to the odd or even number of iterations. That is, when the number of iterations is odd, the initial population is updated using the Black-winged Kite algorithm or the population updated by the Sea Squirt algorithm. When the number of iterations is even, the population updated by the Black-winged Kite algorithm is updated using the Sea Squirt algorithm.

[0045] The Black-winged Kite Algorithm is a novel swarm intelligence optimization algorithm, a metaheuristic optimization algorithm inspired by the migration and predation behaviors of black-winged kites. The basic Black-winged Kite Algorithm process is as follows: Step 1: Initialize the population. Based on the upper and lower bounds of each dimension of the search space, randomly initialize the population. i The position of only the black-winged kite ,Right now:

[0046] in, and These are the individual Black-winged Kite individuals in the 1st... The upper and lower bounds of a dimension, It is a value randomly selected between [0,1].

[0047] Step 2: Calculate the fitness value of each black-winged kite according to the objective function. The half of the black-winged kites with the best fitness value are the leaders and also the best individuals, while the remaining half are the followers. It should be understood that in the first iteration, since the population is the initial population, the leaders formed in this step are the elite subpopulation, and the followers are the exploratory subpopulation.

[0048] Step 3: Attack Strategy. For the leader, the black-winged kite adjusts the angle of its wings and tail according to wind speed while hunting, silently circling to observe its prey before swiftly swooping down and attacking. The following formula is a mathematical model of the leader's black-winged kite attack behavior:

[0049]

[0050] in, and They represent the first t +1 generation and the t The position of the Black-winged Kite. R It is a random number in the interval [0,1]. It is a constant with a value of 0.9. This represents the current iteration number. This represents the total number of iterations.

[0051] Step 4: Migration Strategy. For followers, migration is typically led by a leader. If the fitness value of the current population is lower than that of the random population, the leader will relinquish leadership and join the migrating population, thus becoming a follower. Conversely, if the fitness value of the current population is higher than that of the random population, the leader will continue to guide the population to its destination. The following formula is a mathematical model of the migration behavior of black-winged kites that are followers:

[0052]

[0053] in, Representative at the t The current position of the black-winged kite in the next iteration. This indicates that the current Black-winged Kite is in the [number]th [year]. t Fitness in the next iteration This represents the fitness of a random individual during the iteration. This represents the Cauchy variant.

[0054] It should be understood that the above is the basic Black-winged Kite algorithm flow. In the method for spring optimization design in this application, it is not necessary to perform the initialization of the population operation in step 1.

[0055] As seen in step 4, in the Black-winged Kite algorithm, the position update method for individuals in the population involves generating new individuals near the current individual and the current best individual. This means other individuals in the population move towards the current best individual. If it's a local optimum, as iterations continue, the search individuals in the population will cluster around the best individual, easily leading to premature convergence. Furthermore, the mathematical model of Black-winged Kite migration only considers the impact of Cauchy mutation on the optimal solution, neglecting individuals with poor fitness. Individuals with poor fitness can maintain population diversity and increase the search range. Additionally, the Black-winged Kite algorithm lacks mutation operations. Therefore, the first formula below uses elite individuals to replace the best individual to increase the population search range and considers the impact of the worst individual, completing the mutation processing for individuals. In each iteration, the second formula below selects multiple individuals from the current population to form an elite group, and randomly selects one individual from the elite group as the elite individual. The elite individual refers to the best individual in the population. The specific formula is as follows:

[0056]

[0057] in, It is a mutant individual, shrinkage rate factor It is an important control parameter. It is mainly used to control the magnitude of the difference vector. An elite individual is a random individual among the top 10% of individuals in a population with the highest fitness value. It is the individual with the worst adaptability. , They are two random individuals in the population. express x i Adaptability, This indicates that fitness is sorted.

[0058] Furthermore, the tunicate algorithm uses sine and cosine functions and the optimal individual position to guide the follower position update. Its method for updating the population after the Black-winged Kite algorithm update is as follows: Determine the fitness value of individuals in the population after the black-winged kite algorithm update, sort the individuals in descending order of fitness value, and use the individual with the best fitness value as the food source. Individuals are divided into leaders and followers based on their fitness values; Update the leader based on the food source, and update the current followers based on the leader or the previous follower; Determine if the cutoff condition has been met. If it has, use the individual with the best fitness value as the optimal parameter for the spring design. If not, continue iterating.

[0059] Specifically, the salps swarm algorithm is a novel metaheuristic algorithm proposed by Mirjalili et al. In this algorithm, the salps swarm is searched in the form of chains, resulting in a simple structure, almost no parameter settings, and good solution performance. The goal of the salps swarm algorithm is to find the location of food sources. Mathematically, this is represented as the optimal solution. The basic process of the tunic algorithm is as follows: Step 1: Initialize the population. Randomly initialize the population based on the upper and lower bounds of each dimension of the search space. i Only the sea squirt in the first j Dimensional position ,Right now:

[0060] Step 2: Calculate the fitness value of each tunicate according to the objective function. The location of the tunicate with the best fitness value is the location of the food source.

[0061] Step 3: Determine the leaders and followers. The salps in the first half of the salps chain are the leaders, and the rest are the followers.

[0062] Step 4: Update the leader's position. During the movement and foraging process of the tunic, the leader's position is updated using the following formula:

[0063]

[0064] in Representing dimensions The position of the first individual in the group is the position of the leader. As can be seen from the formula, the leader only updates its position to the food source. For food sources in dimension The position in the middle, Representing dimensions j The upper realm, Representing dimensions j The upper boundary. As a balancing factor, this variable influences the balance between the global exploration and local development phases, as shown in the formula above. Wherein, This represents the current iteration number. This represents the total number of iterations. (Parameter) and It is a uniformly generated random number within the interval [0,1], which determines the dimension. The direction and step size of the next position.

[0065] Step 5: Update the position of the followers using the following formula:

[0066] in, i ≥2, Indicates the first j In the dimension i The position of each follower. It is the position of the individual currently ahead of it.

[0067] Step 6: Perform boundary processing on each dimension of the updated individual. Determine if the required number of iterations has been met. If yes, output the result; otherwise, return to Step 4 to continue iterative evolution.

[0068] It should be understood that the above is the basic tunic algorithm flow. In the method of spring optimization design in this application, it is not necessary to perform the initialization of the population operation in step 1.

[0069] As can be seen from step 5 of the tunic algorithm, the lack of optimal information during the local development phase may hinder rapid convergence in the later stages. Furthermore, when followers move closer to the leader's optimal position, it can cause short-term population aggregation, which, while improving the algorithm's convergence speed, can easily lead to the algorithm getting trapped in local optima.

[0070] This application utilizes sine and cosine functions to update the follower. Sine and cosine functions are not only easy to implement but also exhibit good convergence. Utilizing their oscillatory characteristics for optimization can reduce the risk of the algorithm getting trapped in local optima. The improved follower position is shown in the following formula:

[0071]

[0072]

[0073]

[0074] in, , All are random numbers that follow a uniform distribution. , , It is the globally optimal position at the current iteration number.

[0075] S140: After each iteration, the optimal individual is selected based on its fitness value as the basis for the next generation of the population.

[0076] S150: Determine if the termination condition is met. If it is met, select the optimal individual as the final spring design. If not, continue iterating.

[0077] This application embodiment also provides a spring optimization design system, the system comprising: The function creation module is used to obtain the performance data of the spring and establish the existing objective function and constraints. The initialization module is used to initialize the population using the optimal point set method to obtain an initial population. The initial population contains multiple individuals, and each individual represents a potential spring design scheme. The subpopulation partitioning module is used to evaluate the fitness value of each individual according to the objective function, and to divide the individuals into elite subpopulation and exploratory subpopulation based on the fitness value; The iterative module is used to alternately update individuals using the Black-winged Kite Algorithm and the Salicet Algorithm during the iteration process. During the update, the Black-winged Kite Algorithm uses an elite differential mutation strategy to enhance the diversity of the initial population; the Salicet Algorithm uses a sine and cosine exploration strategy to improve the global search capability; after each iteration, the optimal individual is selected based on the fitness value as the basis for the next generation population. The parameter output module is used to determine whether the termination condition is met. If it is met, the optimal individual is taken as the final spring design scheme; otherwise, the iteration continues.

[0078] The performance of the proposed method will be evaluated using three CEC (Conference on Evolutionary Computation) test suites (CEC2005, CEC2011, and CEC2022). Two different categories of optimization algorithms were used to highlight the performance of the proposed method: the first category includes recently proposed metaheuristic algorithms such as the Black-winged Kite Algorithm (BKA), the Harris Eagle Algorithm (HHO), the Runge-Kutta Algorithm (RUN), the Newton-Raphson Algorithm (NRBO), the Great Wall Construction Algorithm (GWCA), and the Flamingo Search Algorithm (FSA); the second category includes some highly cited classic algorithms such as the Grey Wolf Algorithm (GWO), the Particle Swarm Optimization Algorithm (PSO), the Differential Evolution Algorithm (DE), the Salicylict Algorithm (SSA), the Sine and Cosine Optimization Algorithm (SCA), and the Whale Algorithm (WOA). Performance metrics such as average fitness, standard deviation, convergence curve, Wilcoxon rank-sum test, and Frideman test were also used to highlight the performance differences between the proposed method and other methods.

[0079] 1. Experimental conditions and methods The hardware platform is as follows: CPU is Core i5, 2.1GHz, and memory is 8GB.

[0080] The software platform is MATLAB R2020b.

[0081] 2. Comparison of algorithms and parameter settings The settings for each parameter in the comparison algorithm are shown in Table 1 below.

[0082] Table 1 Algorithm parameter settings

[0083] 3. Parameter sensitivity analysis The experiment aimed to test the impact of changing the parameter values ​​of the method in this application on its performance. The method in this application mainly includes two key parameters: P and F. P is used to adjust the switching between attack strategies of the Blackwing Kite algorithm, significantly affecting the overall accuracy and stability of the algorithm. F is the scaling factor for the elite differential mutation strategy, and its value directly affects the degree of mutation. In the experiment, one parameter was fixed at 0.5, and the other parameter was gradually increased by 0.1 each time until it reached 1, to demonstrate the impact of parameter changes on the performance of the method in this application. The experiment was simulated using CEC-2022, and the results are shown in Tables 2 and 3.

[0084] Table 2. Analysis results of CEC2022 parameter P

[0085] Table 3. Analysis results of CEC2022 parameter F

[0086] As shown in Table 2 of the CEC-2022 test set results, the hybrid algorithm BKASSA, which combines the Black-winged Kite algorithm and the Tulipede algorithm in this application, performs best on 9 functions with P=0.9, achieving 75% success. With P=0.7, it performs best on functions F4 and F7; while with P=1.0, it performs best on function F9. Although P=0.9 does not yield optimal results on functions F4, F7, and F9, its results are similar to those with P=0.7 and P=1.0. Table 3 shows that when F=0.9, BKASSA performs best on 11 functions, accounting for 91.6%. Based on the comprehensive analysis of Tables 2 and 3, it is concluded that the BKASSA algorithm achieves better optimization results when the parameter P=F=0.9.

[0087] 4. CEC-2005 Kit Testing and Evaluation To test the performance of BKASSA, simulation experiments compared the proposed algorithm BKASSA with 23 benchmark problems in the CEC-2005 benchmark suite using seven classic algorithms. The algorithms applied were GWO, PSO, DE, WOA, SCA, basic BKA, and SSA. The maximum number of iterations for all algorithms was 1000. The mean and standard deviation are recorded in Table 4 to evaluate the performance of different optimization algorithms, with the best result for each metric in each function highlighted in bold.

[0088] Table 4. Results of BKASSA on the CEC-2005 benchmark function.

[0089] In CEC-2005, unimodal benchmark functions (F1-F7) have only one global optimum and are often used to evaluate the algorithm's local exploitation capabilities. Multimodal benchmark functions (F8-F23) have one global optimum and many local optima, which increase exponentially with the function's dimension. In most cases, multimodal benchmark functions are difficult to optimize because it is hard to escape many local optima and reach the global optimum. Therefore, multimodal benchmark functions are used to evaluate the algorithm's global exploration capabilities.

[0090] As can be seen from the results in Table 4, BKASSA achieved the best results in all functions except F6, F12, and F13, obtaining 20 optimal results, accounting for 86.9%. The results given by BKASSA demonstrate its advantage over other algorithms.

[0091] 5. CEC-2017 Kit Testing and Evaluation The experiment used CEC-2017 as the test set function for the comparative testing experiments. CEC-2017 contains 29 benchmark test functions, divided into unimodal functions (F1), multimodal functions (F3-F10), mixed functions (F11-F20), and combined functions (F21-F30). To further verify the performance of BKASSA, a second function (F2) was added to the experiment. The problem dimension of all selected functions was 30 (D=30). The experiment compared BKASSA with the standard SSA, BKA, and five other state-of-the-art algorithms: HHO, RUN, NRBO, GWCA, and FSA. The mean and standard deviation of the experimental results are shown in Table 5.

[0092] Table 5. Results of BKASSA on the CEC-2017 benchmark function

[0093] Analysis of Table 5 shows that among the 30 test functions in CEC-2017, the BKASSA algorithm achieved 27 optimal results, accounting for 90%, surpassing the other 7 algorithms. For functions F1-F6, F8-F9, F11-F20, and F22-F30, BKASSA significantly outperformed the standard SSA, BKA, and the other 5 latest algorithms. Regarding standard deviation, BKASSA has a smaller standard deviation while maintaining good optimal results, or it can consistently find the optimal solution. Therefore, BKASSA is superior to HHO, RUN, NRBO, GWCA, and FSA in terms of standard deviation. Standard deviation reflects the dispersion of the dataset; a smaller standard deviation indicates lower dispersion and better stability of the experimental results. This suggests that BKASSA's results have lower dispersion and better algorithm stability.

[0094] Take the average of the rank values ​​in Table 6-7, and... Figure 2-3 The figure shows a complete comparison of the Friedman average rankings of all algorithms on the CEC2017 test function. It can be seen from the figure that BKASSA is the best among all compared algorithms because it has the highest average ranking. Therefore, this algorithm is capable of solving complex problems and can replace previous algorithms.

[0095] Table 6. Friedman Test Ranking of BKASSA and Comparison Algorithm on CEC-2005 Test Set

[0096] Table 7. Friedman Test Ranking of BKASSA and Comparison Algorithm on CEC-2017 Test Set

[0097] 6. Nonparametric statistical analysis To comprehensively evaluate the performance of BKASSA, the Wilcoxon signed-rank test and the Friedman test were used to evaluate BKASSA and its comparison algorithms. The Wilcoxon signed-rank test was used to compare the median deviation of related samples, while the Friedman test was used to compare the median deviation of multiple groups of related samples. The tests were conducted on different test sets at a 95% significance level. The test results are shown in Tables 8 and 9, where "+" indicates that the reference algorithm is superior to the comparison algorithm, "-" indicates that the reference algorithm is inferior to the comparison algorithm, and "=" indicates that there is no significant difference between the two. Observing these results, it was found that the BKASSA algorithm has fewer "-" signs and more "+" and "=" signs, indicating that in most cases, the performance of the BKASSA algorithm is not inferior to the comparison algorithm. Furthermore, Tables 6-7 and... Figure 2-3 The table displays the ranking and average ranking of the Friedman test on different test sets. As can be seen from the table, the BKASSA algorithm ranks first in both the CEC2005 and CEC2017 benchmark functions, and also ranks first in average ranking. These statistics not only validate the superior performance of the BKASSA algorithm on individual benchmark functions, but also reliably assess its practicality in multiple optimization problems.

[0098] Table 8. Wilcoxon Sign-Rank Test Algorithm for BKASSA and its Comparison on the CEC-2005 Test Set

[0099] Table 9. Wilcoxon Sign-Rank Test Algorithm for BKASSA and its Comparison on the CEC-2017 Test Set

[0100] Table 10 shows the optimal solutions of BKASSA, BKA, HHO, SO, DO, WFO, and GOA for solving the tension / compression spring design problem. The results show that BKASSA provides the optimal solution for this problem, and the parameter set used in its design... The values ​​are (0.05168, 0.35672, 11.28826), and the corresponding objective function value is 0.0127. Further analysis shows that BKASSA performs better than the comparison algorithm when optimized. Furthermore, Figure 5 The BKASSA convergence curve illustrates the process of obtaining the optimal values ​​of design variables for a tension / compression spring problem.

[0101] Table 10 Comparison of Results for Tension / Compression Spring Design Problems

[0102] Although preferred embodiments of this application have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of this application.

[0103] Obviously, those skilled in the art can make various modifications and variations to this application without departing from the spirit and scope of this application. Therefore, if such modifications and variations fall within the scope of the claims of this application and their equivalents, this application also intends to include such modifications and variations.

Claims

1. A spring optimization design method, characterized in that, include: Obtain the performance data of the spring and establish the existing objective function and constraints; The population is initialized using the optimal point set method to obtain an initial population, which contains multiple individuals, each of which represents a potential spring design scheme. The fitness value of each individual is evaluated according to the objective function, and the individuals are divided into an elite subpopulation and an exploratory subpopulation based on the fitness value. During the iteration process, the individuals are updated alternately using the Black-winged Kite Algorithm and the Salicet Algorithm. During the update, the Black-winged Kite Algorithm uses an elite differential mutation strategy to enhance the diversity of the initial population; the Salicet Algorithm uses a sine and cosine exploration strategy to improve the global search capability. After each iteration, the optimal individual is selected based on the fitness value as the basis for the next generation of the population; Determine if the termination condition is met. If it is, use the optimal individual as the final spring design. If not, continue iterating.

2. The spring optimization design method according to claim 1, characterized in that, After determining the fitness value of each individual in the initial population, the individuals with the best fitness values ​​are divided into the elite subpopulation, and the individuals with the worst fitness values ​​are divided into the exploratory subpopulation.

3. The spring optimization design method according to claim 1, characterized in that, The Black-winged Kite algorithm utilizes an elite differential mutation strategy to consider the impact of individuals with poor fitness, thereby reducing the risk of premature convergence.

4. The spring optimization design method according to claim 1, characterized in that, The described tunic algorithm uses sine and cosine functions and the optimal individual position to guide the follower position update.

5. The spring optimization design method according to claim 1, characterized in that, When the number of iterations is odd, the initial population or the population updated by the tunic algorithm is updated using the black-winged kite algorithm. When the number of iterations is even, the population updated by the black-winged kite algorithm is updated using the tunic algorithm.

6. A system applying the spring optimization design method according to any one of claims 1-5, characterized in that, include: The function creation module is used to obtain the performance data of the spring and establish the existing objective function and constraints. An initialization module is used to initialize the population using the optimal point set method to obtain an initial population, wherein the initial population contains multiple individuals, and each individual represents a potential spring design scheme. The subpopulation partitioning module is used to evaluate the fitness value of each individual according to the objective function, and to partition the individuals into an elite subpopulation and an exploratory subpopulation based on the fitness value; The iterative module is used to alternately update the individuals using the Black-winged Kite Algorithm and the Salicet Algorithm during the iteration process. During the update, the Black-winged Kite Algorithm uses an elite differential mutation strategy to enhance the diversity of the initial population; the Salicet Algorithm uses a sine and cosine exploration strategy to improve the global search capability. After each iteration, the optimal individual is selected based on the fitness value as the basis for the next generation of the population; The parameter output module is used to determine whether the termination condition is met. If it is met, the optimal individual is taken as the final spring design scheme; otherwise, the iteration continues.

Images