Self-adaptive topological optimization method of bat algorithm in meta-heuristic algorithm based on natural heuristic
Through the adaptive bat algorithm dynamically adjusting the design space and optimization parameters, the 'chessboard' phenomenon and local optimal problems in topological optimization are solved, and the effect of quickly finding the optimal topological structure in complex design space is achieved.
Patent Information
- Application Number
- CN202510525963.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-04-24
- Publication Date
- 2025-09-05
AI Technical Summary
In the existing topology optimization methods, there is unstable optimization results and prone to 'chessboard' phenomenon. In addition, traditional bat algorithms lack flexible adaptive mechanisms, making it difficult to quickly find the optimal topology structure in complex design spaces.
The bat algorithm with an adaptive mechanism is introduced to dynamically adjust the discretization accuracy and constraints of the design space, optimize the bat algorithm parameters, including adjusting the search step size and population size, avoiding local optimal solutions, and improving global search capabilities.
It realizes the rapid finding of the optimal topological structure in complex design space, avoids local optimal solutions, improves the stability and computing efficiency of topological optimization, and obtains smoother material distribution.
Smart Images
Figure CN120600174A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to a topology optimization algorithm, in particular to an adaptive topology optimization method based on a bat algorithm in a nature-inspired meta-heuristic algorithm, which is applied to topology optimization problems in engineering structure design. Background Art
[0002] Topology optimization is a method that determines material distribution within a given design domain and constraints. Its applications span a wide range of fields, including structural design, material layout, and lightweight design. However, among existing topology optimization methods, the Solid Isotropic Material Penalty (SIMP) method is the most commonly used. This method suffers from unstable optimization results and is prone to "checkerboard" artifacts when updating density.
[0003] The bat algorithm is an optimization method that simulates the echolocation behavior of bats. It has global search capabilities and strong local fine-grained search capabilities, effectively avoiding the local optimality problem found in traditional algorithms. However, the application of traditional bat algorithms in topology optimization remains relatively limited, and algorithm parameters (such as search step size and exploration accuracy) are usually fixed, lacking flexible adaptive mechanisms.
[0004] Based on this, a more ideal topology optimization method is needed. Summary of the Invention
[0005] Purpose of the invention: The purpose of the present invention is to provide an adaptive topology optimization method based on the bat algorithm in a nature-inspired metaheuristic algorithm, which can quickly find the optimal topology structure in a complex design space while avoiding falling into a local optimal solution, and is used to solve multi-objective and complex constrained structural optimization problems.
[0006] Technical solution: Adaptive topology optimization of the bat algorithm in the nature-inspired metaheuristic algorithm described in the present invention
[0007] The method comprises the following steps:
[0008] S 1: Initialize the design space of the topology optimization problem and define the design variable - density, objective function - flexibility and constraints;
[0009] S2: Apply the bat algorithm to iteratively optimize the design variables in the design space to obtain a preliminary topology optimization solution;
[0010] S3: By setting an adaptive mechanism, the discretization accuracy or constraint conditions of the design space are dynamically adjusted according to the current iteration results to optimize the topology structure;
[0011] S4: Update bat algorithm parameters based on adaptive mechanism;
[0012] S5: Determine the convergence criterion and terminate the optimization process.
[0013] Furthermore, in step S1, the optimized structure is discretized into grid cells, and the material distribution of each cell can be represented by density, with the density value ranging from [0, 1], where 0 represents that the cell has no material and 1 represents that the cell is completely filled with material;
[0014] Initialize the positions and velocities of a set of bat individuals, where each bat individual is represented by a density vector, and each element of the vector corresponds to the density of a cell in the finite element mesh. Assuming that the design space contains N cells, the representation of the bat individual is a vector X containing N elements. i =[x1, x2, ..., x N ], where X i is the density value of the i-th unit, and 0≤X i ≤1;
[0015] At the same time, a finite element model is established for the optimized structure, the grid units are divided, and the stiffness matrix is assembled to perform finite element analysis. Based on the finite element model, the mathematical model of the topology optimization problem of minimizing the structural flexibility under volume constraints is established as follows:
[0016]
[0017] Where C is the flexibility value, which is defined as the product of the external force and the structural deformation energy, and the unit is N·m.
[0018] Furthermore, in step S2, the speed and position of each individual bat are initialized according to the rules of the bat algorithm, and the flexibility of each individual bat is calculated as its fitness value. The fitness is calculated based on the objective function and the constraints, and is usually the weighted sum of the objective function value and the penalty function of the constraints. Based on the rules of the bat algorithm, the position and speed of the individual bat are updated. The speed update formula of the bat algorithm is as follows:
[0019]
[0020] in: is the flight speed of the kth generation bat, β1 is the factor that controls the flight direction, is the current position (density vector) of the kth generation bat, is the position of the global optimal solution, α is the factor that controls speed fluctuation, and rand(0, 1) is a random number. The position of the individual bat is updated according to the updated speed. The updated density vector is the most important design variable in the optimization process, representing the new material distribution. According to the fitness value, the flight speed and direction of the individual bat are adjusted to converge towards the global optimal solution.
[0021] Furthermore, each bat in the algorithm represents a topology optimization structure. For each topology optimization structure, it is first necessary to update the Young's modulus of each unit in the stiffness matrix according to the following formula, then reassemble the stiffness matrix and solve the displacement value. After obtaining the displacement value, the fitness fit is obtained according to (the calculation formula of C), thereby completing the entire optimization process of the algorithm.
[0022] E=E min +(E max -E min )ρ p
[0023] Among them, E max is the Young's modulus when the material is full, E min is the minimum Young's modulus, which is used to prevent the occurrence of singular stiffness matrix, ρ is the density value, and p is the penalty coefficient.
[0024] Furthermore, in the algorithm, the calculation of the fitness fit includes two parts: the flexibility value and the volume constraint, as described in the following formula: Where U0 is the displacement value obtained by finite element analysis when the initial density is given, and this term serves as a regularization term.
[0025]
[0026] Furthermore, in step S3, during each iteration, an adaptive mechanism is introduced to adjust the discretization accuracy or constraints of the design space based on the quality of the current optimization result. The specific adaptive strategy includes: adjusting the discretization accuracy of the design space according to the quality of the current optimization solution.
[0027] Furthermore, the adaptive strategy: when the optimization result tends to be stable, the discretization accuracy is reduced to reduce the amount of calculation; when the result changes greatly, the discretization accuracy is increased to ensure more refined optimization.
[0028] Furthermore, in step S4, the key parameters of the bat algorithm are updated according to the results of the adaptive adjustment to improve the efficiency and accuracy of the optimization. The specific implementation is as follows: By introducing an adjustment factor λ (k) , adjust the bat's flight speed according to the changes in the current optimization results. The specific update formula is:
[0029]
[0030] Among them, β2 is the guidance coefficient, which controls the direction strength of the individual approaching the global optimal solution when updating the speed, Δf (k) =f (k) -f (k-1) is the objective function change between the current optimization result and the previous generation, λ (k)is an adaptive factor that adjusts the random fluctuation factor α according to the change of the objective function to increase or decrease the randomness of the search:
[0031] α (k+1) =α (k) ·(1+λ (k) Δf (k) )
[0032] Furthermore, the search process is further optimized by dynamically adjusting the population size. When the objective function changes significantly, the population size can be increased to enhance the search capability; when the objective function changes slightly, the population size can be reduced to accelerate the convergence process:
[0033]
[0034] Among them, N (k) is the population size of the kth generation, μ (k) is the factor that controls the population size adjustment, Δf (k) is the change of the optimization result of the current generation, f max is the maximum value of the objective function, when Δf (k) When it is large, it means that the current optimization process has changed a lot, and the population size may need to be increased to better explore the search space. On the contrary, if Δf (k) If is smaller, the optimization is close to convergence and the population size can be reduced, thus reducing the sour cost and accelerating convergence.
[0035] Adjustments to the convergence criteria:
[0036] |f (k) -f (k-1) |<∈ dynamic ·(1+λ (k) Δf (k) )
[0037] Among them, ∈ dynamic is a dynamic threshold that decreases gradually as the optimization progresses, ensuring more fine-tuning of the solution as it approaches the optimal solution.
[0038] Furthermore, in step S5, when the change in the objective function is less than a preset threshold ∈, or when the maximum number of iterations is reached, the optimization process is considered to have converged, and the optimization process can be terminated and the final optimization result can be output. The specific convergence criteria are as follows:
[0039] |f(x (k+1) )-f(x (k) )|<∈
[0040] Or when the number of iterations k reaches the maximum number k max When , the optimization is stopped, and once the optimization process meets the convergence conditions, the final topology design results are output.
[0041] Furthermore, the adaptive mechanism includes adjusting the discretization accuracy or constraint conditions of the design space according to the topology optimization results, thereby optimizing local details of the topology structure.
[0042] Furthermore, the topology optimization problem is to minimize the weight or volume of the structure while satisfying predetermined stiffness or strength constraints.
[0043] Compared with the prior art, the method of the present invention has the following beneficial effects:
[0044] 1. This invention introduces an adaptive mechanism to dynamically adjust the key parameters of the bat algorithm based on the quality of the solution in the current optimization process. The advantage of this mechanism is that during the optimization process, the algorithm parameters can be automatically adjusted according to the quality of the current solution, avoiding ineffective exploration and focusing the search on the area with the highest probability of convergence. In addition, the dynamic adjustment of the adaptive step size and population size enables the algorithm to better perform global search and avoid prematurely falling into local optimal solutions, which is particularly important when the design space is complex.
[0045] 2. Traditional topology optimization methods (such as SIMP) are prone to the "checkerboard" phenomenon, whereby density values in certain areas converge to 0 or 1 during the optimization process, resulting in unstable topology. However, the Bat Algorithm introduces randomness and dynamic adjustments into the optimization process, effectively avoiding this phenomenon and achieving a smoother, more reasonable material distribution.
[0046] 3. This paper proposes a dynamic adjustment strategy based on the quality of the optimization results: the search step size is adjusted according to the magnitude of the objective function's changes, ensuring a detailed search when the solution is close to optimal, avoiding large-scale ineffective searches. When the optimization results fluctuate significantly, the population size is increased to enhance search power; when the results become more stable, the population size is reduced to accelerate convergence. This strategy adjusts the balance between exploration and exploitation at different optimization stages, improving computational efficiency and solution quality.
[0047] 4. The present invention proposes an adaptive topology optimization method based on the bat algorithm. By dynamically adjusting the optimization strategy and algorithm parameters, the density value of the design domain is updated, thereby improving the stability of the topology optimization and solving the "checkerboard" problem. BRIEF DESCRIPTION OF THE DRAWINGS
[0048] Figure 1 For the initial structure of the case, flexibility: 1.59;
[0049] Figure 2 The structure of the case after optimization by this method has a flexibility of 1.52.
[0050] Figure 3The stress conditions of the case. DETAILED DESCRIPTION
[0051] In order to make the purpose, technical solution and advantages of the present invention clearer, the technical solution of the present invention will be further described below.
[0052] Step 1: Initialize the design space of the topology optimization problem
[0053] The design space is represented as a finite element model, where the material distribution of each finite element is represented by a density value in the range [0, 1], where 0 indicates that the element has no material and 1 indicates that the element is completely filled with material. The size and shape of the design space are determined by the specific problem requirements. The density value of each element is represented as an N-dimensional vector, that is, the density value of each element is a variable. Assuming the design space contains N elements, the design variable is an N-dimensional vector:
[0054] X=[x1,x2,...,x N ]
[0055] where x i Represents the density value of the i-th unit, and satisfies 0≤x i ≤1.
[0056] The objective function is flexibility. You can choose to minimize the flexibility of the structure or maximize the stiffness. At the same time, certain structural strength or stiffness constraints need to be met to ensure that the optimization results meet the actual engineering requirements.
[0057] At the same time, a finite element model is established for the optimized structure, the grid units are divided, and the stiffness matrix is assembled to perform finite element analysis. Based on the finite element model, the mathematical model of the topology optimization problem of minimizing the structural flexibility under volume constraints is established as follows:
[0058]
[0059] Where C is the flexibility value, which is defined as the product of the external force and the structural deformation energy, and the unit is N·m
[0060] Step 2: Apply the bat algorithm for topology optimization
[0061] By randomly initializing the positions and velocities of a group of bat individuals, each bat individual represents a density vector
[0062] X i =[x1, x2, ..., x N ]
[0063] Each position corresponds to the material distribution of a unit in the design space. The speed and position of each individual bat are initialized according to the rules of the bat algorithm, usually selected within a random range. The flexibility of each individual bat is calculated as its fitness value. Fitness is calculated based on the objective function and constraints, usually as a weighted sum of the objective function value and the penalty function of the constraints. Based on the rules of the bat algorithm, the position and speed of the individual bat are updated. The speed update formula of the bat algorithm is as follows:
[0064]
[0065] in: is the flight speed of the kth generation bat, β1 is the factor that controls the flight direction, is the current position (density vector) of the kth generation bat, is the position of the global optimal solution, α is the factor that controls speed fluctuation, and rand(0, 1) is a random number.
[0066] Based on the updated speed, the bat's position is updated. The updated density vector is the most important design variable in the optimization process, representing the new material distribution. Based on the fitness value, the bat's flight speed and direction are adjusted to converge toward the global optimal solution.
[0067] Step 3: Adaptive mechanism adjusts the discretization accuracy or constraints of the design space
[0068] During each iteration, an adaptive mechanism is introduced to adjust the discretization accuracy or constraints of the design space based on the quality of the current optimization result. Specific adaptive strategies may include adjusting the discretization accuracy of the design space according to the quality of the current optimization solution. For example, when the optimization results tend to be stable, the discretization accuracy can be appropriately reduced to reduce the computational effort; when the results vary significantly, the discretization accuracy can be increased to ensure more refined optimization. Constraints can be appropriately relaxed or tightened based on the degree of satisfaction of the current design. For example, if certain constraints are tight, their restrictions can be relaxed to enhance optimization flexibility; if certain constraints are loose, their restrictions can be tightened to improve design quality.
[0069] Step 4: Update bat algorithm parameters based on adaptive mechanism:
[0070] According to the results of adaptive adjustment, some key parameters in the bat algorithm are updated to further improve the efficiency and accuracy of optimization. The specific implementation is as follows: By introducing an adjustment factor λ (k) , adjust the bat's flight speed according to the changes in the current optimization results. The specific update formula is:
[0071]
[0072] Among them, β2 is the guidance coefficient, which controls the direction strength of the individual approaching the global optimal solution when updating the speed, Δf (k) =f (k) -f (k-1) is the objective function change between the current optimization result and the previous generation, λ (k) is an adaptive factor. The random fluctuation factor α is adjusted according to the change of the objective function to increase or decrease the randomness of the search:
[0073] α (k+1) =α (k) ·(1+α (k) Δf (k) )
[0074] The search process is further optimized by dynamically adjusting the population size. When the objective function changes significantly, the population size can be increased to enhance the search capability; when the objective function changes slightly, the population size can be reduced to accelerate the convergence process:
[0075]
[0076] Among them, μ (k) is the factor that controls the population size adjustment, Δf (k) is the change of optimization result of the current generation, f max is the maximum value of the objective function.
[0077] Step 5: Determine the convergence criterion and terminate the optimization process
[0078] When the change in the objective function is less than the preset threshold ∈, or when the maximum number of iterations is reached, the optimization process is considered to have converged, and the optimization process can be terminated and the final optimization result can be output. The specific convergence criteria are as follows:
[0079] |f(x (k+1) )-f(x (k) )|<∈
[0080] Or when the number of iterations k reaches the maximum number k max When , the optimization stops. Once the optimization process meets the convergence criteria, the final topology design result is output. This result is usually a density distribution vector that represents the material distribution of each unit.
[0081] To facilitate those skilled in the art to understand the present invention, the present invention is further described below through a specific structural topology optimization example. In addition to the method of the present invention, the SIMP method is also used to solve the problem, thereby demonstrating the advantages of the method of the present invention.
[0082] Case Study: Adaptive Topology Optimization of Beam Structures
[0083] Beam structures are widely used in architecture and mechanical engineering. The optimization goal is often to reduce weight while ensuring load-bearing capacity and structural stiffness. Adaptive topology optimization methods based on the Bat Algorithm optimize material distribution and minimize flexibility while meeting design requirements.
[0084] Design domain dimensions: length L = 3 m, height H = 1 m.
[0085] Mesh division: Divide the design domain into 60×20=1200 finite elements, and the density of each element is ρ i Represents material distribution.
[0086] Material properties: Young's modulus E = 200 GPa, Poisson's ratio v = 0.3.
[0087] Constraints: Volume fraction is limited to 40%.
[0088] Objective function: Minimize structural flexibility C(ρ) = F T U, where F is the load vector and U is the displacement vector.
[0089] Initialization parameters:
[0090] Mesh the design domain: Discretize the beam structure into a finite element mesh and define the initial density value ρ of each element i ∈[0.1, 1].
[0091] Establish a finite element model: According to the structural boundary conditions (such as one end is fixed and the other end is subjected to concentrated force), assemble the stiffness matrix K and calculate the initial displacement vector U.
[0092] Set optimization parameters:
[0093] The number of bats N = 30, the initial speed v i =0
[0094] The search step range is β∈[0.5, 1.5], and the random perturbation factor α is 0.3.
[0095] Initialize the population: randomly generate N density distribution vectors X i =[ρ1,ρ2,...,ρ 1200 ]Each vector represents a solution.
[0096] Optimization results:
[0097] Initial design: The density of all units is 1 and the initial compliance value is C0 = 320 kN·m.
[0098] Optimized design: The final density distribution shows that the material is concentrated in the critical load-bearing areas of the beam, and the flexibility value is reduced to C * =160kN·m.
[0099] Comparative analysis:
[0100] Bat Algorithm: The flexibility value is reduced by 50%, the material distribution is smooth, and there is no "checkerboard" phenomenon.
[0101] SIMP method: The flexibility value is reduced by 45%, and there are some density jump areas.
[0102] The results show that the topology optimization method based on the bat algorithm achieves significantly better results than the SIMP method, showing good results in both flexibility optimization and material distribution. This shows that the topology method based on the bat algorithm has good applicability in solving bracket structure optimization problems.
[0103] The above description is merely a preferred embodiment of the present invention and does not limit the present invention in any way. Any person skilled in the art who, without departing from the scope of the present invention, makes any equivalent substitution, modification, or other changes to the technical solution and technical content disclosed in the present invention shall be deemed to be within the scope of the present invention and still fall within the scope of protection of the present invention.
Claims
1. An adaptive topology optimization method based on the bat algorithm in a nature-inspired metaheuristic algorithm, characterized in that: The following steps are involved: S1: Initialize the design space of the topology optimization problem and define the design variable - density, objective function - flexibility and constraints; S2: Apply the bat algorithm to iteratively optimize the design variables in the design space to obtain a preliminary topology optimization solution; S3: By setting an adaptive mechanism, the discretization accuracy or constraint conditions of the design space are dynamically adjusted according to the current iteration results to optimize the topology structure; S4: Update bat algorithm parameters based on adaptive mechanism; S5: Determine the convergence criterion and terminate the optimization process.
2. The adaptive topology optimization method based on the bat algorithm in the nature-inspired metaheuristic algorithm according to claim 1 is characterized in that: In step S1, the optimized structure is discretized into grid cells. The material distribution of each cell can be represented by density. The density value is in the range of [0, 1], where 0 represents that the cell has no material and 1 represents that the cell is completely filled with material. Initialize the positions and velocities of a set of bat individuals, where each bat individual is represented by a density vector, and each element of the vector corresponds to the density of a cell in the finite element mesh. Assuming that the design space contains N cells, the representation of the bat individual is a vector X containing N elements. i =[x1, x2, ..., x N ], where X i is the density value of the i-th unit, and 0≤X i ≤1; At the same time, a finite element model is established for the optimized structure, the grid units are divided, and the stiffness matrix is assembled to perform finite element analysis. Based on the finite element model, the mathematical model of the topology optimization problem of minimizing the structural flexibility under volume constraints is established as follows: Where C is the flexibility value, which is defined as the product of the external force and the structural deformation energy, and the unit is N·m.
3. The adaptive topology optimization method based on the bat algorithm in the nature-inspired metaheuristic algorithm according to claim 2, characterized in that: In step S2, the speed and position of each individual bat are initialized according to the rules of the bat algorithm, and the flexibility of each individual bat is calculated as its fitness value. The fitness is calculated based on the objective function and the constraints, and is usually the weighted sum of the objective function value and the penalty function of the constraints. Based on the rules of the bat algorithm, the position and speed of the individual bat are updated. The speed update formula of the bat algorithm is as follows: in: is the flight speed of the kth generation bat, β1 is the factor that controls the flight direction, is the current position (density vector) of the kth generation bat, is the position of the global optimal solution, α is the factor that controls speed fluctuation, and rand(0, 1) is a random number. The position of the individual bat is updated according to the updated speed. The updated density vector is the most important design variable in the optimization process, representing the new material distribution. According to the fitness value, the flight speed and direction of the individual bat are adjusted to converge towards the global optimal solution.
4. The adaptive topology optimization method based on the bat algorithm in the nature-inspired metaheuristic algorithm according to claim 3 is characterized in that: Each bat in the algorithm represents a topology optimization structure. For each topology optimization structure, it is first necessary to update the Young's modulus of each unit in the stiffness matrix according to the following formula, then reassemble the stiffness matrix and solve the displacement value. After obtaining the displacement value, the fitness fit is obtained according to (the calculation formula of C), thus completing the entire optimization process of the algorithm. E=E min +(And max -AND min )ρ p Among them, E max is the Young's modulus when the material is full, E min is the minimum Young's modulus, which is used to prevent the occurrence of singular stiffness matrix, ρ is the density value, and p is the penalty coefficient.
5. The adaptive topology optimization method based on the bat algorithm in the nature-inspired metaheuristic algorithm according to claim 4 is characterized in that: In the algorithm, the calculation of the fitness fit includes two parts: flexibility value and volume constraint, as described in the following formula, where U0 is the displacement value obtained by finite element analysis when the initial density is given, and this term serves as a regularization term.
6. The adaptive topology optimization method based on the bat algorithm in the nature-inspired metaheuristic algorithm according to claim 5, characterized in that: In step S3, during each iteration, an adaptive mechanism is introduced to adjust the discretization accuracy or constraint conditions of the design space based on the quality of the current optimization result. The adaptive strategy includes: adjusting the discretization accuracy of the design space according to the quality of the current optimization solution.
7. The adaptive topology optimization method based on the bat algorithm in the nature-inspired metaheuristic algorithm according to claim 6, characterized in that: The adaptive strategy is as follows: when the optimization result tends to be stable, the discretization accuracy is reduced to reduce the amount of calculation; when the result changes greatly, the discretization accuracy is increased to ensure more refined optimization.
8. The adaptive topology optimization method based on the bat algorithm according to claim 7, characterized in that: In step S4, the key parameters of the bat algorithm are updated according to the results of the adaptive adjustment to improve the efficiency and accuracy of the optimization. The specific implementation is as follows: By introducing an adjustment factor λ (k) , adjust the bat's flight speed according to the changes in the current optimization results. The specific update formula is: Among them, β2 is the guidance coefficient, which controls the direction strength of the individual approaching the global optimal solution when updating the speed, Δf (k) =f (k) -f (k-1) is the objective function change between the current optimization result and the previous generation, λ (k) is an adaptive factor that adjusts the random fluctuation factor α according to the change of the objective function to increase or decrease the randomness of the search: a (k+1) =a (k) ·(1+λ (k) ·Δf (k) ) The search process is further optimized by dynamically adjusting the population size. When the objective function changes significantly, the population size can be increased to enhance the search capability; when the objective function changes slightly, the population size can be reduced to accelerate the convergence process: Among them, N (k) is the population size of the kth generation, μ (k) is the factor that controls the population size adjustment, Δf (k) is the change of the optimization result of the current generation, f max is the maximum value of the objective function, when Δf (k) When it is large, it means that the current optimization process has changed a lot, and the population size may need to be increased to better explore the search space. On the contrary, if Δf (k) If is smaller, the optimization is close to convergence and the population size can be reduced, thus reducing the sour cost and accelerating convergence.
9. The adaptive topology optimization method based on the bat algorithm according to claim 8, characterized in that: Adjustments to the convergence criteria: |f (k) -f (k-1) |< ∈dynamic ·(1+λ (k) ·Δf (k) ) Among them, ∈ dvnamic is a dynamic threshold that decreases gradually as the optimization progresses, ensuring more fine-tuning of the solution as it approaches the optimal solution.
10. The adaptive topology optimization method based on the bat algorithm according to claim 9, characterized in that: In step S5, when the change in the objective function is less than a preset threshold ∈, or when the maximum number of iterations is reached, the optimization process is considered to have converged, and the optimization process can be terminated and the final optimization result can be output. The specific convergence criteria are as follows: |f(x (k+1) )-f(x (k) )|<∈ Or when the number of iterations k reaches the maximum number k max When , the optimization is stopped, and once the optimization process meets the convergence conditions, the final topology design results are output.