A welding beam design method based on a hybrid swarm intelligence optimization algorithm

Abstract

The application provides a welding beam design method based on a hybrid swarm intelligence optimization algorithm, and relates to the technical field of welding beam design.The method firstly defines a mathematical model of the welding beam, determines an objective function and constraint conditions of the welding beam design;optimizes ChOA and AOA algorithms to obtain an improved chimpanzee optimization algorithm IChOA and an arithmetic optimization algorithm IAOA, and builds a hybrid swarm intelligence optimization algorithm based on the IChOA and the IAOA;then the hybrid swarm intelligence optimization algorithm is used to solve the objective function of the welding beam design to obtain an optimal scheme of the welding beam design; the method makes up for the shortcomings of the chimpanzee algorithm and the arithmetic optimization algorithm, integrates the two algorithms into one, and obtains a hybrid swarm intelligence optimization algorithm with superior performance; and the algorithm is applied to the welding beam design optimization problem, so that the design cost of the welding beam is greatly reduced.

Description

Technical Field

[0001] This invention relates to the field of welded beam design technology, and in particular to a welded beam design method based on a hybrid swarm intelligence optimization algorithm. Background Technology

[0002] The welded beam design problem is a minimization problem among many engineering problems, aiming to reduce the design cost of welded beams. It is characterized by multiple constraints and nonlinearity. Existing algorithms for solving this problem include Genetic Algorithms (GA), Grey Wolf Optimizer (GWO), and Whale Optimization Algorithm (WOA), but the design solutions found by these algorithms need further optimization.

[0003] The Chimpanzee Optimization Algorithm (ChOA) is an optimization technique developed by M. Khishe et al. in 2020, which abstracts and simplifies individual intelligence and sexual motivation during group hunting in chimpanzees. Compared to other swarm intelligence algorithms, ChOA has two main characteristics: first, it employs differentiated strategies to update positions based on the different hunting responsibilities of chimpanzees, thereby improving global search capabilities; second, social incentives and sexual motivations influence their encirclement behavior in the final stage, and introducing probability and chaos factors to simulate this phenomenon accelerates the algorithm's convergence speed. However, ChOA still suffers from drawbacks such as low solution accuracy, slow convergence speed, and susceptibility to getting trapped in local optima.

[0004] The Arithmetic Optimization Algorithm (AOA) was proposed by Laith Abualigah et al. in 2021, inspired by arithmetic operators in mathematical computation methods. This algorithm utilizes the distribution characteristics of operator computation results for global optimization, exhibiting excellent optimization performance. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to address the shortcomings of the prior art by providing a welding beam design method based on a hybrid swarm intelligence algorithm. The method uses an improved ChOA and AOA hybrid search mechanism to solve the welding beam design problem, obtains a better welding beam design scheme, and reduces the welding beam design cost.

[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is: a welding beam design method based on a hybrid swarm intelligence optimization algorithm, comprising the following steps:

[0007] Step 1: Determine the objective function and constraints for the welded beam design;

[0008] The mathematical model of the welded beam is defined as follows: The welded beam has four physical properties: weld width h, clamp length l, beam length t, and beam width b. These four physical properties are used as design variables for the welded beam design, denoted as x1, x2, x3, and x4. This represents a set of design schemes consisting of 4 design variables;

[0009] The objective function for the welded beam design is set as follows:

[0010]

[0011] The constraints are:

[0012]

[0013]

[0014]

[0015]

[0016]

[0017]

[0018]

[0019] in, For welding stress, For bending stress, For the deformation of the beam, For the instability critical load, τ max σ max δ max These represent the maximum values ​​of welding stress, bending stress, and beam deformation, respectively, with P being the torque.

[0020] Step 2: Optimize the ChOA and AOA algorithms respectively to obtain the improved chimpanzee optimization algorithm IChOA and the arithmetic optimization algorithm IAOA. Construct a hybrid swarm intelligence optimization algorithm based on IChOA and IAOA, denoted as H-IChOA-IAOA.

[0021] First, for the ChOA algorithm, on the one hand, the nonlinear decay mode of the convergence factor f is optimized, and on the other hand, a preference weight λ based on Manhattan distance is introduced into the position update formula. Second, for the AOA algorithm, sine and cosine functions are used to map the spatial location information of the current individual and replace the multipliers determined by the interval vector in AOA. Finally, 20% of the individuals in the chimpanzee population are selected to perform the IAOA optimization process.

[0022] The nonlinear decay mode of the convergence factor f is optimized using the following equation:

[0023]

[0024] In the formula, f inital Let f be the initial value of the convergence factor f. final Let f be the final value of the convergence factor, and t and T be the current iteration number and the maximum iteration number, respectively.

[0025] Step 3: Based on the hybrid swarm intelligence optimization algorithm based on IChOA and IAOA built in Step 2, solve the objective function of the welded beam design to obtain the optimal solution for the welded beam design;

[0026] Step 3.1: Initialize the chimpanzee population. The initial positions of the chimpanzees are the initial values ​​of the four design variables in Step 1. Construct an N×4 matrix as the solution space, where N is the population size.

[0027] Step 3.2: Using the objective function of the welded beam design as the fitness function, calculate the fitness of the chimpanzee population and determine the positions of attackers, obstructors, expellers, and pursuers.

[0028] Step 3.3: Update the location of individual chimpanzees;

[0029] (1) 80% of chimpanzee individuals updated their positions using the following formula:

[0030] X(t+1)=λ1X1+λ2X2+λ3X3+λ4X4

[0031]

[0032] Among them, X A X B X C X D X1, X2, X3, and X4 represent the position information of the attacker, obstacle, expeller, and pursuer before the update, respectively. X1, X2, X3, and X4 represent the position information of the attacker, obstacle, expeller, and pursuer after the update, respectively. X is the current position of the chimpanzee individual. A = f·(2·r1-1), C = 2·r2, m = Chaotic_value is the chaotic mapping. The values ​​of r1 and r2 are randomly generated between [0,1] by the rand function. m is the chaotic mapping vector representing the influence of motivation.

[0033] (2) 20% of chimpanzee individuals updated their location in the following manner:

[0034] Random numbers r3, r4, and r5 are randomly generated within the interval [0,1]. When r3 > MOA, the chimpanzee individual's position is updated according to the following formula:

[0035]

[0036] When r3 < MOA, the chimpanzee individual's position is updated according to the following formula:

[0037]

[0038] In the formula, X(t) is the position of the t-th generation chimpanzee, X(t+1) is the position of the (t+1)-th generation chimpanzee, and X... best (t) represents the optimal individual position, i.e., the attacker's position. λ1, λ2, λ3, and λ4 are the current individual's preference weights relative to the attacker, obstructor, expelr, and pursuer, respectively, based on the Manhattan distance. MOA is the mathematical optimizer acceleration function, MOP is the mathematical optimizer probability, r3, r4, and r5 are random numbers in the interval [0,1], ε is an infinitesimal positive number, and g(·) is the mapping function, representing a mapping relationship.

[0039] The definition of the mathematical optimizer speedup function MOA is:

[0040]

[0041] In the formula, Max = 0.7 represents the maximum value of the acceleration function, and Max = 0.2 represents the minimum value of the acceleration function;

[0042] The definition of the mathematical optimizer probability (MOP) is:

[0043]

[0044] In the formula, α is a sensitive parameter, which defines the development accuracy in the iteration, and is set to 5 here;

[0045] The formulas for calculating the preference weights λ1, λ2, λ3, and λ4 are as follows:

[0046]

[0047] In the formula, d1, d2, d3, and d4 represent the Manhattan distances of the current individual relative to the attacker, obstructor, expeller, and pursuer, respectively.

[0048] The expression for the mapping function is:

[0049] g(X(t))=sin(X(t))+cos(X(t))

[0050] Step 3.4: Check whether the range of the solution after updating the position of the chimpanzee individual exceeds the range of the design variable. If it does, replace the solution that exceeds the range with the boundary value of the design variable.

[0051] Step 3.5: Determine if the current iteration number is less than the maximum iteration number. If so, return to step 3.2. Otherwise, output the minimum value of the objective function as the minimum design cost of the welded beam, and the currently found solution as the optimal welded beam design scheme.

[0052] The beneficial effects of adopting the above technical solution are as follows: The welding beam design method based on the hybrid swarm intelligence optimization algorithm provided by the present invention makes up for the shortcomings of the chimpanzee algorithm and the arithmetic optimization algorithm, and integrates the two into one, proposing a high-performance hybrid swarm intelligence optimization algorithm. Applying the proposed algorithm to the welding beam design optimization problem greatly reduces the design cost of welding beams. Attached Figure Description

[0053] Figure 1 A flowchart illustrating a welded beam design method based on a hybrid swarm intelligence optimization algorithm, provided as an embodiment of the present invention.

[0054] Figure 2 This is a schematic diagram of the welded beam structure provided in an embodiment of the present invention. Detailed Implementation

[0055] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0056] In this embodiment, a welded beam design method based on a hybrid swarm intelligence optimization algorithm is described, such as... Figure 1 As shown, it includes the following steps:

[0057] Step 1: Determine the objective function and constraints for the welded beam design;

[0058] Will as Figure 2 The mathematical model of the welded beam shown is defined as follows: The welded beam has four physical properties: weld width h, clamp length l, beam length t, and beam width b. These four physical properties are used as design variables for the welded beam design, denoted as x1, x2, x3, and x4. This represents a set of design schemes consisting of 4 design variables;

[0059] The objective function for the welded beam design is set as follows:

[0060]

[0061] The constraints of the mathematical model of the welded beam are:

[0062]

[0063]

[0064]

[0065]

[0066]

[0067]

[0068]

[0069] in, For welding stress, For bending stress, For the deformation of the beam, For the instability critical load, τ max σ max δ max These represent the maximum values ​​of welding stress, bending stress, and beam deformation, respectively, with P being the torque.

[0070]

[0071]

[0072] J is the polar moment of inertia, E is Young's modulus, and G is the shear modulus.

[0073] In this embodiment, τ max =13600psi, σ max =30000psi, L=14in, δ max =0.25in, E = 30 × 10 6 psi, G = 12 × 10 6 psi, P = 6000 lb;

[0074] It should be noted that: constraints This indicates that the shear stress borne by the welded beam must not exceed the maximum limit value; constraint condition. This indicates that the bending stress on the welded beam must not exceed the maximum limit value; constraint condition. This indicates that the end deviation of the welded beam must not exceed 0.25, a constraint condition. This indicates that the weld width h must not exceed the beam width b, which is a constraint condition. This indicates that the critical load for instability of the welded beam must not be less than a fixed value P, constraining the condition. This indicates that the weld width h must not be less than the minimum limit value of 0.125;

[0075] Step 2: Optimize the ChOA and AOA algorithms respectively to obtain the improved chimpanzee optimization algorithm IChOA and the arithmetic optimization algorithm IAOA. Construct a hybrid swarm intelligence optimization algorithm based on IChOA and IAOA, denoted as H-IChOA-IAOA.

[0076] First, to address the shortcomings of the ChOA algorithm, we optimize the nonlinear decay mode of the convergence factor f and introduce a preference weight λ based on Manhattan distance in the position update formula. Second, to address the shortcomings of the AOA algorithm, we use sine and cosine functions to map the spatial location information of the current individual and replace the multipliers determined by the interval vector in AOA. Finally, we extract 20% of the individuals in the chimpanzee population to perform the IAOA optimization process.

[0077] The nonlinear decay mode of the convergence factor f is optimized using the following equation:

[0078]

[0079] In the formula, f inital Let f be the initial value of the convergence factor f. final Let f be the final value of the convergence factor f, and t and T be the current iteration number and the maximum iteration number, respectively; in this embodiment, f inital =2.5, f final =0;

[0080] The convergence factor exhibits the following pattern: in the early stages, the rate of change is relatively large, meaning that the step size between neighboring chimpanzees increases, accelerating their approach to prey; in the later stages, the decay is gradual, with a smaller rate of change, ensuring that chimpanzees have sufficient time to conduct a detailed search near the prey, effectively balancing the exploration and development phases of the algorithm.

[0081] Step 3: Based on the hybrid swarm intelligence optimization algorithm (H-IChOA-IAOA) built in Step 2, solve the objective function of the welded beam design to obtain the optimal solution for the welded beam design;

[0082] Step 3.1: Initialize the chimpanzee population. The initial positions of the chimpanzees are the initial values ​​of the four design variables in Step 1. Construct an N×4 matrix as the solution space, where N is the population size.

[0083] Step 3.2: Use the objective function of the welded beam design as the fitness function to calculate the fitness of the chimpanzee population (calculating fitness means substituting the values ​​of the design variables into the objective function to obtain the value) and determine the positions of attackers, obstructors, expellers and pursuers.

[0084] Step 3.3: Update the location of individual chimpanzees;

[0085] (1) 80% of chimpanzee individuals updated their positions using the following formula:

[0086] X(t+1)=λ1X1+λ2X2+λ3X3+λ4X4

[0087]

[0088] Among them, X A X B X C X D X1, X2, X3, and X4 represent the position information of the attacker, obstacle, expeller, and pursuer before the update, respectively. X1, X2, X3, and X4 represent the position information of the attacker, obstacle, expeller, and pursuer after the update, respectively. X is the current position of the chimpanzee individual. A = f·(2·r1-1), C = 2·r2, m = Chaotic_value is the chaotic mapping. The values ​​of r1 and r2 are randomly generated between [0,1] by the rand function. m is the chaotic mapping vector representing the influence of motivation.

[0089] (2) 20% of chimpanzee individuals updated their location in the following manner:

[0090] Random numbers r3, r4, and r5 are randomly generated within the interval [0,1]. When r3 > MOA, the chimpanzee individual's position is updated according to the following formula:

[0091]

[0092] When r3 < MOA, the chimpanzee individual's position is updated according to the following formula:

[0093]

[0094] In the formula, X(t) is the position of the t-th generation chimpanzee, X(t+1) is the position of the (t+1)-th generation chimpanzee, and X... best (t) represents the optimal individual position, i.e., the attacker's position. λ1, λ2, λ3, and λ4 are the current individual's preference weights relative to the attacker, obstructor, expelr, and pursuer, respectively, based on the Manhattan distance. MOA is the mathematical optimizer acceleration function, MOP is the mathematical optimizer probability, r3, r4, and r5 are random numbers in the interval [0,1], ε is an infinitesimal positive number, and g(·) is the mapping function, representing a mapping relationship.

[0095] The definition of the mathematical optimizer speedup function MOA is:

[0096]

[0097] In the formula, Max = 0.7 represents the maximum value of the acceleration function, and Max = 0.2 represents the minimum value of the acceleration function;

[0098] The definition of the mathematical optimizer probability (MOP) is:

[0099]

[0100] In the formula, α is a sensitive parameter, which defines the development accuracy in the iteration, and is set to 5 here;

[0101] The formulas for calculating the preference weights λ1, λ2, λ3, and λ4 are as follows:

[0102]

[0103] In the formula, d1, d2, d3, and d4 represent the Manhattan distances of the current individual relative to the attacker, obstructor, expeller, and pursuer, respectively.

[0104] The expression for the mapping function is:

[0105] g(X(t))=sin(X(t))+cos(X(t))

[0106] Step 3.4: Check whether the range of the solution after updating the chimpanzee individual's position exceeds the range of the design variable. If it does, replace the solution that exceeds the range with the boundary value of the design variable.

[0107] Step 3.5: Determine if the current iteration number is less than the maximum iteration number. If so, return to step 3.2. Otherwise, output the minimum value of the objective function as the minimum design cost of the welded beam, and the currently found solution as the optimal welded beam design scheme.

[0108] In this embodiment, the pseudocode for the hybrid swarm intelligence optimization algorithm is as follows:

[0109]

[0110] This embodiment uses a hybrid swarm intelligence optimization algorithm (H-IChOA-IAOA) to design welded beams with the lowest relative cost. The design performance is compared with that of genetic algorithm (GA), particle swarm optimization algorithm (PSO), gray wolf algorithm (GWO), whale algorithm (WOA), basic chimpanzee algorithm (ChOA), basic arithmetic optimization algorithm (AOA), and two improved chimpanzee algorithms (SChOA and IChOA). The data results are shown in Table 1.

[0111] In this embodiment, the parameters are set to a population size of 30, a maximum number of iterations of 500, and the program is run repeatedly 30 times. The output results are shown in the table below:

[0112]

[0113] As shown in the table above, H-IChOA-IAOA finds and filters the relatively optimal values ​​of the four main factors of the welded beam through group parallelism, and finally calculates a relatively lower design cost. The ranking of the solutions to the welded beam design problem by each algorithm is as follows: GA>ChOA>WOA>PSO>GWO>SChOA>IChOA>AOA>H-IChOA-IAOA. Therefore, it is evident that the method of this invention breaks through the optimal lower bound of recent algorithms for solving the welded beam design problem, and can obtain a relatively better design solution.

[0114] This embodiment employs a hybrid optimization algorithm (H-IChOA-IAOA) combining an improved chimpanzee algorithm and an improved arithmetic optimization algorithm to solve the minimum cost problem in welded beam design. IChOA refers to the improved chimpanzee optimization algorithm using a nonlinear convergence factor and preference weights, while IAOA refers to the improved arithmetic optimization algorithm replacing multipliers determined by variable intervals with sine and cosine function mappings. The welded beam design problem can be abstracted into a mathematical programming problem with four structural parameters, seven constraints, and one objective function for algorithm optimization. The optimized data shows that the hybrid optimization algorithm H-IChOA-IAOA of this invention can effectively design welded beams with lower costs.

[0115] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope defined by the claims of the present invention.

Claims

1. A method for designing welded beams based on a hybrid swarm intelligence algorithm, characterized in that: Includes the following steps: Step 1: Determine the objective function and constraints for the welded beam design; The mathematical model of the welded beam is defined as follows: The welded beam comprises four physical properties: weld width h, clamp length l, beam length t, and beam width b. These four physical properties are used as design variables for the welded beam design, denoted as... ; This represents a set of design schemes consisting of 4 design variables; The objective function for the welded beam design is set as follows: ； The constraints are: ； ； ； ； ； ； ； in, For welding stress, For bending stress, For the deformation of the beam, This is the critical load for instability. , , These represent the maximum values ​​of welding stress, bending stress, and beam deformation, respectively, with P being the torque. Step 2: Optimize the ChOA and AOA algorithms respectively to obtain the improved chimpanzee optimization algorithm IChOA and the arithmetic optimization algorithm IAOA. Construct a hybrid swarm intelligence optimization algorithm based on IChOA and IAOA, denoted as H-IChOA-IAOA. Firstly, for the ChOA algorithm, on the one hand, the convergence factor is optimized. On the one hand, it exhibits a nonlinear decay mode; on the other hand, it introduces preference weights based on Manhattan distance into the location update formula. Secondly, for the AOA algorithm, sine and cosine functions are used to map the spatial location information of the current individual and replace the multipliers determined by the interval vector in AOA; finally, 20% of the individuals in the chimpanzee population are selected to perform the IAOA optimization process. The convergence factor The nonlinear decay mode is optimized using the following formula: ； wherein is an initial value of the convergence factor , is a final value of the convergence factor , and are the current iteration number and the maximum iteration number, respectively. Step 3: Based on the hybrid swarm intelligence optimization algorithm based on IChOA and IAOA built in Step 2, solve the objective function of the welded beam design to obtain the optimal solution for the welded beam design; Step 3.1: Initialize the chimpanzee population. The initial positions of the chimpanzees are the initial values ​​of the four design variables from Step 1. The matrix is ​​used as the solution space, where N is the population size; Step 3.2: Using the objective function of the welded beam design as the fitness function, calculate the fitness of the chimpanzee population and determine the positions of attackers, obstructors, expellers, and pursuers. Step 3.3: Update the location of individual chimpanzees; (1) 80% of chimpanzee individuals updated their positions using the following formula: ； ； in, , , , These represent the position information of the attacker, obstacle, expeller, and pursuer before the update, respectively. , , , These represent the updated location information of the attacker, the obstacle, the expeller, and the pursuer, respectively. This indicates the current location of the individual chimpanzee. , , For chaotic mapping, and The value is determined by the rand function. Randomly generated between It is a chaotic mapping vector representing the influence of representative motivations; (2) 20% of chimpanzee individuals updated their locations using the following method: Randomly generated intervals random numbers , and ,when At that time, the chimpanzee individual updates its position according to the following formula: ； When the chimpanzee individual updates its position according to the following equation: ； In the formula, It is the first The location of the chimpanzee It is the first The location of the chimpanzee This represents the optimal individual position, which is also the attacker's position. These represent the Manhattan distance-based preference weights of the current individual relative to the attacker, obstructor, expelr, and pursuer, respectively; MOA (Modal of Interest) is the speedup function for the exponential optimizer; and MOP (Modal of Opposition) is the probability of the exponential optimizer. , and For interval random numbers, It is an infinite decimal. A mapping function represents a mapping relationship; The definition of the mathematical optimizer acceleration function MOA is: ； wherein denotes the maximum value of the acceleration function, denotes the minimum value of the acceleration function; The definition of the mathematical optimizer probability (MOP) is: ； wherein is a sensitive parameter and defines the development accuracy in iterations, which is taken as 5 here; Preference weight The calculation formula is: ； In the formula, These represent the Manhattan distances of the current individual relative to the attacker, the obstructor, the expeller, and the pursuer, respectively. The expression for the mapping function is: ； Step 3.4: Check whether the range of the solution after updating the position of the chimpanzee individual exceeds the range of the design variable. If it does, replace the solution that exceeds the range with the boundary value of the design variable. Step 3.5: Determine if the current iteration number is less than the maximum iteration number. If so, return to step 3.

2. Otherwise, output the minimum value of the objective function as the minimum design cost of the welded beam, and the currently found solution as the optimal welded beam design scheme.

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