Theoretical-numerical fusion driven composite cap-shaped stiffened plate optimization method

Abstract

The invention relates to a theory-numerical value fusion driven composite hat-shaped stiffened plate optimization method, belongs to the field of thin-wall structure optimization, and provides a theory-numerical value fusion driven structure optimization method which combines finite element numerical value simulation calculation and a correction theory algorithm. According to the method, not only can the requirements of poor theoretical solution precision and low numerical simulation solution efficiency be solved, but also parameter optimization of the composite material thin-wall stiffened structure can be realized, so that the composite material cap-shaped stiffened wall plate structure is optimized, the limitation of experimental means is made up, and the experimental efficiency of the composite material cap-shaped stiffened wall plate structure is improved. Therefore, a more reliable design basis is provided for engineering application.

Description

Technical Field

[0001] This invention relates to a structural optimization method, particularly a structural optimization method for composite material cap-shaped stiffened plates, specifically a theory-numerical fusion driven optimization method for composite material cap-shaped stiffened plates. Background Technology

[0002] Composite material reinforced thin-walled structures, due to their high specific strength and high specific stiffness, have gradually become the mainstream load-bearing structures for aircraft. The stability and load-bearing capacity of these structures have a profound impact on the service performance of aircraft.

[0003] Composite cap-shaped stiffened panels have a huge post-buckling load capacity, and finding the optimal design scheme for stiffened panels quickly is an urgent problem for engineering designers. However, due to the complex material characteristics and structural features of composite stiffened thin-walled structures, it is difficult to predict their load-bearing capacity, and structural optimization is even more difficult.

[0004] To leverage the material-level performance of composite materials and maximize the potential of the structure's post-buckling load-bearing range, this invention proposes a theory-numerical fusion-driven structural optimization method for cap-shaped stiffened panels. By combining finite element numerical simulation calculations with modified engineering theory, a dual-loop optimization framework is established, consisting of an inner loop correction coefficient and an outer loop optimized structure. This results in a composite material cap-shaped stiffened panel optimization method that is both efficient and accurate.

[0005] This method can not only significantly reduce the need for poor theoretical solution accuracy and low numerical solution efficiency, and improve research efficiency and accuracy, but also achieve parameter optimization of composite thin-walled stiffened structures, thereby optimizing composite cap-shaped stiffened wall panel structures, making up for the limitations of experimental methods, and thus providing a more reliable design basis for engineering applications. Summary of the Invention

[0006] To address the aforementioned technical problems, this invention proposes a theory-numerical fusion-driven optimization method for composite material cap-shaped stiffeners, thereby resolving the issues present in the prior art.

[0007] The purpose of this invention is to propose a theory-numerical fusion-driven optimization method for composite material hat-shaped stiffened panels, targeting the field of composite material stiffened panel structure optimization. This method aims to establish a dual-loop optimization framework—inner loop correction coefficients and outer loop optimization structure—by combining finite element numerical simulation with modified engineering theory. This addresses the issues of poor theoretical solution accuracy and low numerical simulation efficiency, enabling the optimization of composite material hat-shaped stiffened panel structures and providing a more reliable design basis for thin-walled structure engineering applications in aircraft.

[0008] The present invention adopts the following technical solution:

[0009] A theory-numerical fusion-driven optimization method for composite material cap-shaped stiffeners, characterized by the following steps:

[0010] S1: Establish a buckling finite element model of composite cap-shaped stiffened wall panel: Construct a buckling finite element model based on the dimensions of the composite cap-shaped stiffened wall panel structure and the composite material layup information, calculate the buckling load, and compare and correct it with the experiment. Correct the finite element model until the buckling load error is ≤5%.

[0011] S2: Establish a modified engineering algorithm for skin local buckling load: Based on the boundary conditions of the composite cap-shaped stiffened panel skin, and using the composite laminate buckling analysis method, calculate the skin local buckling load. Considering the reinforcement of the skin boundary conditions by the stiffeners, a modified engineering algorithm for skin buckling load with modified bending stiffness and calculation width is proposed.

[0012] S3: Sensitivity analysis of structural parameters of composite material cap-shaped stiffened wall panel: The orthogonal experimental design method was used to conduct sensitivity analysis on the buckling performance parameters of the structure, obtain the contribution of each design variable to the objective function, and select variables with a contribution rate >5% as optimization parameters;

[0013] S4: Establish a multi-objective optimization algorithm for composite material cap-shaped stiffened plates driven by theory-numerical fusion: Based on the adaptive simulated annealing algorithm and the non-dominated sorting genetic algorithm, a dual-loop optimization architecture is built to optimize the inner loop correction coefficient and the outer loop structure, so as to obtain the structural optimization parameters.

[0014] Furthermore: The aforementioned theory-numerical fusion driven optimization method for composite cap-shaped stiffened plates is characterized in that the optimization structure is a dual-loop optimization architecture, the inner loop optimization is based on the numerical simulation algorithm to optimize the correction coefficient in the theoretical algorithm of skin buckling load, and the outer loop optimization is based on the optimal solution of buckling load correction coefficient and theoretical algorithm to optimize the structure of composite cap-shaped stiffened plates;

[0015] Furthermore, the process of establishing the buckling finite element model of the composite material cap-shaped stiffened panel in step S1 specifically includes:

[0016] (a) In finite element software, a shell element model of the cap-shaped stiffened wall panel is established based on the actual component dimensions;

[0017] (b) Divide the grid according to different material regions, and set the composite material layup according to the actual layup sequence of the skin and the reinforcing ribs, with the 0° direction of the layup aligned with the axial direction of the reinforcing ribs;

[0018] (c) A shared node mesh is established in the skin-stiffener connection area, and the stiffener-frame beam interface is constrained by Tie;

[0019] (d) Establish reference points in the extended regions of the loading end and the fixed end, and couple the nodes in the end connection area (i.e., the potting area) to the reference points, so that the reference points control the fixing and loading of the boundary conditions;

[0020] (e) The fixed end reference point constrains all degrees of freedom, the loading end reference point constrains all degrees of freedom except for the z-direction translation, and the two sides of the stiffened wall plate constrain the y-direction translation degree of freedom.

[0021] (f) Perform linear perturbation buckling analysis in the finite element software and apply axial compressive load at the loading end to obtain the first-order buckling load and mode;

[0022] Furthermore, the process of establishing the modified skin local buckling load engineering algorithm in step S2 specifically includes:

[0023] (a) Calculation using traditional skin local buckling load engineering algorithm;

[0024] (b) Considering the reinforcement of the skin bending stiffness by the lower flange stringers, the skin bending stiffness is modified accordingly: D R D is the skin-corrected bending stiffness matrix, and D is the uncorrected bending stiffness matrix of the skin. f Here is the bending stiffness matrix of the lower flange stringer, and the bending stiffness coefficient d = (b ss -b s ) / b ss .

[0025] (c) A modified buckling load calculation method for composite stiffened panels is proposed: N cr This represents the corrected buckling load for the cap-shaped stiffened panel. This represents the buckling load calculated for side-fixed supports. This represents the buckling load calculated using a simply supported side panel, where D is the ratio of the stiffened panel's bending stiffness to that of the skin.

[0026] Furthermore, the process of analyzing the sensitivity of the structural parameters of the composite material cap-shaped stiffened wall panel specifically includes:

[0027] (a) Design variables include: stiffener spacing (b) s Lower flange width b f Web width b w Hat top width b c Web inclination angle α, distance s between two transverse frames, total length L of stiffened panel, number of stiffeners N, skin stiffness correction factor d x And calculate the width b ss ;

[0028] (b) Set the constraints as follows: b s ≤b ss ≤bs +2b f The remaining design variables take values ​​within the range of ±20%;

[0029] (c) An orthogonal experimental design method was used to establish a quadratic polynomial response surface model: Where: y A Let x be the objective function; i To design the sample points of the variables, β represents the main effect of each term in the multinomial model on the objective function, including the linear term effect β. i Quadratic effect β j and interaction effect β ij ;

[0030] (d) Regularize the coefficients of the polynomial, fit the model through ridge regression, calculate the standardized regression coefficients, and screen variables with an absolute contribution of >5% as key influencing factors; (e) Input the key variables and response surface model into step S4 for target optimization;

[0031] Furthermore: the process of establishing a theory-numerical fusion-driven multi-objective optimization algorithm for composite material cap-shaped stiffened plates specifically includes:

[0032] (a) Based on the adaptive simulated annealing algorithm, the difference between the simulation results of S1 and the theoretical results of S2 is used as the optimization objective to optimize the skin width b in the theoretical calculation. ss Multi-objective optimization is performed using the skin bending stiffness correction factor d to obtain the computational width b. ss The optimal solution for the skin bending stiffness correction factor d;

[0033] (b) Using traditional post-buckling load engineering algorithms, a multi-objective optimization algorithm for the structural parameters of composite cap-shaped stiffened panels is established based on a non-dominated sorting genetic algorithm.

[0034] (c) The objective function for optimizing the inner loop correction coefficient is defined as: min ABS(diff)=|σ 数值解 -σ 理论解 The objective function for optimizing the outer ring structure is: And set weighting factors and define the optimization range of the optimization variables;

[0035] (d) Perform iterations, output the Pareto optimal solution set, and obtain the structural optimization parameters. Attached Figure Description

[0036] Figure 1 This is a flowchart of the method for optimizing the shape of a helicopter fairing based on numerical simulation, as proposed in this invention.

[0037] Figure 2 This is a schematic diagram of the dimensions of the hat-shaped reinforced wall panel structure in the implementation method of the present invention.

[0038] Figure 3 This is a schematic diagram of the boundary conditions in the implementation method of the present invention.

[0039] Figure 4 This is a schematic diagram of the finite element results of skin buckling in the implementation method of the present invention.

[0040] Figure 5 This is a schematic diagram of the sensitivity results of the cap-shaped reinforced wall panel in the implementation method of the present invention.

[0041] Figure 6 This is a schematic diagram of the algorithm structure in the implementation method of the present invention.

[0042] Figure 7 This is a schematic diagram of the Pareto optimal solution set in the implementation method of the present invention.

[0043] Figure 8 This is a schematic diagram of the load-displacement curve verified by finite element software in the implementation method of the present invention. Specific implementation methods

[0044] The technical solution of the present invention will be further described clearly and completely below with reference to the accompanying drawings and specific embodiments in the invention examples.

[0045] like Figure 1 As shown.

[0046] A theory-numerical fusion-driven optimization method for composite cap-shaped stiffened panels is proposed. Its structural framework consists of: establishing a buckling finite element model of the composite cap-shaped stiffened panel; establishing a modified skin local buckling load engineering algorithm; analyzing the sensitivity of structural parameters of the composite cap-shaped stiffened panel; establishing a theory-numerical fusion-driven multi-objective optimization algorithm for the calculation width and skin bending stiffness correction coefficient; and establishing a multi-objective optimization algorithm for the composite cap-shaped stiffened panel structure based on a non-dominated sorting genetic algorithm.

[0047] This implementation method takes a composite material cap-shaped stiffened panel as an example. The total length of the experimental specimen is 890mm, the total width is 1155mm, the stiffener spacing is 155.14mm, the spacing between the two horizontal frames is 554mm, the glue-filling area on both sides is 60mm long, the 0° layup direction is the stiffener axis, the skin layup parameters are [45 / -45 / -45 / 90 / 45 / 0]s, and the stiffener layup parameters are all... The thickness of each layer is 0.185 mm, and the engineering constant of the composite material is: E 11 =162GPa, E 22 =9.14 GPa, G 12 = 4.57 GPa, v 12 =0.331, structural dimensions as follows Figure 2As shown, the implementation process of the optimization method for composite cap-shaped stiffened plates driven by theory-numerical fusion is presented:

[0048] The first step involves constructing a buckling finite element model based on the dimensions of the composite cap-shaped stiffened panel structure and the composite layup information. The buckling load is then calculated and compared with experimental data for correction. The finite element model is corrected until the buckling load error is ≤5%. The specific steps include:

[0049] ① In finite element software, a shell element model of the cap-shaped stiffened panel is created based on the actual component dimensions, such as... Figure 2 As shown;

[0050] ② Divide the grid according to different material areas, and set the composite material layup according to the actual layup sequence of the skin and the reinforcing ribs. The 0° direction of the layup is consistent with the axial direction of the reinforcing ribs.

[0051] ③ Establish a shared node mesh in the skin-stiffener connection area, and use Tie constraints for the stiffener-frame beam interface;

[0052] ④ Establish reference points in the extended regions of the loading and fixing ends, and couple the nodes in the end connection area (i.e., the potting area) to the reference points. The reference points control the fixing and loading of the boundary conditions, such as... Figure 3 As shown;

[0053] ⑤ The fixed end reference point constrains all degrees of freedom, the loading end reference point constrains all degrees of freedom except for the z-direction translation, and the two sides of the stiffened wall plate constrain the y-direction translation degree of freedom;

[0054] ⑥ In the finite element software, perform linear perturbation buckling analysis and apply axial compressive load at the loading end to obtain the first-order buckling load and modes, such as... Figure 4 As shown.

[0055] ⑦ The numerical simulation results are compared with the experimental results, with an error of 3.76%. The comparison results are shown in Table 1 below:

[0056] Table 1 Comparison of numerical simulation results and experimental results

[0057]

[0058] The second step involves calculating the local buckling load of the skin based on the boundary conditions of the composite cap-shaped stiffened panel skin and the buckling analysis method for composite laminates. Considering the reinforcement of the skin boundary conditions by the stiffeners, an engineering algorithm for adjusting the bending stiffness and calculation width of the skin buckling load is proposed. The specific steps include:

[0059] ① Material stiffness information is shown in Table 2:

[0060] Table 2. Calculation of material stiffness for composite stiffened plates:

[0061]

[0062] ② With the loading edge fixed and the side simply supported: α=2, the calculated buckling load is 35.64MPa, which is 37.89% different from the experimental value (57.38MPa).

[0063] ③Consider the reinforcement of skin bending stiffness by the lower flange stringers, and modify the skin bending stiffness accordingly: Bending stiffness coefficient d=(b ss -b s ) / b ss =0.23,

[0064] The corrected skin stiffness is shown in Table 3:

[0065] Table 3 Skin Correction Stiffness Coefficients

[0066]

[0067] A modified buckling load calculation method for composite stiffened panels is proposed: The error compared to the experimental value (57.38 MPa) was 5.2% (a reduction of 32.49%).

[0068] The third step involves using orthogonal experimental design to conduct sensitivity analysis on the structural buckling performance parameters, obtaining the contribution of each design variable to the objective function, and selecting variables with a contribution rate >5% as optimization parameters. The specific steps include:

[0069] ① The initial design variables are: X = (b s ,b f ,b w ,b c ,α,s,L,N,d x ,b ss ), Spacing b s Lower flange width b f Web width b w Hat top width b c Web inclination angle α, distance s between two transverse frames, total length L of stiffened panel, number of stiffeners N, skin stiffness correction factor d x And calculate the width b ss ;

[0070] ② Set the constraints as follows:

[0071] ③ An orthogonal experimental design method was used to establish a quadratic polynomial response surface model:

[0072] ④ Regularize the coefficients of the polynomial, fit the model using ridge regression, calculate the standardized regression coefficients, and select variables with an absolute contribution greater than 5% as key influencing factors. The sensitivity results are as follows: Figure 5 As shown in Table 5, the trends of the design variables under the objective function are summarized as follows:

[0073] Table 5. Value trends of design variables under each objective function.

[0074]

[0075] In the table, a design variable marked with "×" indicates that it has little effect on the objective function, "↑" indicates that the value should be increased, "↓" indicates that the value should be decreased, and "—" indicates that it cannot be determined.

[0076] ⑤ The number of design variables has been reduced from 8 to 5, namely the web width b w Hat top width b c The key variables, including the web inclination angle α, the total length L of the stiffened wall panel, and the number of stiffeners N, are input into step S4 for target optimization.

[0077] The fourth step involves constructing a dual-loop optimization architecture based on adaptive simulated annealing and non-dominated sorting genetic algorithm, with inner-loop correction coefficient optimization and outer-loop structure optimization. The inner-loop correction coefficient process uses the difference between the simulation results of S1 and the theoretical results of S2 as the optimization objective, and adjusts the skin width b calculated in the theoretical calculation. ss Multi-objective optimization is performed on the skin bending stiffness correction factor d to obtain the optimal solution for the calculated width and bending stiffness correction factor. The outer ring optimization structure is calculated based on the optimal solution output by S4 to calculate the width b. ss Based on the traditional post-buckling load engineering algorithm and a non-dominated sorting genetic algorithm, a multi-objective optimization algorithm for the structural parameters of composite cap-shaped stiffened panels is established, along with the skin bending stiffness correction factor d. This algorithm outputs the Pareto optimal solution set to obtain the structural optimization parameters. The algorithm structure is as follows: Figure 6 As shown, the specific steps include:

[0078] ① Using the adaptive simulated annealing algorithm, the difference between the simulation results of S1 and the theoretical results of S2 is used as the optimization objective. Multi-objective optimization is performed on the calculated skin width bss and the skin bending stiffness correction coefficient d in the theoretical calculation to obtain the optimal solutions for both.

[0079] ② The objective function for optimizing the inner loop correction coefficient is defined as: min ABS(diff)=|σ 数值解 -σ 理论解 The objective function for optimizing the outer ring structure is: The weight of the unit density mass function m(X) is set to 0.1, σcr (X) The weight of the buckling stress function is 0.001, σ co (X) The weight of the overall failure stress function is 0.001, and the scaling factor is 1 for all of them;

[0080] ③ Set the optimization range of the optimization variables as shown in Table 6:

[0081] Table 6 Optimization Range of Optimization Variables

[0082]

[0083] ④ Iterative execution outputs the Pareto optimal solution set. After 240 iterations, the calculation results tend to stabilize in the objective function region, such as... Figure 7 As shown in the figure, red iteration points represent infeasible solutions, i.e., solutions that do not meet the constraints; blue points represent feasible solutions; and green points represent optimal solutions.

[0084] The final optimized design variables and objective function results are shown in Table 7:

[0085] Table 7. Optimized design variables and objective function

[0086]

[0087] ⑤ The optimization results show that after optimization, the mass per unit density decreased by 41.03%, the buckling stress increased by 97.91%, and the failure stress increased by 80.93%.

[0088] Based on the above optimization results, the results were re-verified using the finite element method. The optimized model was then remodeled in the finite element software, and the load-displacement curves were output as follows: Figure 8 As shown, the stress value corresponding to the inflection point of first-order buckling is 127.54 MPa, and the final ultimate stress value is 369.15 MPa, with the error between the result and the optimization result within 10%.

[0089] This invention elucidates the principles and implementation methods of a helicopter fairing shape optimization method based on numerical simulation. The above description of the implementation scheme is intended to aid in understanding the method and core ideas of this invention. However, those skilled in the art will recognize that, based on the ideas of this invention, there will be variations in specific implementation methods and application scope. Therefore, the content of this specification should not be construed as limiting the invention.

[0090] The parts not covered in this invention are the same as or can be implemented using existing technologies.

Claims

1. A theory-numerical fusion-driven optimization method for composite material cap-shaped stiffened plates, characterized in that, Includes the following steps: S1: Establish a buckling finite element model of composite cap-shaped stiffened wall panel: Construct a buckling finite element model based on the dimensions of the composite cap-shaped stiffened wall panel structure and the composite material layup information, calculate the buckling load, and compare and correct it with the experiment. Correct the finite element model until the buckling load error is ≤5%. S2: Establish a modified engineering algorithm for skin local buckling load: Based on the boundary conditions of the composite cap-shaped stiffened panel skin, and using the composite laminate buckling analysis method, calculate the skin local buckling load. Considering the reinforcement of the skin boundary conditions by the stiffeners, a modified engineering algorithm for skin buckling load with modified bending stiffness and calculation width is proposed. S3: Sensitivity analysis of structural parameters of composite material cap-shaped stiffened wall panel: The orthogonal experimental design method was used to conduct sensitivity analysis on the buckling performance parameters of the structure, obtain the contribution of each design variable to the objective function, and select variables with a contribution rate >5% as optimization parameters; S4: Establish a multi-objective optimization algorithm for composite material cap-shaped stiffened plates driven by theory-numerical fusion: Based on the adaptive simulated annealing algorithm and the non-dominated sorting genetic algorithm, a dual-loop optimization architecture is built to optimize the inner loop correction coefficient and the outer loop structure, so as to obtain the structural optimization parameters.

2. The method for optimizing composite material cap-shaped stiffened plates driven by theory-numerical fusion according to claim 1 is characterized in that, The optimized structure is a dual-loop optimization architecture. The inner loop optimization is based on numerical simulation algorithm to find the correction coefficient in the theoretical algorithm of skin buckling load. The outer loop optimization is based on the optimal solution of buckling load correction coefficient and theoretical algorithm to optimize the structure of composite cap-shaped stiffened panel.

3. The method for optimizing composite material cap-shaped stiffened plates driven by theory-numerical fusion according to claim 1 is characterized in that, The process of establishing the buckling finite element model of the composite cap-shaped stiffened wall panel in step S1 specifically includes: (a) In finite element software, a shell element model of the cap-shaped stiffened wall panel is established based on the actual component dimensions; (b) Divide the grid according to different material regions, and set the composite material layup according to the actual layup sequence of the skin and the reinforcing ribs, with the 0° direction of the layup aligned with the axial direction of the reinforcing ribs; (c) Establish a shared node mesh in the skin-stiffener connection area, and use Tie constraints for the stiffener-frame beam interface; (d) Establish reference points in the extended regions of the loading end and the fixed end, and couple the nodes in the end connection area (i.e., the potting area) to the reference points, so that the reference points control the fixing and loading of the boundary conditions; (e) The fixed end reference point constrains all degrees of freedom, the loading end reference point constrains all degrees of freedom except for the z-direction translation, and the two sides of the stiffened wall plate constrain the y-direction translation degree of freedom. (f) In the finite element software, perform linear perturbation buckling analysis and apply axial compressive load at the loading end to obtain the first-order buckling load and mode.

4. The method for optimizing composite material cap-shaped stiffened plates driven by theory-numerical fusion according to claim 1 is characterized in that, The process of establishing the modified skin local buckling load engineering algorithm in step S2 specifically includes: (a) Calculation using traditional skin local buckling load engineering algorithm; (b) Considering the reinforcement of the skin bending stiffness by the lower flange stringers, the skin bending stiffness is modified accordingly: D R D is the skin-corrected bending stiffness matrix, and D is the uncorrected bending stiffness matrix of the skin. f Here is the bending stiffness matrix of the lower flange stringer, and the bending stiffness coefficient d = (b ss -b s ) / b ss ; (c) Considering the reinforcing effect of stiffeners on the skin boundary, a modified method for calculating buckling loads is proposed: N cr This represents the corrected buckling load for the cap-shaped stiffened panel. This represents the buckling load calculated for side-fixed supports. This represents the buckling load calculated using a simply supported side panel, where D is the ratio of the stiffened panel's bending stiffness to that of the skin.

5. The method for optimizing composite material cap-shaped stiffened plates driven by theory-numerical fusion according to claim 1 is characterized in that, The process of analyzing the sensitivity of structural parameters of composite material cap-shaped stiffened wall panels specifically includes: (a) Design variables include: stiffener spacing (b) s Lower flange width b f Web width b w Hat top width b c Web inclination angle α, distance s between two transverse frames, total length L of stiffened panel, number of stiffeners N, skin stiffness correction factor d x And calculate the width b ss ; (b) Set the constraints as follows: b s ≤b ss ≤b s +2b f The remaining design variables take values ​​within the range of ±20%; (c) An orthogonal experimental design method was used to establish a quadratic polynomial response surface model: Where: y A Let x be the objective function; i To design the sample points of the variables, β represents the main effect of each term in the multinomial model on the objective function, including the linear term effect β. i Quadratic effect β j and interaction effect β ij ; (d) The coefficients of the polynomial are regularized, and the standardized regression coefficients are calculated by fitting the model through ridge regression. Variables with an absolute contribution of >5% are selected as key influencing factors. (e) Input the key variables and response surface model into step S4 for target optimization.

6. The method for optimizing composite material cap-shaped stiffened plates driven by theory-numerical fusion according to claim 1 is characterized in that, The process of establishing a multi-objective optimization algorithm for composite material cap-shaped stiffened plates driven by theory-numerical fusion specifically includes: (a) Based on the adaptive simulated annealing algorithm, the difference between the simulation results of S1 and the theoretical results of S2 is used as the optimization objective to optimize the skin width b in the theoretical calculation. ss Multi-objective optimization is performed using the skin bending stiffness correction factor d to obtain the computational width b. ss The optimal solution for the skin bending stiffness correction factor d; (b) Using traditional post-buckling load engineering algorithms, a multi-objective optimization algorithm for the structural parameters of composite cap-shaped stiffened panels is established based on a non-dominated sorting genetic algorithm. (c) The objective function for optimizing the inner loop correction coefficient is defined as: min ABS(diff)=|σ 数值解 -σ 理论解 The objective function for optimizing the outer ring structure is: And set weighting factors and define the optimization range of the optimization variables; (d) Perform iterations, output the Pareto optimal solution set, and obtain the structural optimization parameters.

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