Multi-objective welding beam design method based on grey prediction evolutionary algorithm

Abstract

The invention relates to the technical field of welded beams, in particular to a multi-target welded beam design method based on a grey prediction evolutionary algorithm, which comprises the following steps: randomly generating a first-generation solution set, a second-generation solution set and a third-generation solution set of a multi-target welded beam mathematical model; mining evolutionary trends of three solutions randomly selected from the first-generation solution set, the second-generation solution set and the third-generation solution set by using a grey prediction propagation operator, and predicting a solution in the intermediate-generation solution set; after the intermediate generation solution set is generated, constructing a candidate solution set, and performing non-dominated sorting on the candidate solution set to obtain a non-dominated leading edge set; selecting a fourth-generation solution set from the non-dominated leading edge set by using an adaptive environment selection mechanism, namely, autonomously switching a selection strategy according to the evolution condition of the solution; and finally, through algorithm iteration, a group of Pareto optimal solutions which are fully converged and are diverse and uniform in distribution are obtained. According to the method, the flexibility and adaptability of designing the multi-target welding beam are improved, and the application effect and economic benefits of the welding beam in actual engineering are promoted.

Description

Technical Field

[0001] The present invention relates to the technical field of welded beams, and in particular to a multi-objective welded beam design method based on a grey prediction evolutionary algorithm. Background Art

[0002] In modern industrial manufacturing, welded beams, as a key component of structural parts, are widely used in fields such as bridges, buildings, and mechanical engineering. To meet different engineering requirements, the design of welded beams needs to comprehensively consider multiple objectives. Traditional single-objective welded beam design methods usually only focus on minimizing the total cost of welded beams. Although this method can effectively reduce manufacturing costs, it may lead to poor performance of welded beams in actual applications. For example, when only considering cost, designers may choose more economical materials or simplify the design, which often results in excessive deflection of the welded beam when it bears a load, that is, the deformation amount at the beam end exceeds the safe range. Excessive deflection not only affects the overall stability of the structure but also may cause additional safety hazards such as stress concentration and fatigue damage, ultimately shortening the service life of the welded beam and increasing maintenance costs.

[0003] In contrast, multi-objective welded beam design methods particularly focus on the balance between the total cost and the beam end deflection under a specific load. Appropriate deflection control can ensure that when the welded beam bears the expected load, it will neither affect the use function due to excessive deformation nor waste material resources due to an overly rigid design. By optimizing the beam end deflection under a specific load, designers can achieve a lightweight design while ensuring structural safety. In addition, a reasonable deflection design can also improve the dynamic response characteristics of the welded beam, making it show better stability and durability in a vibrating environment.

[0004] Multi-Objective Evolutionary Algorithm (MOEA) is a commonly used solution method for solving multi-objective welded beam design problems. From the perspective of meta-heuristics, MOEAs can be divided into: MOEAs based on natural evolution mechanisms, such as NSGA-II that uses genetic operators and MOEA / D that uses differential evolution operators; MOEAs inspired by biological social behaviors, such as MOPSO inspired by bird flock foraging and MOACO inspired by ant foraging. R; Physical phenomenon-based MOEAs, such as EMOSA that simulates the metal annealing process and MOGSA that simulates gravity; math model-inspired MOEAs, such as MVMO-EDA that introduces a probability model. However, the convergence and diversity of the Pareto optimal solution sets obtained by the above algorithms still need to be improved, so that the solution sets of these algorithms may not provide sufficient diversity and uniformity, specifically manifested as: in some mechanical engineering welding beam design projects, the solution sets obtained by some algorithms may be overly concentrated in certain local areas and cannot fully cover the entire feasible solution space, resulting in the omission of some potential excellent design schemes; or in the later stage of iteration of some algorithms, the improvement of the solution set is extremely small, making it difficult to accurately approach the true optimal solution front, unable to provide designers with rich and high-quality choices, limiting the optimization potential of welding beam design in terms of performance-cost balance, resulting in insufficient flexibility of the design scheme and difficulty in meeting the complex and changing actual engineering requirements. Summary of the Invention

[0005] To this end, the technical problem to be solved by the present invention is to overcome the problem that the single-objective welding beam design method in the prior art only aims at minimizing the total cost of the welding beam, resulting in poor performance of the welding beam in actual applications; and the problem that using a multi-objective evolutionary algorithm to solve the multi-objective welding beam design problem, the convergence and diversity of the obtained Pareto optimal solution set are insufficient, resulting in insufficient flexibility of the welding beam design scheme and inability to well meet the actual engineering requirements.

[0006] To solve the above technical problems, the present invention provides a multi-objective welding beam design method based on a grey prediction evolutionary algorithm, including:

[0007] S1: Construct a multi-objective welding beam mathematical model;

[0008] S2: Randomly obtain the first-generation solution set, the second-generation solution set, and the third-generation solution set of the multi-objective welding beam mathematical model; wherein, the number of solutions in each generation of the solution set is N; each solution in each generation of the solution set contains the values of all decision variables of the multi-objective welding beam mathematical model.

[0009] S3: Randomly select a solution from the first-generation solution set, the second-generation solution set, and the third-generation solution set respectively, and use the grey prediction reproduction operator to obtain a solution of the intermediate-generation solution set, and repeat this step until the number of solutions in the intermediate-generation solution set is N;

[0010] S4: Combine the intermediate-generation solution set and the third-generation solution set to obtain a candidate solution set;

[0011] S5: Perform non-dominated sorting on the candidate solution set to obtain a non-dominated front set;

[0012] S6: If the number of solutions in the first non-dominated front of the non-dominated front set is greater than N and at least a preset proportion of the solutions in the third-generation solution set are included, then the uniformity selection strategy is adopted to select the fourth-generation solution set from the first non-dominated front; otherwise, the crowding degree selection strategy is adopted to select the fourth-generation solution set from the non-dominated front set;

[0013] S7: Update the second-generation solution set as the new first-generation solution set, update the third-generation solution set as the new second-generation solution set, update the fourth-generation solution set as the new third-generation solution set, and return to execute S3 until the maximum number of iterations is reached. Take the fourth-generation solution set obtained in the final round as the Pareto optimal solution set for designing the multi-objective welded beam; among them, each solution in the Pareto optimal solution set of the multi-objective welded beam is a design scheme for the multi-objective welded beam.

[0014] Preferably, the step of randomly selecting one solution from each of the first-generation solution set, the second-generation solution set, and the third-generation solution set and using the grey prediction reproduction operator to obtain a solution in the intermediate-generation solution set includes:

[0015] Randomly select one solution from each of the first-generation solution set, the second-generation solution set, and the third-generation solution set, and denote them as the first solution, the second solution, and the third solution in sequence;

[0016] Based on the values of each decision variable in the first solution, the second solution, and the third solution, calculate the maximum value and the minimum value of each decision variable; based on the difference between the upper and lower bounds of each decision variable, calculate the difference threshold of each decision variable;

[0017] The grey prediction reproduction operator consists of a mean grey model, a linear fitting model, and a random perturbation model;

[0018] If the minimum value of the current decision variable is greater than or equal to its difference threshold, then use the mean grey model to predict the value of the current decision variable in a solution of the intermediate-generation solution set;

[0019] If the maximum value of the current decision variable is less than its difference threshold, then use the linear fitting model to predict the value of the current decision variable in a solution of the intermediate-generation solution set;

[0020] If the minimum value of the current decision variable is less than its difference threshold or the maximum value of the current decision variable is greater than or equal to its difference threshold, then use the random perturbation model to predict the value of the current decision variable in a solution of the intermediate-generation solution set.

[0021] Preferably, the process of obtaining the nth solution in the intermediate-generation solution set includes:

[0022] When obtaining the nth solution in the intermediate-generation solution set, the expressions for the maximum value and the minimum value of each decision variable are respectively:

[0023]

[0024] Among them, respectively represent the maximum and minimum values of the k-th decision variable when obtaining the n-th solution in the intermediate generation solution set; represents the value of the k-th decision variable in the first solution when obtaining the n-th solution in the intermediate generation solution set; represents the value of the k-th decision variable in the second solution when obtaining the n-th solution in the intermediate generation solution set; represents the value of the k-th decision variable in the third solution when obtaining the n-th solution in the intermediate generation solution set; n = 1, 2,..., N;

[0025] The calculation of the difference threshold for each decision variable has the following expression:

[0026] θ k = 0.01×(up k - low k ) ;

[0027] Among them, θ k represents the difference threshold of the k-th decision variable; up k represents the upper bound of the k-th decision variable; low k represents the lower bound of the k-th decision variable;

[0028] The expression for the value of the k-th decision variable in the n-th solution of the intermediate generation solution set is:

[0029]

[0030] Among them, represents the value of the k-th decision variable in the n-th solution of the intermediate generation solution set; α represents the grey development coefficient, β represents the grey control coefficient, and ω represents the random perturbation range control parameter. Their expressions are respectively:

[0031]

[0032] Among them, G represents the maximum number of iterations.

[0033] Preferably, after predicting the value of the current decision variable in a solution of the intermediate generation solution set, it further includes:

[0034] Judging whether the value of the current decision variable in the predicted n-th solution of the intermediate generation solution set exceeds its upper and lower boundaries. If the value of the current decision variable in the predicted n-th solution of the intermediate generation solution set exceeds its boundary, boundary processing is performed on the value of the current decision variable in the predicted n-th solution of the intermediate generation solution set. The expression for the boundary processing is:

[0035]

[0036] Among them, represents the value of the k-th decision variable in the n-th solution of the intermediate generation solution set; up k represents the upper bound of the k-th decision variable; low k represents the lower bound of the k-th decision variable.

[0037] Preferably, the construction of the multi-objective welding beam mathematical model includes:

[0038] Taking the weld thickness, the length of the clip, the cross-sectional length of the welding beam, and the cross-sectional width of the welding beam as decision variables, aiming at minimizing the manufacturing cost of the welding beam and minimizing the deflection at the beam end under the action of the load, a multi-objective function is constructed; determining the welding stress constraint condition, the bending stress constraint condition, the weld thickness constraint condition, and the buckling bearing capacity constraint condition in the vertical direction; based on the two objective functions and all constraint conditions, a multi-objective welding beam mathematical model is constructed.

[0039] Preferably, the expression of the multi-objective welding beam mathematical model includes:

[0040] The expression of the multi-objective function is:

[0041]

[0042] Among them, the first expression in the multi-objective function represents the first objective function aiming at minimizing the manufacturing cost of the welding beam; the second expression in the multi-objective function represents the second objective function aiming at minimizing the deflection at the beam end under the action of the load; Z 1 (x) represents the first objective value; Z 2 (x) represents the second objective value; x 1 represents the value of the first decision variable in the solution x, that is, the value of the weld thickness; x 2 represents the value of the second decision variable in the solution x, that is, the value of the clip length; x 3 represents the value of the third decision variable in the solution x, that is, the value of the cross-sectional length of the welding beam; x 4 represents the value of the fourth decision variable in the solution x, that is, the value of the cross-sectional width of the welding beam; w 1 represents the first coefficient, w 1 = 1.1047; w 2 represents the second coefficient, w 2 = 0.04811; w 3 represents the third coefficient, w 3 = 14; P represents the load, P = 6000; C represents the intermediate parameter, C = 4(14 3 ) / (30×10 6 )≈3.6587×10 -4 ;

[0043] The expressions of all constraint conditions are:

[0044]

[0045] Among them, g 1 (x) represents the welding stress constraint condition; g 2 (x) represents the bending stress constraint condition; g 3 (x) represents the weld thickness constraint condition; g 4 (x) represents the buckling bearing capacity constraint condition in the vertical direction; τ max represents the maximum value of the welding stress, τ max = 13600 psi; τ(x) represents the welding stress corresponding to the solution x, and its expression is: τ” = HRJ -1 , H = P(L + 0.5x 2 ), H, R, and J respectively represent the first intermediate variable, the second intermediate variable, and the third intermediate variable; σ max represents the maximum value of the bending stress, σ max = 30000 psi; σ(x) represents the bending stress corresponding to the solution x, and its expression is: σ(x) = 6PL(x 4 (x 3 ) 2 ) -1 ; L represents the length of the welded beam, L = 14 in; P represents the load, P = 6000 lbs; P c (x) represents the buckling bearing capacity in the vertical direction, and its expression is: E represents the elastic modulus of the welded beam material, E = 30×10 6 psi; G represents the shear modulus of the welded beam material, G = 12×10 6 psi.

[0046] Preferably, the uniformity selection strategy includes:

[0047] Substitute each solution in the first non-dominated front into the multi-objective function in turn, and calculate the respective objective values corresponding to each solution in the first non-dominated front; perform normalization processing on the respective objective values corresponding to each solution in the first non-dominated front to obtain the normalized first non-dominated front; among them, the expression of the normalization processing is:

[0048]

[0049] Among them, F 1 represents the first non-dominated front; represents all solutions in the first non-dominated front; represents the value after normalization of the m-th objective value corresponding to the u-th solution in the first non-dominated front; represents the m-th objective value corresponding to the u-th solution in the first non-dominated front; respectively represent the maximum and minimum values of the m-th objective values of all solutions in the first non-dominated front;

[0050] Move all the solutions belonging to the third-generation solution set in the first non-dominated front into the solution set Q new ; If the number of solutions in the solution set Q new at this time is less than N, randomly select solutions from the first non-dominated front to complete the solution set Q new ; Remove all the solutions belonging to the third-generation solution set in the first non-dominated front to obtain the target first non-dominated front, denoted as

[0051] Based on the calculation formula of the uniformity of the solution set, calculate the uniformity of the solution set Q new , and its expression is:

[0052]

[0053] where f PU (Q new ) represents the uniformity of the solution set Q new ; x new,i represents the i-th solution in the solution set Q new ; x new,j represents the j-th solution in the solution set Q new ; Q new \{x new,i} represents the solution set Q new,u after removing x new , that is, x new,j ∈Q new and x new,j ≠x new,i ; ||·|| represents the Manhattan distance; M represents the number of objectives;

[0054] Find the solution from the target first non-dominated solution set and find the solution x new from the solution set Q new,h such that the uniformity of the solution set U is the smallest, and denote x new,h and as and respectively, and call the solution set U the best neighbor solution set of the solution set Q new ; where, represents the solution set obtained by adding new,h to the solution set Q new after removing x ; x new,h is the solution set Qnew the h-th solution in represent the target first non-dominated front the l-th solution in

[0055] Calculate the uniformity of the solution set U based on the uniformity calculation formula of the solution set.

[0056] If the uniformity of the solution set U is less than that of the solution set Q new then replace the new in the solution set Q with the in the target first non-dominated front to obtain the updated solution set Q new and the updated target first non-dominated front, and continue to find the best neighbor solution set of the updated solution set Q new ;

[0057] If the uniformity of the solution set U is greater than or equal to that of the solution set Q new then stop updating the solution set Q new , and end the loop, taking the current solution set Q new as the final solution set Q new ;

[0058] Perform denormalization on the normalized values of each objective value corresponding to each solution in the final solution set Q new to restore each objective value corresponding to each solution, obtaining the fourth-generation solution set.

[0059] Preferably, the crowding degree selection strategy includes:

[0060] The non-dominated front set contains multiple non-dominated fronts. Denote the first v non-dominated fronts in the non-dominated front set as F 1 , F 2 , …, F v ; where the sum of the number of solutions in F 1 to F v is greater than N;

[0061] Put all the solutions in F 1 to F v-1 into the solution set Q new ; Calculate the crowding degree of each solution in the v-th non-dominated front F v based on the crowding degree calculation formula of the solution, and its expression is:

[0062]

[0063] where represents the crowding degree of the t-th solution in F v ; M represents the number of objectives; respectively represent F vThe (t + 1)-th solution and the (t - 1)-th solution in, i.e., the previous solution and the next solution of the t-th solution; denotes F v the m-th objective value corresponding to the (t + 1)-th solution in; Z m (x v,t-1 ) denotes F v the m-th objective value corresponding to the (t - 1)-th solution in; respectively denote the maximum value and the minimum value of the m-th objective value of all solutions in the v-th non-dominated front F v ;

[0064] Sort the crowding degree of all solutions in F v in descending order, and put the first N - |Q new | solutions with the largest crowding degree into the solution set Q new to obtain the solution set Q new containing N solutions, as the fourth-generation solutions; where, |Q new | represents the number of solutions in the solution set Q new .

[0065] Preferably, the non-dominated sorting of the candidate solution set is performed to obtain a non-dominated front set including:

[0066] Calculate the constraint violation value of each solution in the candidate solution set based on the constraint violation value expression of the solution; based on the constraint violation value of each solution, perform non-dominated sorting on the candidate set according to the dominance relationship, that is: taking all solutions in the candidate solution set as a benchmark, find all non-dominated solutions and form the first non-dominated front; remove all solutions in the first non-dominated front from the candidate solution set, take all solutions in the current candidate solution set after removal as a benchmark, find all non-dominated solutions and form the second non-dominated solution, and so on, until all solutions in the candidate solution set are assigned to different non-dominated fronts to obtain a non-dominated sorting set;

[0067] Among them, the expression of the constraint violation value CV(x) of the solution is:

[0068]

[0069] Among them, A represents the number of inequality constraints; B represents the number of equality constraints; a represents the index of the inequality constraint; b represents the index of the equality constraint; D a (x) < 0 represents the a-th inequality constraint; T b (x) = 0 represents the b-th equality constraint;

[0070] The dominance relationship is: for any two solutions y 1 and y 2 , if the constraint violation value of y 1 is less than that of y 2If the constraint violation value of y 1 dominates y 2 , and vice versa, y 2 dominates y 1 ; when the constraint violation values of both are equal, if and only if the objective value of y 1 on all objective functions is not less than that of y 2 , and the objective value on at least one objective function is greater than that of y 2 , then y 1 dominates y 2 , and vice versa, y 2 dominates y 1 .

[0071] Preferably, a random initialization method is adopted to randomly obtain the first-generation solution set, the second-generation solution set, and the third-generation solution set of the multi-objective welded beam mathematical model, including:

[0072] Determine the value range of each decision variable according to the upper and lower bounds of each decision variable; randomly select a value from the value range of each decision variable as the initial value; according to the random initialization expression, obtain the value of each decision variable in each solution;

[0073] The random initialization expression is: x s,k = low k + rand × (up k - low k );

[0074] where x s,k represents the initial value of the k-th decision variable of solution x; rand represents a random number uniformly distributed from 0 to 1; up k represents the upper bound of the k-th decision variable; low k represents the lower bound of the k-th decision variable.

[0075] The above technical solution of the present invention has the following beneficial effects compared with the prior art:

[0076] A multi-objective welded beam design method based on a grey prediction evolutionary algorithm according to the present invention comprehensively considers two objectives, namely the manufacturing cost of the beam and the deflection at the beam end under load. Through multi-objective optimization, it can effectively reduce the manufacturing cost while ensuring the safety and stability of the welded beam structure (deflection within a reasonable range), achieving a good balance between performance and cost, and avoiding sacrificing performance due to excessive pursuit of cost or causing cost waste due to excessive emphasis on performance. It strictly restricts the welding stress, bending stress, the relationship between the weld thickness and the beam cross-section width, and the vertical buckling bearing capacity, ensuring that the welded part is firm and reliable during the actual use of the welded beam and will not have welding failure due to excessive stress. The beam body can withstand the applied load and will not deform or be damaged due to excessive bending stress. The weld size is reasonable and meets the structural and technological requirements. It has sufficient stability in the vertical direction and will not have buckling instability, thus comprehensively improving the structural reliability of the welded beam, extending its service life, reducing maintenance costs and potential safety hazards. It uses a grey prediction reproduction operator to discover the evolutionary law of three consecutive generations of solutions, and then predicts the next generation of solutions, improving the accuracy and speed of model convergence. It uses an adaptive environmental selection mechanism to adaptively switch different selection strategies according to the evolutionary status of the solution set. When the solution set is still in a rapid convergence state, the crowding degree selection strategy is used, and when the solution set converges slowly, the uniformity selection strategy is used. The adaptive environmental selection mechanism can effectively ensure the diversity and uniformity of the final solution set, further improving the quality of the solution set. Through the grey prediction reproduction operator and the adaptive environmental selection mechanism, the obtained final solution set has the characteristics of sufficient convergence, diverse and uniform distribution. Each solution in the final solution set corresponds to a design scheme of the multi-objective welded beam. Therefore, this method can provide a rich variety of high-quality design schemes for welded beam designers, enabling them to select the most suitable scheme from the Pareto optimal solution set according to the different requirements and actual situations of specific projects, better meeting the personalized requirements of welded beams in different fields such as bridges, buildings, and mechanical engineering, improving the flexibility and adaptability of the design, and promoting the application effect and economic benefits of welded beams in actual projects. BRIEF DESCRIPTION OF THE DRAWINGS

[0077] In order to make the content of the present invention easier to be clearly understood, the following further details the present invention according to specific embodiments of the present invention in conjunction with the accompanying drawings, wherein:

[0078] Figure 1 is a flowchart of a multi-objective welded beam design method based on a grey prediction evolutionary algorithm provided by the present invention;

[0079] Figure 2 is a schematic diagram of the welded beam structure;

[0080] Figure 3It is a schematic diagram of the process of predicting the intermediate generation solution set using the grey prediction reproduction operator;

[0081] Figure 4 It is a comparison graph of the Pareto fronts of the solution sets obtained by the grey prediction evolutionary algorithm and other existing algorithms; among them, Figure 4 (a) in it represents the Pareto front graph of the solution set obtained by the GPEA-AES algorithm; Figure 4 (b) in it represents the Pareto front graph of the solution set obtained by the NSGA3 algorithm; Figure 4 (c) in it represents the Pareto front graph of the solution set obtained by the MOPSO / vPF algorithm; Figure 4 (d) in it represents the Pareto front graph of the solution set obtained by the PFG-MOEA algorithm. Specific implementation manners

[0082] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, so that those skilled in the art can better understand the present invention and be able to implement it, but the embodiments cited do not limit the present invention.

[0083] Referring to Figure 1 as shown, Figure 1 is a flowchart of a multi-objective welded beam design method based on the grey prediction evolutionary algorithm provided by the present invention; specifically including:

[0084] S1: Figure 2 It shows the welded beam structure in the multi-objective welded beam design problem. There are a total of four decision variables to be solved, namely the weld thickness, the length of the clamp bar, the cross-sectional length and width of the welded beam. These four values form a solution to the multi-objective welded beam design problem;

[0085] Therefore, the present invention takes the weld thickness, the length of the clamp bar, the cross-sectional length of the welded beam and the cross-sectional width of the welded beam as decision variables, and takes the minimum manufacturing cost of the welded beam and the minimum deflection at the beam end under the action of the load as objectives to construct a multi-objective function; determine the welding stress constraint condition, the bending stress constraint condition, the weld thickness constraint condition and the buckling bearing capacity constraint condition in the vertical direction; based on the two objective functions and all constraint conditions, construct a multi-objective welded beam mathematical model, including:

[0086] The expression of the multi-objective function is:

[0087]

[0088] Among them, the first expression in the multi-objective function represents the first objective function with the minimum manufacturing cost of the welded beam as the objective; the second expression in the multi-objective function represents the second objective function with the minimum deflection at the beam end under the action of the load as the objective; Z 1 (x) represents the first objective value; Z2 (x) represents the second target value; x 1 represents the value of the first decision variable in the solution x, that is, the value of the weld thickness, and its value range is [0.125, 5]; x 2 represents the value of the second decision variable in the solution x, that is, the value of the length of the clip, and its value range is [0.1, 10]; x 3 represents the value of the third decision variable in the solution x, that is, the value of the cross-sectional length of the welded beam, and its value range is [0.1, 10]; x 4 represents the value of the fourth decision variable in the solution x, that is, the value of the cross-sectional width of the welded beam, and its value range is [0.125, 5]; w 1 represents the first coefficient, w 1 = 1.1047; w 2 represents the second coefficient, w 2 = 0.04811; w 3 represents the third coefficient, w 3 = 14; P represents the load, P = 6000; C represents the intermediate parameter, C = 4(14 3 ) / (30×10 6 )≈3.6587×10 -4 ;

[0089] The expressions of all constraint conditions are:

[0090]

[0091] Among them, g 1 (x) represents the welding stress constraint condition; g 2 (x) represents the bending stress constraint condition; g 3 (x) represents the weld thickness constraint condition; g 4 (x) represents the buckling bearing capacity constraint condition in the vertical direction; τ max represents the maximum value of the welding stress, τ max = 13600 psi; τ(x) represents the welding stress corresponding to the solution x, and its expression is: τ” = HRJ -1 , H = P(L + 0.5x 2 ), H, R, and J respectively represent the first intermediate variable, the second intermediate variable, and the third intermediate variable; σ max represents the maximum value of the bending stress, σ max = 30000 psi; σ(x) represents the bending stress corresponding to the solution x, and its expression is: σ(x) = 6PL(x 4 (x 3 ) 2 )-1 ; L represents the length of the welded beam, L = 14 in; P represents the load, P = 6000 lbs; P c (x) represents the buckling capacity in the vertical direction, and its expression is: E represents the elastic modulus of the welded beam material, E = 30×10 6 psi; G represents the shear modulus of the welded beam material, G = 12×10 6 psi;

[0092] S2: Randomly obtain the first-generation solution set, the second-generation solution set, and the third-generation solution set of the multi-objective welded beam mathematical model; where, the number of solutions in each generation of solution sets is N; each solution in each generation of solution sets contains the values of all decision variables of the multi-objective welded beam mathematical model;

[0093] In a specific embodiment of the present invention, a random initialization method is adopted to randomly obtain the first-generation solution set, the second-generation solution set, and the third-generation solution set of the multi-objective welded beam mathematical model, that is, for each decision variable of the solution, a value is randomly selected from a certain value range as the initial value, including:

[0094] According to the upper and lower bounds of each decision variable, determine the value range of each decision variable; randomly select a value from the value range of each decision variable as the initial value; according to the random initialization expression, obtain the value of each decision variable in each solution;

[0095] The random initialization expression is: x s,k = low k + rand×(up k - low k );

[0096] Where, x s,k represents the initial value of the kth decision variable of the solution x; rand represents a random number uniformly distributed from 0 to 1; up k represents the upper bound of the kth decision variable; low k represents the lower bound of the kth decision variable;

[0097] S3: Randomly select a solution from the first-generation solution set, the second-generation solution set, and the third-generation solution set respectively, and use the grey prediction reproduction operator to obtain a solution of the intermediate-generation solution set, and repeat this step until the number of solutions in the intermediate-generation solution set is N, including:

[0098] Randomly select a solution from the first-generation solution set, the second-generation solution set, and the third-generation solution set respectively, and denote them as the first solution, the second solution, and the third solution in sequence;

[0099] Based on the values of each decision variable in the first solution, the second solution, and the third solution, calculate the maximum and minimum values of each decision variable; based on the difference between the upper and lower bounds of each decision variable, calculate the difference threshold of each decision variable.

[0100] The gray prediction reproduction operator consists of a mean gray model, a linear fitting model, and a random perturbation model.

[0101] If the minimum value of the current decision variable is greater than or equal to its difference threshold, use the mean gray model to predict the value of the current decision variable in a solution of the intermediate generation solution set.

[0102] If the maximum value of the current decision variable is less than its difference threshold, use the linear fitting model to predict the value of the current decision variable in a solution of the intermediate generation solution set.

[0103] If the minimum value of the current decision variable is less than its difference threshold or the maximum value of the current decision variable is greater than or equal to its difference threshold, use the random perturbation model to predict the value of the current decision variable in a solution of the intermediate generation solution set.

[0104] Among them, the process of obtaining the nth solution in the intermediate generation solution set includes:

[0105] When obtaining the nth solution in the intermediate generation solution set, the expressions for the maximum and minimum values of each decision variable are respectively:

[0106]

[0107] Among them, respectively represent the maximum and minimum values of the kth decision variable when obtaining the nth solution in the intermediate generation solution set; represents the value of the kth decision variable in the first solution when obtaining the nth solution in the intermediate generation solution set; represents the value of the kth decision variable in the second solution when obtaining the nth solution in the intermediate generation solution set; represents the value of the kth decision variable in the third solution when obtaining the nth solution in the intermediate generation solution set; n = 1, 2,..., N;

[0108] The expression for calculating the difference threshold of each decision variable is:

[0109] θ k = 0.01×(up k - low k );

[0110] Among them, θ k represents the difference threshold of the kth decision variable; up k represents the upper bound of the kth decision variable; low k represents the lower bound of the kth decision variable;

[0111] The expression for the value of the k-th decision variable in the n-th solution of the intermediate generation solution set is as follows:

[0112]

[0113] Among them, represents the value of the k-th decision variable in the n-th solution of the intermediate generation solution set; α represents the grey development coefficient, β represents the grey control coefficient, and ω represents the random disturbance range control parameter, and their expressions are respectively:

[0114]

[0115] Among them, G represents the maximum number of iterations;

[0116] After predicting the value of the current decision variable in a solution of the intermediate generation solution set, it further includes:

[0117] Judge whether the value of the current decision variable in the n-th solution of the predicted intermediate generation solution set exceeds its upper and lower bounds. If the value of the current decision variable in the n-th solution of the predicted intermediate generation solution set exceeds its bounds, perform boundary processing on the value of the current decision variable in the n-th solution of the predicted intermediate generation solution set; the expression for the boundary processing is:

[0118]

[0119] Among them, represents the value of the k-th decision variable in the n-th solution of the intermediate generation solution set; up k represents the upper bound of the k-th decision variable; low k represents the lower bound of the k-th decision variable;

[0120] In summary, the process of predicting the intermediate generation solution set using the grey prediction reproduction operator is as Figure 3 shown;

[0121] S4: Combine the intermediate generation solution set with the third-generation solution set to obtain the candidate solution set Q candi ;

[0122] S5: Perform non-dominated sorting on the candidate solution set to obtain the non-dominated front set (F 1 , F 2 , …, F v , …), including:

[0123] Based on the constraint violation value expression of the solution, calculate the constraint violation value of each solution in the candidate solution set; based on the constraint violation value of each solution, according to the dominance relationship, perform non-dominated sorting on the candidate set, that is: taking all solutions in the candidate solution set as a benchmark, find all non-dominated solutions and form the first non-dominated front; remove all solutions in the first non-dominated front from the candidate solution set, taking all solutions in the current candidate solution set after removal as a benchmark, find all non-dominated solutions and form the second non-dominated solution, and so on, until all solutions in the candidate solution set are assigned to different non-dominated fronts to obtain the non-dominated sorting set (F 1 , F 2 , …, F v , …); where the sum of the number of solutions in F 1 ~F v just exceeds N, that is, there are more than v non-dominated fronts in the non-dominated front set. Cumulatively sum the number of solutions in each non-dominated front in order until the cumulative sum of the number of solutions reaches the v-th non-dominated front, and the cumulative sum of the number of solutions just exceeds N; since there are N solutions in a solution set of the multi-objective welded beam mathematical model, in the subsequent adaptive environmental selection mechanism, only consider selecting the fourth-generation solution set from these non-dominated fronts of F 1 ~F v whose sum of the number of solutions just exceeds N;

[0124] Among them, the expression of the constraint violation value CV(x) of the solution is:

[0125]

[0126] where A represents the number of inequality constraints; B represents the number of equality constraints; a represents the index of the inequality constraint; b represents the index of the equality constraint; D a (x) < 0 represents the a-th inequality constraint; T b (x) = 0 represents the b-th equality constraint;

[0127] Among them, the definition of a non-dominated solution is: in a solution set, if a certain solution is not dominated by any solution in the solution set, then this solution is a non-dominated solution;

[0128] The dominance relationship is: for any two solutions y 1 and y 2 , if the constraint violation value of y 1 is less than the constraint violation value of y 2 , then y 1 dominates y 2 , and vice versa, y 2 dominates y 1 ; in the case where their constraint violation values are equal, if and only if y 1 is not inferior to y2 and is strictly better than y on at least one objective function 2 when y 1 dominates y 2 Conversely, y 2 dominates y 1 That is, if and only if y 1 The objective values on all objective functions are not less than those of y 2 and the objective value on at least one objective function is greater than that of y 2 when y 1 dominates y 2 Conversely, y 2 dominates y 1 ;

[0129] In a specific embodiment of the present invention, the fast non-dominated sorting algorithm is used to perform non-dominated sorting on the candidate solution set, and its pseudo-code is shown in Table 1;

[0130] Table 1 Pseudo-code of the fast non-dominated sorting algorithm

[0131]

[0132]

[0133] S6: Using the adaptive environmental selection mechanism, select the fourth-generation solutions from the non-dominated front set, including:

[0134] If the number of solutions in the first non-dominated front in the non-dominated front set is greater than N, and at least a preset proportion of the solutions in the third-generation solution set are included, then adopt the uniformity selection strategy to select the fourth-generation solution set from the first non-dominated front; otherwise, adopt the crowding degree selection strategy to select the fourth-generation solution set from the non-dominated front set; In a specific embodiment of the present invention, the preset proportion is 96%;

[0135] The uniformity selection strategy includes:

[0136] Substitute each solution in the first non-dominated front into the multi-objective function in turn, and calculate the respective objective values corresponding to each solution in the first non-dominated front; perform normalization processing on the respective objective values corresponding to each solution in the first non-dominated front to obtain the normalized first non-dominated front; where, the expression of the normalization processing is:

[0137]

[0138] where, F 1 represents the first non-dominated front; represents all solutions in the first non-dominated front; represents the value after normalization of the m-th objective value corresponding to the u-th solution in the first non-dominated front; denotes the m-th objective value corresponding to the u-th solution in the first non-dominated front; denote the maximum and minimum values of the m-th objective values of all solutions in the first non-dominated front respectively;

[0139] Move all solutions in the first non-dominated front that belong to the third-generation solution set into the solution set Q new ; If the number of solutions in the solution set Q new at this time is less than N, randomly select solutions from the first non-dominated front to complete the solution set Q new ; Remove all solutions in the first non-dominated front that belong to the third-generation solution set to obtain the target first non-dominated front, denoted as

[0140] Based on the calculation formula of the uniformity of the solution set, calculate the uniformity of the solution set Q new , and its expression is:

[0141]

[0142] where, f PU (Q new ) represents the uniformity of the solution set Q new ; x new,i represents the i-th solution in the solution set Q new ; x new,j represents the j-th solution in the solution set Q new ; Q new \{x new,i} represents the solution set Q new,i after removing x new , that is, x new,j ∈Q new and x new,j ≠x new,i ; ||·|| represents the Manhattan distance; M represents the number of objectives;

[0143] Find the solution from the target first non-dominated solution set and find the solution x new from the solution set Q new,h such that the uniformity of the solution set U is the smallest, and denote x new,h and as and respectively, and call the solution set U the best neighbor solution set of the solution set Q new ; where, represents the solution set obtained by adding new,h to the solution set Q new after removing x ; x new,h is the h-th solution in the solution set Q new ; Indicates the target first non-dominated front the l-th solution in;

[0144] Based on the uniformity calculation formula of the solution set, calculate the uniformity of the solution set U;

[0145] If the uniformity of the solution set U is less than that of the solution set Q new then replace the new in the solution set Q with that in the target first non-dominated front to obtain the updated solution set Q new and the updated target first non-dominated front, and continue to search for the best neighbor solution set of the updated solution set Q new ;

[0146] If the uniformity of the solution set U is greater than or equal to that of the solution set Q new then stop updating the solution set Q new , and end the loop, taking the current solution set Q new as the final solution set Q new ;

[0147] Denormalize the normalized values of each objective value corresponding to each solution in the final solution set Q new to restore each objective value corresponding to each solution, that is, restore each objective value corresponding to each solution to its original value, and finally obtain the fourth-generation solution set;

[0148] The crowding degree selection strategy includes:

[0149] The non-dominated front set contains multiple non-dominated fronts. Denote the first v non-dominated fronts in the non-dominated front set as F 1 , F 2 , …, F v ; where the sum of the number of solutions in F 1 to F v is greater than N;

[0150] Put all the solutions in F 1 to F v-1 into the solution set Q new ; Based on the crowding degree calculation formula of the solution, calculate the crowding degree of each solution in the v-th non-dominated front F v , and its expression is:

[0151]

[0152] where, represents the crowding degree of the t-th solution in F v ; M represents the number of objectives; respectively represent F vThe (t + 1)-th solution and the (t - 1)-th solution in, that is, the solution before and after the t-th solution; Denote F v The m-th objective value corresponding to the (t + 1)-th solution in F; Denote F v The m-th objective value corresponding to the (t - 1)-th solution in F; Denote the maximum and minimum values of the m-th objective values of all solutions in the v-th non-dominated front F v respectively;

[0153] Arrange the crowding degree of all solutions in F v in descending order, and put the first N - |Q new | solutions with the largest crowding degree into the solution set Q new to obtain a solution set Q containing N solutions new , as the fourth-generation solutions; where, |Q new | represents the number of solutions in the solution set Q new ;

[0154] In a specific embodiment of the present invention, the pseudo-code of the adopted adaptive environmental selection mechanism is shown in Table 2;

[0155] Table 2 Pseudo-code of the adaptive environmental selection mechanism

[0156]

[0157] S7: Update the second-generation solution set to the new first-generation solution set, update the third-generation solution set to the new second-generation solution set, update the fourth-generation solution set to the new third-generation solution set, and return to execute S3 until the maximum number of iterations is reached. Take the fourth-generation solution set obtained in the final round as the Pareto optimal solution set for designing the multi-objective welded beam; where, each solution in the Pareto optimal solution set of the multi-objective welded beam is a design scheme for the multi-objective welded beam.

[0158] In summary, the present invention proposes a multi-objective welded beam design method based on a grey prediction evolutionary algorithm. First, a mathematical model for the multi-objective welded beam design problem is defined. Subsequently, a grey prediction reproduction operator is introduced, which can explore the evolutionary trend of solutions in three consecutive generations to predict the new generation of solutions. After each generation of new solutions is generated, an adaptive environmental selection mechanism is used to select excellent solutions for the next generation, that is, the selection strategy is autonomously switched according to the evolutionary situation of the solutions. Finally, a set of Pareto optimal solutions with sufficient convergence, diverse distribution, and uniformity can be obtained through algorithm iteration. Among them, the grey prediction evolutionary algorithm mainly consists of two parts: the grey prediction reproduction operator and the adaptive environmental selection mechanism, which integrates the advantages of grey system theory and evolutionary computation. It can not only effectively handle the uncertainty problems under small sample and poor information conditions but also be good at finding Pareto optimal solutions in a large-scale search space. The present invention uses the grey prediction reproduction operator to explore the evolutionary law of solutions in three consecutive generations and then predicts the next generation of solutions. Compared with other evolutionary operators, the grey prediction reproduction operator has obvious advantages in both the accuracy of convergence and the convergence speed. The present invention adopts an adaptive environmental selection mechanism, which adaptively switches different selection strategies according to the evolutionary status of the solution set. When the solution set is still in a rapid convergence state, the crowding degree selection strategy is used, and when the solution set converges slowly, the uniformity selection strategy is used. The adaptive environmental selection mechanism can effectively ensure the diversity and uniformity of the final solution set and further improve the quality of the solution set.

[0159] Experiments are conducted to verify the effectiveness of the method of the present invention. In the experiments, the method provided by the present invention (GPEA-AES) is compared with the non-dominated sorting genetic algorithm (NSGA3), the multi-objective evolutionary algorithm guided by the Pareto front grid (PFG-MOEA), and the adaptive multi-objective particle swarm optimization algorithm based on the virtual Pareto front (MOPSO / vPF). To ensure fairness, the population sizes of the four algorithms are uniformly set to 150, the number of evaluation times is uniformly set to 90000, and all run independently 30 times on the multi-objective welded beam design problem.

[0160] The evaluation index of the final solution set is the hypervolume (HV). HV measures the range of the objective space covered by the solution set by calculating the volume between the final solution set and the reference point. The larger the HV value, the better the convergence of the solution set, and the wider and more uniform the range covered by the solution set in the objective space. The HV calculation formula for the solution set Q is as follows:

[0161]

[0162] where r = (r 1 , r 2 , …, r M ) Tis the reference point; when setting the reference point, it must be ensured that all solutions in the final solution set Q dominate r, and the reference points of all test algorithms must be the same to ensure fairness.

[0163] Figure 4 Shows the Pareto fronts of the final solution sets obtained by four algorithms for the multi-objective welded beam design problem. It can be clearly seen from the figure the diversity of the final solution sets obtained by the four algorithms. Diversity mainly examines the coverage range and uniformity of the solutions; in terms of the coverage range of the solutions, the distribution intervals of the Pareto optimal solution sets obtained by NSGA3, MOPSO / vPF, and PFG-MOEA on the deflection objective are approximately (0, 0.009), while that of GPEA-AES reaches (0, 0.016); obviously, the coverage range of the Pareto optimal solution set obtained by GPEA-AES is wider; in terms of the uniformity of the solutions, the solutions of NSGA3 are mainly concentrated in the middle region, and the sparser the solutions are distributed towards both ends; there are uneven distributions in local regions for MOPSO / vPF and PFG-MOEA; only the final solution set of GPEA-AES is the most evenly distributed.

[0164] Table 3 shows the average values and standard deviations of the HV indicators of the four algorithms for the multi-objective welded beam design problem; it is not difficult to see from the table that the average HV value of GPEA-AES running 30 times is the highest among the four algorithms, which indicates that the Pareto optimal solution set obtained by GPEA-AES is superior to other algorithms in terms of both convergence and diversity. Moreover, the standard deviation of HV of GPEA-AES is also the lowest among the four algorithms, which shows that GPEA-AES has strong stability.

[0165] Table 3 Average values and standard deviations of the HV indicators of the four algorithms for the multi-objective welded beam design problem

[0166]

[0167] Obviously, the above embodiments are merely examples given for clear illustration and are not limitations on the implementation manners. For those of ordinary skill in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to list all the implementation manners here. And the obvious changes or modifications derived therefrom are still within the protection scope of the present invention.

Claims

1. A multi-objective welding beam design method based on grey prediction evolutionary algorithm, characterized in that: include: S1: Construct a multi-objective mathematical model of welded beams; S2: Randomly obtain the first generation solution set, the second generation solution set and the third generation solution set of the multi-objective welded beam mathematical model; wherein the number of solutions in each generation solution set is N; each solution in each generation solution set contains the values ​​of all decision variables of the multi-objective welded beam mathematical model; S3: Randomly select a solution from the first generation solution set, the second generation solution set and the third generation solution set respectively, use the grey prediction breeding operator to obtain a solution in the intermediate generation solution set, and repeat this step until the number of solutions in the intermediate generation solution set is N; S4: Merge the intermediate generation solution set with the third generation solution set to obtain the candidate solution set; S5: Perform non-dominated sorting on the candidate solution set to obtain the non-dominated frontier set; S6: If the number of solutions in the first non-dominated front in the non-dominated front set is greater than N, and it contains at least a preset proportion of solutions in the third-generation solution set, then the uniformity selection strategy is adopted to select the fourth-generation solution set from the first non-dominated front; otherwise, the congestion selection strategy is adopted to select the fourth-generation solution set from the non-dominated front set; S7: Update the second-generation solution set to a new first-generation solution set, update the third-generation solution set to a new second-generation solution set, update the fourth-generation solution set to a new third-generation solution set, return to execute S3, until the maximum number of iterations is reached, and use the fourth-generation solution set obtained in the final round as the Pareto optimal solution set for designing a multi-objective welded beam; wherein each solution in the Pareto optimal solution set of the multi-objective welded beam is a design scheme for the multi-objective welded beam.

2. The multi-objective welding beam design method based on grey prediction evolutionary algorithm according to claim 1 is characterized in that: The method of randomly selecting a solution from the first generation solution set, the second generation solution set and the third generation solution set, and using the grey prediction breeding operator to obtain a solution of the intermediate generation solution set includes: Randomly select a solution from the first generation solution set, the second generation solution set, and the third generation solution set, respectively, and record them as the first solution, the second solution, and the third solution; Based on the values ​​of each decision variable in the first solution, the second solution, and the third solution, the maximum value and the minimum value of each decision variable are calculated; based on the difference between the upper and lower bounds of each decision variable, the difference threshold of each decision variable is calculated; The grey prediction reproduction operator is composed of a mean grey model, a linear fitting model and a random disturbance model; If the minimum value of the current decision variable is greater than or equal to its difference threshold, the mean grey model is used to predict the value of the current decision variable in a solution of the intermediate generation solution set; If the maximum value of the current decision variable is less than its difference threshold, the linear fitting model is used to predict the value of the current decision variable in a solution of the intermediate generation solution set; If the minimum value of the current decision variable is less than its difference threshold or the maximum value of the current decision variable is greater than or equal to its difference threshold, the random perturbation model is used to predict the value of the current decision variable in a solution of the intermediate generation solution set.

3. The multi-objective welding beam design method based on grey prediction evolutionary algorithm according to claim 2 is characterized in that: The process of obtaining the nth solution in the intermediate generation solution set includes: When obtaining the nth solution in the intermediate solution set, the expressions for the maximum and minimum values ​​of each decision variable are: in, They represent the maximum and minimum values ​​of the kth decision variable when obtaining the nth solution in the intermediate solution set; Indicates the value of the kth decision variable in the first solution when obtaining the nth solution in the intermediate solution set; Indicates the value of the kth decision variable in the second solution when obtaining the nth solution in the intermediate solution set; Indicates the value of the kth decision variable in the third solution when obtaining the nth solution in the intermediate solution set; n = 1, 2, ..., N; The difference threshold of each decision variable is calculated as follows: θ k =0.01×(up k -low k ); Among them, θ k represents the difference threshold of the k-th decision variable; up k represents the upper bound of the k-th decision variable; low k represents the lower bound of the kth decision variable; The expression for the value of the kth decision variable in the nth solution in the intermediate generation solution set is: in, represents the value of the kth decision variable in the nth solution in the intermediate generation solution set; α represents the grey development coefficient, β represents the grey control coefficient, and ω represents the random disturbance range control parameter. Their expressions are: Where G represents the maximum number of iterations.

4. The multi-objective welding beam design method based on grey prediction evolutionary algorithm according to claim 2 is characterized in that: After predicting the value of the current decision variable in a solution of the intermediate generation solution set, it also includes: Determine whether the value of the current decision variable in the nth solution of the predicted intermediate generation solution set exceeds its upper and lower boundaries. If the value of the current decision variable in the nth solution of the predicted intermediate generation solution set exceeds its boundary, perform boundary processing on the value of the current decision variable in the nth solution of the predicted intermediate generation solution set; the expression of the boundary processing is: in, represents the value of the kth decision variable in the nth solution in the intermediate generation solution set; up k represents the upper bound of the k-th decision variable; low k represents the lower bound of the kth decision variable.

5. The multi-objective welding beam design method based on grey prediction evolutionary algorithm according to claim 1 is characterized in that: The construction of the multi-objective welded beam mathematical model comprises: With weld thickness, clamp length, cross-sectional length of welded beam and cross-sectional width of welded beam as decision variables, and with the goals of minimizing the manufacturing cost of welded beam and minimizing the disturbance of beam end under load, a multi-objective function is constructed; welding stress constraints, bending stress constraints, weld thickness constraints and buckling bearing capacity constraints in the vertical direction are determined; based on the two objective functions and all constraints, a multi-objective welded beam mathematical model is constructed.

6. The multi-objective welding beam design method based on grey prediction evolutionary algorithm according to claim 5 is characterized in that: The expression of the multi-objective welded beam mathematical model includes: The expression of the multi-objective function is: Among them, the first expression in the multi-objective function represents the first objective function with the minimum manufacturing cost of the welded beam as the goal; the second expression in the multi-objective function represents the second objective function with the minimum disturbance at the beam end under the load as the goal; Z1(x) represents the first objective value; Z2(x) represents the second objective value; x 1 represents the value of the first decision variable in solution x, that is, the value of the weld thickness; x 2 represents the value of the second decision variable in the solution x, that is, the value of the length of the clamp; x 3 represents the value of the third decision variable in the solution x, that is, the value of the cross-sectional length of the welded beam; x 4 represents the value of the fourth decision variable in the solution x, that is, the value of the cross-sectional width of the welded beam; w1 represents the first coefficient, w1=1.1047; w2 represents the second coefficient, w2=0.04811; w3 represents the third coefficient, w3=14; P represents the load, P=6000; C represents the intermediate parameter, C=4(14 3 ) / (30×10 6 )≈3.6587×10 -4 ; The expressions for all constraints are: Among them, g1(x) represents the welding stress constraint; g2(x) represents the bending stress constraint; g3(x) represents the weld thickness constraint; g4(x) represents the vertical buckling bearing capacity constraint; τ max represents the maximum value of welding stress, τ max =13600psi; τ(x) represents the welding stress corresponding to the solution x, and its expression is: τ”=HRJ -1 , H=P(L+0.5x 2 ), H, R, and J represent the first intermediate variable, the second intermediate variable, and the third intermediate variable, respectively; σ max represents the maximum value of bending stress, σ max =30000psi; σ(x) represents the bending stress corresponding to the solution x, and its expression is: σ(x) = 6PL(x 4 (x 3 ) 2 ) -1 ; L represents the length of the welding beam, L = 14in; P represents the load, P = 6000lbs; P c (x) represents the buckling capacity in the vertical direction, and its expression is: E represents the elastic modulus of the welding beam material, E = 30 × 10 6 psi; G represents the shear modulus of the welded beam material, G = 12 × 10 6 psi.

7. The multi-objective welding beam design method based on grey prediction evolutionary algorithm according to claim 5 is characterized in that: The uniformity selection strategy includes: Substitute each solution in the first non-dominated front into the multi-objective function in turn, and calculate each objective value corresponding to each solution in the first non-dominated front; perform normalization on each objective value corresponding to each solution in the first non-dominated front to obtain the normalized first non-dominated front; wherein the normalization expression is: Among them, F1 represents the first non-dominated frontier; represents all solutions in the first non-dominated front; represents the normalized value of the mth target value corresponding to the uth solution in the first non-dominated frontier; represents the mth objective value corresponding to the uth solution in the first non-dominated frontier; They represent the maximum and minimum values ​​of the mth objective values ​​of all solutions in the first non-dominated frontier respectively; Move all solutions belonging to the third generation solution set in the first non-dominated frontier into the solution set Q new In; if the solution set Q new If the number of solutions is less than N, then randomly select solutions from the first non-dominated frontier to complete the solution set Q new ; Eliminate all solutions belonging to the third generation solution set in the first non-dominated frontier and obtain the target first non-dominated frontier, denoted as Based on the uniformity calculation formula of the solution set, calculate the solution set Q new The uniformity of is expressed as: Among them, fP U( Q new ) represents the solution set Q new Uniformity; x new,i Represents the solution set Q new The i-th solution in new,j Represents the solution set Q new The jth solution in Q new \{x new,i } means to remove x new,i The solution set Q new , that is, x new,j ∈Q new And x new,j ≠x new,i ;||·|| represents Manhattan distance; M represents the number of targets; Find the solution from the target first non-dominated solution set And from the solution set Q new Find the solution x in new,h , so that the uniformity of the solution set U is minimized, and x new,h and Denoted as and And the solution set U is called solution set Q new The best neighbor solution set of ; among them, Indicates that when removing x new,h The solution set Q new Add The solution set obtained after that; x new,h For the solution set Q new The h-th solution in ; represents the first non-dominated frontier of the target The lth solution in ; Based on the uniformity calculation formula of the solution set, calculate the uniformity of the solution set U; If the uniformity of solution set U is less than that of solution set Q new The uniformity of the solution set Q new In and the target first non-dominated front Replace each other and get the updated solution set Q new and the updated target first non-dominated frontier, and continue to find the updated solution set Q new The best neighbor solution set of ; If the uniformity of solution set U is greater than or equal to solution set Q new If the uniformity of new , and end the loop, and set the current solution set Q new As the final solution set Q new ; For the final solution set Q new The normalized values ​​of each target value corresponding to each solution in are denormalized to restore each target value corresponding to each solution, and the fourth generation solution set is obtained.

8. The multi-objective welding beam design method based on grey prediction evolutionary algorithm according to claim 5 is characterized in that: The congestion degree selection strategy includes: The non-dominated frontier set contains multiple non-dominated frontiers. The first v non-dominated frontiers in the non-dominated frontier set are denoted as F1, F2, ..., F v ; Among them, F1 to F v The sum of the number of solutions is greater than N; F1 to F v-1 All solutions in are put into the solution set Q new In; Based on the solution crowding calculation formula, calculate the vth non-dominated frontier F v The congestion degree of each solution in is expressed as: in, Indicates F v The congestion degree of the t-th solution in; M represents the number of targets; Respectively represent F v The (t+1)th solution and the (t-1)th solution are the previous solution and the next solution of the tth solution; Indicates F v The mth target value corresponding to the (t+1)th solution in Z m (x v,t-1 ) indicates F v The mth target value corresponding to the (t-1)th solution in ; They represent the vth non-dominated front F v The maximum and minimum values ​​of the mth objective value of all solutions in ; F v The congestion of all solutions in the solution are arranged in descending order, and the top N-|Q solutions with the largest congestion are selected. new |A liberation into the solution set Q new In the solution set Q, we get N solutions. new , as the fourth generation solution; among them, |Q new | represents the solution set Q new The number of solutions.

9. The multi-objective welding beam design method based on grey prediction evolutionary algorithm according to claim 5 is characterized in that: The non-dominated sorting of the candidate solution set to obtain the non-dominated frontier set includes: Based on the constraint violation value expression of the solution, the constraint violation value of each solution in the candidate solution set is calculated; based on the constraint violation value of each solution, the candidate set is non-dominated sorted according to the dominance relationship, that is: taking all solutions in the candidate solution set as the benchmark, all non-dominated solutions are found and the first non-dominated front is formed; all solutions in the first non-dominated front are eliminated from the candidate solution set, and all solutions in the candidate solution set after the elimination are taken as the benchmark, all non-dominated solutions are found and the second non-dominated solution is formed, and so on, until all solutions in the candidate solution set are assigned to different non-dominated fronts, and a non-dominated sorted set is obtained; The constraint violation value CV(x) of the solution is expressed as: Where A represents the number of inequality constraints; B represents the number of equality constraints; a represents the index of the inequality constraint; b represents the index of the equality constraint; D a (x)<0 indicates the ath inequality constraint; T b (x) = 0 represents the bth equality constraint; The dominance relationship is: for any two solutions y1 and y2, if the constraint violation value of y1 is less than the constraint violation value of y2, then y1 dominates y2, otherwise, y2 dominates y1; when the constraint violation values ​​of the two are equal, y1 dominates y2 if and only if the target value of y1 on all objective functions is not less than y2, and the target value on at least one objective function is greater than y2, otherwise, y2 dominates y1.

10. The multi-objective welding beam design method based on grey prediction evolutionary algorithm according to claim 1 is characterized in that: Using the random initialization method, the first generation solution set, the second generation solution set and the third generation solution set of the multi-objective welded beam mathematical model are randomly obtained, including: According to the upper and lower bounds of each decision variable, determine the value range of each decision variable; randomly select a value from the value range of each decision variable as the initial value; according to the random initialization expression, obtain the value of each decision variable in each solution; The random initialization expression is: s,k =low k +rand×(up k -low k ); Among them, x s,k represents the initial value of the kth decision variable of the solution x; rand represents a random number uniformly distributed from 0 to 1; up k represents the upper bound of the k-th decision variable; low k represents the lower bound of the kth decision variable.

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