ChOA pressure vessel optimization design method based on hawk algorithm exploration

Abstract

The invention discloses a ChOA pressure vessel optimization design method based on eagle algorithm exploration. The method comprises the steps that 1, parameters are set, wherein the thickness Ts of a pressure vessel shell, the thickness Th of an end socket, the inner radius R and the cylindrical section length L without considering the end socket are set; step 2, according to the parameters in the step 1, by taking the manufacturing cost of the pressure vessel as a target, constructing a pressure vessel target function; and step 3, solving the objective function in the step 2 by adopting an improved chimpanzee optimization algorithm to obtain an optimal parameter solution of the pressure vessel, and randomly exploring preys by adopting two methods of using eagle in an exploration stage of the improved chimpanzee optimization algorithm. According to the ChOA pressure vessel optimization design method based on the hawk algorithm exploration, the parameter design of the optimal solution of the pressure vessel can be rapidly and accurately realized, and the problem of poor parameter design accuracy caused by local optimum is avoided.

Description

Technical Field

[0001] The present invention relates to the technical field of pressure vessel cost evaluation, and in particular to a pressure vessel optimization design method based on ChOA (ChOA) algorithm exploration. Background Art

[0002] With the continuous advancement of scientific research, various engineering design optimization problems are becoming increasingly complex. Traditional optimization methods are no longer able to meet the needs of practical design problems, and optimization difficulties have become a bottleneck in scientific research and engineering design. Against this backdrop, swarm intelligence algorithms have been proposed. These algorithms use computers to optimize complex problems, inspired by the motion patterns of swarms in nature. These include the classic particle swarm optimization (PSO) algorithm, widely used for engineering constrained optimization problems due to its simple principles. The butterfly optimization algorithm (WOA), with its rapid convergence, has been applied to finite element model modification problems. The more novel salp swarm algorithm (SSA) has also been introduced into the field of structural optimization and has achieved promising results. Other algorithms include the artificial colony optimization (ACO) algorithm, which is often used in path planning. Due to the flexibility of swarm intelligence algorithms, developing and improving them based on problem characteristics is gaining increasing attention.

[0003] In 2020, the paper "Khishe M, Mosavi M R. Chimp optimization algorithm [J]. Expert Systems with Applications, 2020, 149:113338" developed the Chimp Optimization Algorithm (ChOA), inspired by the differences in intelligence and abilities of individuals in chimpanzee populations and their predatory behavior. While the ChOA algorithm has fewer parameters, is easier to implement, is relatively stable, and has no gradient restrictions, making it widely used for parameter optimization of functional links, it still suffers from some drawbacks, such as premature convergence, a single convergence method, and low optimization accuracy.

[0004] To address the above problems, scholars have improved the chimpanzee optimization algorithm with different strategies. The literature "KaurM, Kaur R, Singh N, et al. SChoA: a newly fusion of sine and cosine with chimp optimization algorithm for HLS of datapaths in digital filters and engineering applications [J]. Engineering with Computers, 2021 (2–4)" proposed a sine-cosine chimp optimization algorithm to reduce the slow convergence speed of the chimpanzee algorithm and the tendency to fall into local optimality; the literature "Liu Chenghan, He Qing. Golden sine chimpanzee optimization algorithm integrating multiple strategies [J]. Acta Automatica Sinica: 1-14" introduced Halton sequence and improved nonlinear factors, and combined with the golden sine correlation idea, proposed a multi-strategy golden sine chimpanzee algorithm to improve the algorithm's optimization performance; the literature "He Qing, Luo Shihang. Hybrid improved strategy chimpanzee optimization algorithm and its mechanical application [J]. Control and Decision: 1-11" adopted a water wave dynamic adaptive factor in position update to avoid the convergence of the group; the literature "WEI KA, MK B, MM C. Dynamic Levy Flight Chimp Optimization[J].2021[2022-03-31]” introduced a hybrid time-varying continuous Levy flight strategy to improve the search and exploration capabilities of the algorithm; the literature "Luo Shihang, He Qing. Chaos elite pool collaborative teaching and learning improved ChOA and its application[J]. Computer Engineering and Applications: 1-13" introduced an adaptive oscillation strategy at the convergence factor to enhance the exploration capability of the algorithm.

[0005] Although the above literature has improved ChOA through various means, which has improved the algorithm performance to a certain extent, there is still room for improvement in local exploration capabilities, escaping local optimality, and optimization speed, which makes it difficult to accurately calculate the optimal parameters of pressure vessels. Summary of the Invention

[0006] The technical problem to be solved by the present invention is: the pressure vessel optimization design method based on ChOA explored by the Sky Eagle algorithm can achieve the optimal and precise design of the pressure vessel parameters and improve the accuracy.

[0007] The technical solution adopted by the present invention is: a pressure vessel optimization design method based on ChOA explored by the Sky Eagle algorithm, the method comprising the following steps:

[0008] Step 1: Parameter setting: Set the shell thickness of the pressure vessel Ts and the head thickness T h, inner radius R, ignoring the length L of the cylindrical section of the head, that is, assume:

[0009]

[0010] Step 2: Based on the parameters in step 1 and taking the manufacturing cost of the pressure vessel as the target, construct the pressure vessel objective function, namely, the objective function:

[0011]

[0012] Objective function constraints:

[0013]

[0014] Value range: x1 and x2 are integer multiples of 0.0625, 10≤x3,x4≤200

[0015] Step 3: Use the improved chimpanzee optimization algorithm to solve the objective function in step 2 and obtain the optimal solution for the parameters of the pressure vessel. The exploration phase of the improved chimpanzee optimization algorithm uses two methods of Sky Eagle to randomly explore prey.

[0016] Furthermore, the optimization solution process of the improved Chimpanzee optimization algorithm is as follows:

[0017] 3.1) Population Initialization: The Latin Square initialization method is used to generate an initial population, which is divided into a chimpanzee population and a hawk population. For the chimpanzee population and the hawk population, a control factor B is set to 0.8 to control the ratio of the two populations.

[0018] 3.2) Development: If Rand[0,1]>B, the search agent will update the position like the chimpanzee through formula (7);

[0019] X2=X barrier -A2*|C2*X barrier -m2*X| (7)

[0020] Use the improved chimpanzee algorithm to solve the objective function to obtain the optimal solution, and narrow the range of the f value to [0,1] to focus more on the development ability of chimpanzees;

[0021] 3.3) Exploration: If Rand[0,1]≤B, the search agent will update its position by Equation (11) / Equation (22) like the Skyhawk;

[0022]

[0023] Where, represents the position after the (t+1)th iteration in the first way, represents the best position obtained after the tth iteration, corresponding to the position of the optimal solution of the Chimpanzee optimization algorithm; This means exploring the position after the (t+1)th iteration in the second way. Levy(D) is the levy(D) flight distribution function, where D represents the dimension. The calculation equation for this distribution function is as follows:

[0024]

[0025] Where s is a fixed constant with a value of 0.01, λ is a fixed constant with a value of 1.5, μ and v are random numbers in the interval [0,1]. The calculation is as follows:

[0026]

[0027] Where M is the gamma function, and λ is 1.5. is a random solution in the range [1N] at the tth iteration;

[0028] If Rand[0,1]>0.5, then use formula (22) to update its position, otherwise use formula (11) to update its position.

[0029] Using the two methods of the eagle, the prey is randomly explored (i.e., Equation (11) / Equation (22)). If the eagle can find a better location than the chimpanzees, all the chimpanzees will move to the eagle to explore. If no better location is found, the chimpanzee group will continue exploring in the original location. This operation is to avoid falling into a local optimal solution. This emphasizes the exploration ability of the eagle and the exploitation ability of the chimpanzees. We can change the exploration and exploitation capabilities of the algorithm by simply adjusting the value of B.

[0030] Furthermore, the method for generating the initialization population using the above Latin square initialization method is:

[0031] Step 1: Determine the population size N;

[0032] Step 2: Set the upper and lower bounds of each dimensional variable x [x min ,x max ] Divide the population size N into equal parts, and obtain N×dim intervals of different dimensions, where x corresponds to the thickness Ts of the pressure vessel and the thickness T h , inner radius R, not considering the length L of the cylindrical section of the head, that is,

[0033]

[0034] Step 3: Set up an N×dim random full permutation matrix A, corresponding to N population individuals;

[0035] Step 4: Randomly select a point in each interval corresponding to each element of matrix A to obtain N diverse population individuals.

[0036] Furthermore, the above N×dim is 40×2.

[0037] Furthermore, the convergence factor f in the above-mentioned ChOA algorithm adopts a nonlinear convergence factor based on the tangent function. The calculation formula of the nonlinear convergence factor based on the tangent function is as follows:

[0038]

[0039] Where,

[0040] f initial The initial convergence factor (constant), which represents the maximum exploration ability of the algorithm at the beginning;

[0041] ε is a parameter that controls the speed of change of the tan function. The larger it is, the slower the decay, and the smaller it is, the faster the change. It is a nonlinear adjustment factor.

[0042] T max Maximum number of iterations;

[0043] t is the current iteration number.

[0044] The beneficial effects of the present invention are as follows: Compared with the existing technology, the pressure vessel optimization design method of the present invention based on the ChOA exploration of the Eagle algorithm can quickly and accurately achieve the optimal solution parameter design of the pressure vessel, avoiding the problem of poor parameter design accuracy caused by local optimality; the Eagle optimization algorithm (AO) enables some chimpanzees to fly, expands the search range to improve global search capabilities, and reduces the possibility of falling into local optimality. At the same time, a nonlinear convergence factor is introduced, which mainly emphasizes the development ability of the chimpanzee and the exploration ability of the Eagle to balance the two stages of exploration and development. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 A diagram of the social hierarchy and characteristics of chimpanzees;

[0046] Figure 2 Update the rule graph for the location;

[0047] Figure 3 is an individual distribution scatter plot; in the figure, (a) is a random sampling scatter plot; (b) is a Latin hypercube sampling scatter plot;

[0048] Figure 4 This is the flow chart of the AChOA algorithm;

[0049] Figure 5 is the convergence curve of the F1-F13 test function;

[0050] Figure 6 Schematic diagram of the pressure vessel. DETAILED DESCRIPTION

[0051] Aiming at the problems existing in the conventional ChOA algorithm for optimizing pressure vessels, the pressure vessel parameters are optimized based on the ChOA algorithm explored by the Sky Eagle algorithm, making the pressure vessel parameter optimization more accurate.

[0052] The conventional ChOA algorithm is an optimization algorithm developed based on the differences in individual intelligence and abilities and predation activities in chimpanzee populations. Figure 1 As shown in Figure 1, chimpanzee groups are divided into drivers, blockers, chasers, and attackers based on their individual intelligence and abilities to complete the four main steps of the hunting process: driving, blocking, chasing, and attacking prey. The mathematical formula for chimpanzees driving prey is shown in Equation (1), and the formula for chasing prey is shown in Equation (2):

[0053] D(t)=CX p (t)-mX c (t) (1)

[0054] X c (t+1)=X p (t)-AD (2)

[0055] Where: t represents the current iteration number; X p is the prey position vector; X c is the current chimpanzee position vector; A, C, and m are coefficient vectors, i.e., A = [A1 A2 A3 A4], C = [C1 C2 C3 C4], m = [m1 m2 m3 m4], and the solution formulas for each parameter are shown in Equations (3)-(5):

[0056] A=2f*r1-f (3)

[0057] c=2*r2 (4)

[0058] m=chaotic_value (5)

[0059] Where: r1 and r2 are random vectors between [0,1]; f is the linear convergence factor, which decreases from 2.5 to 0 as the iteration proceeds; A is a random variable between [-f,f], such as Figure 2 As shown in the figure, when |A|<1, the chimpanzee individual approaches the prey, otherwise it moves away from the prey; m is the chaotic mapping vector; c is the factor affecting the prey position on the chimpanzee's expulsion and pursuit of prey, which is randomly generated between [0,2].

[0060] After the population is initialized, the four best solutions are selected in turn as the positions of the attacker, obstacle, driver, and chaser. The positions of the other chimpanzees in the population are updated around the following four chimpanzee positions:

[0061] X1=X attacker -A1*|C1*X attacker -m1*X| (6)

[0062] X2=X blocker -A2*|C2*X blocker -m2*X| (7)

[0063] X3=X chase -A3*|C3*X chase -m3*X| (8)

[0064] X4=X driver -A4*|C4*X driver -m4*X| (9)

[0065]

[0066] Where X represents the current position vector of the currently updated individual (i.e., common chimpanzee), that is, the current position of this individual in the solution space, representing a current candidate solution, X attacker is the position vector of the group optimal solution; X barrier is the position vector of the group suboptimal solution; X chaser is the position vector of the third optimal solution of the group; X driver is the position vector of the fourth optimal solution of the group; X t+1 Represents the updated position vector of the current chimpanzee.

[0067] Example 1: Figures 1 to 3 As shown in FIG, the pressure vessel optimization design method of ChOA based on the Sky Eagle algorithm exploration includes the following steps:

[0068] Step 1: Parameter setting: Set the shell thickness of the pressure vessel Ts and the head thickness T h , inner radius R, ignoring the length L of the cylindrical section of the head, that is, assume:

[0069]

[0070] Step 2: Based on the parameters in step 1 and taking the manufacturing cost of the pressure vessel as the target, construct the pressure vessel objective function, namely, the objective function:

[0071]

[0072] Objective function constraints:

[0073]

[0074]

[0075] Value range: x1 and x2 are integer multiples of 0.0625, 10≤x3,x4≤200

[0076] Step 3: Use the improved chimpanzee optimization algorithm to solve the objective function in step 2 and obtain the optimal solution for the parameters of the pressure vessel. The exploration phase of the improved chimpanzee optimization algorithm uses two methods of Sky Eagle to randomly explore prey.

[0077] Skyhawk search strategy: The ChOA algorithm's exploration is guided by the top four chimpanzees, but they don't know where the prey is, so this search method can easily get stuck in a local optimum. Therefore, the present invention considers introducing a mechanism that allows Skyhawk to explore prey in mid-air, so that the position updates of some chimpanzees will not be affected by the top four chimpanzees, thus avoiding getting stuck in a local optimum. In this invention, two ways of finding prey are mainly introduced by Skyhawk. The following is a detailed introduction to the two ways of finding prey:

[0078] Search method 1: Wide range search (X1)

[0079] When the eagle flies high in the sky, it uses a wider field of view to explore the location of its prey and determine its approximate range. This behavior can be expressed in a mathematical formula:

[0080]

[0081] Where, represents the position after the (t+1)th iteration in the first way, represents the best position obtained after the tth iteration, corresponding to the position of the optimal solution of the Chimpanzee optimization algorithm;

[0082] Search method 2: Small range search (X2)

[0083] In this method, after discovering its prey, the eagle will surround it in the sky and track it at the same altitude as the ground. The mathematical expression is:

[0084]

[0085] Where, This means exploring the position after the (t+1)th iteration in the second way. x is the position vector of the current individual, representing a candidate solution for the current position of a hawk in the population. y is the current position of the prey, which can also be understood as the known optimal solution or local target point in the current search. The difference between the two represents the direction vector between the individual and the prey, indicating the search direction "toward the prey." Levy(D) is the levy(D) flight distribution function, where D represents the dimension. The calculation equation for this distribution function is as follows:

[0086]

[0087] Where s is a fixed constant with a value of 0.01, λ is a fixed constant with a value of 1.5, μ and v are random numbers in the interval [0,1]. The calculation is as follows:

[0088]

[0089] Where M is the gamma function, and λ is 1.5. is a random solution in the range [1N] at iteration t, and for the other letters, follow these instructions:

[0090] y1=r×cos(θ) (16)

[0091] x1=r×sin(θ) (17)

[0092] r=r1+U+D1 (18)

[0093] θ=-K×D1+θ1 (19)

[0094] θ1=(3×π) / 2 (20)

[0095] Where y1 and x1 are used to represent the spiral shape in the search; r1 is a fixed number of search cycles, ranging from 1 to 20, U is a small value fixed at 0.00565; D1 is an integer from 1 to the length of the search space (Dim), and K is a small value fixed at 0.005;

[0096] Since the eagle has found the prey in the second exploration phase, its trajectory is constrained by the prey, which is similar to the way chimpanzees surround their prey. To make the algorithm more exploratory, formula (11) is updated to formula (22):

[0097]

[0098] Since the search mode of the Chimpanzee algorithm is still used when the AO algorithm is introduced into the search mechanism, it can be seen from formula (10) that the priorities of the top four Chimpanzee algorithms are the same when updating the position. Therefore, an adaptive weighted position update strategy is proposed. The expression of this strategy is as follows:

[0099]

[0100] Based on the above theory, the optimization solution process of the improved chimpanzee optimization algorithm is as follows:

[0101] 3.1) Population Initialization: Use the Latin Square initialization method to generate an initial population. This initial population is divided into a chimpanzee population and a hawk population (if the total number of individuals generated is 100, B = 0.8, based on this ratio, the first 80 individuals are directly classified as chimpanzees and the last 20 individuals are classified as hawks). For the chimpanzee population and the hawk population, set a control factor B to control the ratio of the two populations, set to 0.8;

[0102] The quality of the initial population is crucial to the success of the iterations. The initial chimpanzee algorithm randomly generates individuals in the initial population, which makes it difficult to guarantee population diversity. Latin hypercube sampling is a stratified sampling method based on the space-filling concept. Compared to random sampling, it has the following characteristics:

[0103] (1) The sampling points are random, non-overlapping, and well distributed in the solution space.

[0104] (2) The sampling points are stable, and samples are drawn from each layer to ensure the comprehensiveness of the samples.

[0105] Therefore, the Latin Hypercube strategy is introduced to ensure that the individuals in the initial population are random, and the individuals generated are diverse, which can improve the optimization performance of the algorithm to a certain extent. The initial population dimension of the algorithm is set to dim, and the population size is set to N. The population initialization steps of the Chimpanzee algorithm combined with the Latin Hypercube strategy are:

[0106] Step 1: Determine the population size N;

[0107] Step 2: Set the upper and lower bounds of each dimensional variable x [x min ,x max ] Divide the population size N into equal parts, and obtain N×dim intervals of different dimensions, where x corresponds to the thickness Ts of the pressure vessel and the thickness T h , inner radius R, not considering the length L of the cylindrical section of the head, that is,

[0108]

[0109] Step 3: Set up an N×dim random full permutation matrix A, corresponding to N population individuals;

[0110] Step 4: Randomly select a point in each interval corresponding to each element of matrix A to obtain N diverse population individuals;

[0111] Figure 3 The following is a scatter plot of randomly generated individuals and a scatter plot of individuals generated using the Latin Hypercube population initialization strategy, with the parameters set to N = 40 and dim = 2. It can be seen that the initial population constructed using the Latin Hypercube strategy has no overlapping individuals and is relatively evenly distributed across the solution space, resulting in greater diversity than the randomly generated initial population. This, to a certain extent, avoids distortion in the algorithm's optimization results.

[0112] 3.2) Development: If Rand[0,1]>B, the search agent will update the X2 position like the chimpanzee through formula (7);

[0113] X2=X barrier -A2*|C2*X barrier -m2*X| (7)

[0114] Use the improved chimpanzee algorithm to solve the objective function to obtain the optimal solution, and narrow the range of the f value to [0,1] to focus more on the development ability of chimpanzees;

[0115] 3.3) Exploration: If Rand[0,1]≤B, the search agent will update its position by Equation (11) / Equation (22) like the Skyhawk;

[0116]

[0117] If Rand[0,1]>0.5, then use formula (22) to update its position, otherwise use formula (11) to update its position.

[0118] Using the two methods of the eagle, the prey is randomly explored (i.e., Equation (11) / Equation (22)). If the eagle can find a better location than the chimpanzees, all the chimpanzees will migrate to the eagle to explore. If no better location is found, the chimpanzee group will continue exploring in the original location. This operation is to avoid falling into a local optimal solution. This emphasizes the exploration ability of the eagle and the exploitation ability of the chimpanzees. We can change the exploration and exploitation capabilities of the algorithm by simply adjusting the value of B.

[0119] The basic ChOA algorithm uses a linear reduction strategy to adjust the convergence factor f. However, this strategy has certain problems, namely, it cannot fully balance the global search and local development capabilities of the algorithm, and thus cannot fully reflect the actual optimization process, resulting in a decrease in the performance of the algorithm. Therefore, a nonlinear convergence factor based on the tangent function is introduced in the ChOA algorithm to optimize the performance of the algorithm. This new convergence factor strategy makes the convergence factor decay more slowly in the early stages of the iteration to enhance the global search capability, and accelerates the decay rate in the later stages of the iteration to enhance the local development capability. The convergence factor f in the ChOA algorithm uses a nonlinear convergence factor based on the tangent function. The calculation formula of the nonlinear convergence factor based on the tangent function is as follows:

[0120]

[0121] In the formula, in the formula,

[0122] f initial The initial convergence factor (constant), which represents the maximum exploration ability of the algorithm at the beginning;

[0123] ε is a parameter that controls the speed of change of the tan function. The larger it is, the slower the decay, and the smaller it is, the faster the change. It is a nonlinear adjustment factor.

[0124] T max Maximum number of iterations;

[0125] t is the current iteration number.

[0126] In order to illustrate the effect of the present invention, the following simulation experiments and result analysis are carried out

[0127] 1. Experimental design and test functions: 23 benchmark test functions were selected to verify the performance of the algorithm. These functions have been tested in many literatures, such as "Long Wen, Wu Tiebin. Improved Grey Wolf Optimization Algorithm for Coordinated Exploration and Exploitation Capabilities [J]. Control and Decision, 2017, 32(10): 1749-1757", as shown in Table 1, where Dim represents its dimension, range represents its domain, and f min is the theoretical optimal value. The population size is set to 100, and the number of iterations (maximum value) is T max It is set to 500. 30 simulations were performed under the same operating environment, and the mean and variance of each test function were recorded.

[0128] Table 1 Optimization setting parameters for different test functions

[0129]

[0130]

[0131]

[0132] 2. Comparison of optimization results of each algorithm test function

[0133] 2.1 Parameter settings

[0134] In order to verify the performance of the proposed algorithm, some mature algorithms with various performance characteristics in exploration and exploitation were selected and tested with the same benchmark function. These algorithms include ChOA, ALO "MIRJALILI S.The AntLion Optimizer[J / OL].Advances in Engineering Software,2015,83:80-98", TACPSO "AModified Particle Swarm Optimization with an Adaptive Acceleration Coefficients.[C]Proceedings of the 2009Asia-Pacific Conference on InformationProcessing-Volume 02Available online:https: / / dl.acm.org / doi / 10.1109 / APCIP.2009.217(accessed on 18May 2023)", MPSO "Tian,D.;Shi,Z.MPSO:ModifiedParticle Swarm Optimization and Its Applications[J].Swarm and Evolutionary Computation 2018,41,49–68", MPA "Faramarzi,A.;Heidarinejad,M.;Mirjalili,S.;Gandomi,AHMarine Predators Algorithm:A Nature-Inspired Metaheuristic[J].Expert Systems with Applications2020,152,113377,doi:10.1016 / j.eswa.2020.113377", NGO" Dehghani, M.; Hubálovsky, Trojovsky, P. Northern Goshawk Optimization: A New Swarm-Based Algorithm for Solving Optimization Problems[J]. IEEE Access2021,9,162059–162080".

[0135] The present invention sets parameters according to the references corresponding to different algorithms. The specific information is shown in Table 2.

[0136] Table 2 Parameter settings of each algorithm

[0137]

[0138]

[0139] 2.2 Test function F1-F13 solution results

[0140] To validate the algorithm's optimization capabilities across various dimensions, this section sets the test function dimensions to 30, 50, and 100, respectively. This multi-dimensional analysis provides a more comprehensive understanding of the algorithm's performance across different problem sizes. The results shown in Table 3 show that AChOA generally outperforms other algorithms on average when solving functions F1 to F13. This demonstrates that AChOA achieves excellent optimization results in multi-dimensional environments. In particular, AChOA maintains outstanding performance in high-dimensional problems (with a dimension of 100). This reflects the algorithm's high adaptability and solver capabilities, enabling it to handle optimization problems of various dimensions. Furthermore, attention was paid to the algorithm's stability, assessed using standard deviation. Notably, the proposed AChOA exhibits more stable performance on most test functions, meaning that the algorithm not only performs well on average but also exhibits less fluctuation between runs. This is an important performance metric, especially in practical applications, where stability is crucial for ensuring algorithm reliability. Taken together, AChOA offers significant advantages in optimization accuracy, stability, and robustness. It not only achieves excellent results in different dimensions, but also maintains consistent performance over multiple runs. This makes AChOA a powerful optimization tool that can be applied to various practical problems, thereby improving the efficiency and credibility of problem solving.

[0141] In order to further verify the superiority of the AChOA algorithm, for the sake of brevity, Figure 5The convergence curves of various algorithms are shown. It can be clearly seen from the figure that although the convergence speed of the AChOA algorithm is not outstanding in the initial stage, its advantages gradually become apparent as the optimization search progresses. This significant improvement can be attributed to the nonlinear convergence factor introduced in the AChOA algorithm, which can effectively balance the algorithm's global search and local development capabilities. In terms of optimization results, AChOA has shown significant advantages in the vast majority of benchmark functions, far ahead of other algorithms, thanks to the successful fusion of AO and ChOA. In summary, the AChOA algorithm has achieved significant improvements in search performance, demonstrating its excellent performance in complex optimization problems.

[0142] Table 3 Solution results of test functions F1-F13 (Dim=30,50,100)

[0143]

[0144]

[0145]

[0146] 2.3 Test function F14-F23 solution results

[0147] As shown in Table 4, F14-F23 is a set of fixed-dimensional multimodal functions, each with a fixed dimension. These functions are widely used in research across various fields because they can simulate a variety of complex real-world problems. In particular, in the performance evaluation of optimization algorithms, the F14-F23 function set is often used to test the robustness and efficiency of the algorithms. Because the dimensions of F14-F23 are fixed, multidimensional analysis is not required. This feature allows researchers to focus on improving algorithm performance without worrying about the complexity introduced by dimensionality changes. However, precisely because these functions have fixed dimensions, they place higher demands on the algorithms, requiring them to accurately find the global optimal solution within the given dimensions. Experimental results show that many optimization algorithms can obtain optimal solutions for fixed-dimensional multimodal functions without the need to calibrate the true optimal value. Comparing the results of the proposed algorithm with other algorithms clearly demonstrates its seamless integration across the development, exploration, and development phases. This capability is a unique feature of the proposed algorithm, enabling it to automatically adjust its behavior at different stages to better adapt to the nature of the problem, resulting in more accurate and faster solutions to the objective function of pressure vessels. This also highlights the algorithm's adaptability and robustness, two features that are particularly important when solving real-world problems.

[0148] Table 4 Solution results of test functions F14-F23

[0149]

[0150]

[0151] Significance Analysis: To verify the differences between the AChOA algorithm and other algorithms, we used the Wilcoxon rank-sum test to calculate p-values ​​and compare their performance. The results in Table 5 show that the p-values ​​of the Wilcoxon rank-sum test for the AChOA algorithm are almost all less than the 5% significance level. This finding clearly demonstrates from a statistical perspective that the AChOA algorithm has a significant advantage in solving basic functions. This not only strengthens the reliability of the AChOA algorithm but also further demonstrates its superior performance in solving complex optimization problems.

[0152] Table 5 Comparison results of significance analysis

[0153] ID ChOA ALO TACPSO MSPO MPA NGO F1 3.31E-20 3.31E-20 3.31E-20 3.31E-20 3.31E-20 3.31E-20 F2 2.83E-10 2.84E-05 3.78E-18 4.13E-17 1.30E-03 4.45E-09 F3 3.31E-20 3.31E-20 3.31E-20 3.31E-20 3.31E-20 3.31E-20 F4 3.06E-18 4.25E-04 3.75E-06 4.32E-15 4.35E-10 9.36E-07 F5 5.45E-15 3.75E-09 4.12E-11 8.45E-15 4.36E-12 1.31E-10 F6 8.45E-10 4.32E-01 5.21E-07 6.12E-09 1.12E-03 3.78E-08 F7 1.54E-07 3.65E-04 7.32E-04 9.45E-06 5.64E-04 9.12E-03 F8 7.06E-18 7.21E-09 7.06E-18 7.06E-18 6.45E-03 9.21E-10 F9 3.31E-20 #DIV / 0 3.31E-20 3.31E-20 #DIV / 0 #DIV / 0 F10 2.91E-20 2.62E-23 3.31E-20 3.31E-20 3.48E-24 7.45E-06 F11 3.31E-20 #DIV / 0 3.31E-20 3.31E-20 #DIV / 0 #DIV / 0 F12 7.96E-18 9.37E-10 7.96E-09 7.06E-13 8.25E-03 1.43E-06 F13 4.25E-16 7.38E-09 6.25E-19 2.24E-18 3.75E-2 2.32E-09 12 / 0 / 0 9 / 2 / 1 12 / 0 / 0 12 / 0 / 0 10 / 2 / 0 10 / 2 / 0

[0154] Conclusion: This study proposes a chimpanzee optimization algorithm (AChOA) based on a flying eagle search strategy for pressure vessel parameter optimization. This algorithm aims to grant some chimpanzees the ability to fly, thereby expanding the search scope. Furthermore, improvements are made to the existing chimpanzee and flying eagle algorithms. A nonlinear convergence factor is introduced into the chimpanzee algorithm, and the search mechanism is adjusted in the flying eagle algorithm to achieve a better balance between development and exploration. This enhances the development potential of the gray wolf and the exploration capability of the flying eagle, thereby achieving more accurate optimal pressure vessel parameter optimization. The performance of the AChOA algorithm was evaluated on 23 standard benchmark functions. Experimental results show that compared with multiple swarm intelligent optimization algorithms, the AChOA algorithm outperforms in terms of convergence speed and accuracy. The main innovation of this study is that the introduction of the flying chimpanzee algorithm expands the search domain, enabling the algorithm to more effectively explore the problem space and achieve more accurate pressure vessel parameter optimization. Furthermore, the improvements to the development and exploration strategies enhance the algorithm's global search and local optimization capabilities, making it more adaptable to different types of optimization problems.

[0155] Pressure vessel design problem: The main idea is to transform the actual optimization problem into a mathematical model, and then use various algorithms to find the optimal solution. is the fitness function in the original algorithm; Represents the search space. x1,x2…x nRepresents different dimensions (four dimensions in the present invention). The ultimate goal is to minimize the use of materials while satisfying various mechanical properties. Therefore, they have some identical or different constraints. The algorithm of the present invention should have a method for handling constraints so that it can be applied to these engineering problems. Therefore, the simplest constraint handling method (penalty function) can be effectively used to handle the constraints in the algorithm. That is, if the search agent violates any constraint, it will be assigned a large objective function value. In this way, after the next iteration, it will be automatically replaced by a new search agent.

Claims

1. The pressure vessel optimization design method based on ChOA explored by Sky Eagle algorithm is characterized by: The method comprises the following steps: Step 1: Parameter setting: Set the shell thickness of the pressure vessel Ts and the head thickness T h , inner radius R, ignoring the length L of the cylindrical section of the head, that is, assume: Step 2: Based on the parameters in step 1 and taking the manufacturing cost of the pressure vessel as the target, construct the pressure vessel objective function, namely, the objective function: Objective function constraints: Value range: x1 and x2 are integer multiples of 0.0625, 10≤x3,x4≤200 Step 3: Use the improved chimpanzee optimization algorithm to solve the objective function in step 2 and obtain the optimal solution for the parameters of the pressure vessel. The exploration phase of the improved chimpanzee optimization algorithm uses two methods of Sky Eagle to randomly explore prey.

2. The pressure vessel optimization design method based on ChOA of Sky Eagle algorithm exploration according to claim 1 is characterized in that: The optimization solution process of the improved chimpanzee optimization algorithm is: 3.1) Population Initialization: The Latin Square initialization method is used to generate an initial population, which is divided into a chimpanzee population and a hawk population. For the chimpanzee population and the hawk population, a control factor B is set to 0.8 to control the ratio of the two populations. 3.2) Development: If Rand[0,1]>B, the search agent will update the X2 position like the chimpanzee through formula (7); X2=X barrier -A2*|C2*X barrier -m2*X| (7) Use the improved Chimpanzee algorithm to solve the objective function to obtain the optimal solution and narrow the range of f value to [0,1]; 3.3) Exploration: If Rand[0,1]≤B, the search agent updates its position by equation (11) / equation (22); Where, represents the position after the (t+1)th iteration in the first way, represents the best position obtained after the tth iteration, corresponding to the position of the optimal solution of the Chimpanzee optimization algorithm; This means exploring the position after the (t+1)th iteration in the second way. Levy(D) is the levy(D) flight distribution function, where D represents the dimension. The calculation equation for this distribution function is as follows: Where s is a fixed constant with a value of 0.01, λ is a fixed constant with a value of 1.5, μ and v are random numbers in the interval [0,1]. The calculation is as follows: Where M is the gamma function, and λ is 1.

5. is a random solution in the range [1N] at the tth iteration; If Rand[0,1]>0.5, then use formula (22) to update its position, otherwise use formula (11) to update its position.

3. The pressure vessel optimization design method based on ChOA of Sky Eagle algorithm exploration according to claim 2 is characterized in that: The method of generating the initialization population by the Latin square initialization method is: Step 1: Determine the population size N; Step 2: Set the upper and lower bounds of each dimensional variable x [x min ,x max ] Divide the population size N into equal parts, and obtain N×dim intervals of different dimensions, where x corresponds to the thickness Ts of the pressure vessel and the thickness T h , inner radius R, not considering the length L of the cylindrical section of the head, that is, Step 3: Set up an N×dim random full permutation matrix A, corresponding to N population individuals; Step 4: Randomly select a point in each interval corresponding to each element of matrix A to obtain N diverse population individuals.

4. The pressure vessel optimization design method based on ChOA of Sky Eagle algorithm exploration according to claim 3 is characterized in that: N×dim is 40×2.

5. The pressure vessel optimization design method based on ChOA of Sky Eagle algorithm exploration according to claim 3 is characterized in that: The convergence factor f in the ChOA algorithm adopts a nonlinear convergence factor based on the tangent function. The calculation formula of the nonlinear convergence factor based on the tangent function is as follows: Where, f initial is the initial convergence factor; ε is the parameter that controls the speed of change of the tan function; T max Maximum number of iterations; t is the current iteration number.

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