A topology optimization method and system for minimizing local relative displacement differences

Abstract

This invention discloses a topology optimization method and system for minimizing local relative displacement differences. The method includes the following steps: presetting the input parameters required for topology optimization; defining the design domain and non-design domain of the geometric model; performing finite element analysis on the geometric model; constructing an objective function for minimizing local relative displacement differences; solving the sensitivity of the objective function using the adjoint vector method; setting volume constraints and stress constraints for the geometric model, and calculating the element sensitivity corresponding to the stress constraints; performing optimization based on the moving asymptote method, updating the density variables of each element within the design domain; presetting convergence conditions; if the convergence conditions are not met, returning to finite element analysis for all elements; if the convergence conditions are met, outputting the final topology optimization result. This invention reduces the local relative displacement differences of the design object, achieving the minimization of local relative displacement differences in the structure, and obtaining a clear optimization result image.

Description

Technical Field

[0001] This invention relates to the field of topology optimization technology, and specifically to a topology optimization method and system for minimizing local relative displacement differences. Background Technology

[0002] Topology optimization refers to seeking a layout and arrangement within a design domain that achieves optimal results on one or more performance metrics, based on one or more constraints.

[0003] Traditional topology optimization often uses compliance as the objective function under a single constraint, which has limitations for many real-world engineering problems. To broaden the application of topology optimization, various constraints can be incorporated, such as volume constraints, stress constraints, and strain constraints, and different objective functions can be designed to meet different performance indices. Currently, existing design objectives in engineering mainly focus on minimizing compliance and volume, with less attention paid to minimizing displacement. Existing displacement-based design objectives only minimize the global average displacement and cannot satisfy the goal of optimizing local flatness in some engineering scenarios. For example, aircraft wings and engine blades need to maintain good aerodynamic shape under external loads, which places high demands on the uniformity of local deformation. Therefore, how to use minimizing displacement as the design objective and obtain the flatness of the equipment's local area under required operating conditions to meet design requirements remains a current technical challenge. Summary of the Invention

[0004] To overcome the defects and shortcomings of existing technologies, this invention provides a topology optimization method and system for minimizing local relative displacement differences. This invention adds volume constraints and stress constraints to solve the structural problems that cannot be optimized to minimize local relative displacement differences in existing engineering practice. It includes the objective function and its sensitivity, and the calculation methods of volume, stress constraints and their sensitivity. The design domain is updated using a moving progressive method. The resulting optimization image is clear, without obvious gray-scale units and meshed units, and has good manufacturability.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] This invention provides a topology optimization method for minimizing local relative displacement differences, comprising the following steps:

[0007] Preset the input parameters required for topology optimization;

[0008] Define the design domain and non-design domain of the geometric model;

[0009] The geometric model is solved using the finite element method.

[0010] Construct an objective function that minimizes the local relative displacement difference, whereby the objective function for minimizing the local relative displacement difference represents minimizing the relative difference between the displacement of each node on the target surface and the mean value of each updated node displacement;

[0011] The sensitivity of the objective function is solved using the adjoint vector method;

[0012] Set volume constraints and stress constraints for the geometric model. Based on the volume constraints, limit the weighted average density of each element to within the set volume constraints. Based on the stress constraints, set the stress of each element to be less than the allowable stress of the material. Calculate the element sensitivity corresponding to the stress constraints.

[0013] The optimization solution is based on the moving asymptote method, and the density variables of each element in the design domain are updated.

[0014] The system presets convergence conditions. If the convergence conditions are not met, it returns to solving the geometric model using finite element methods. It iterates through all elements to solve the finite element method. If the convergence conditions are met, it outputs the final topology optimization result.

[0015] As a preferred technical solution, the preset parameters required for topology optimization include geometric parameters, material parameters, load magnitude, constraint location, and filter radius.

[0016] As a preferred technical solution, the design domain and non-design domain of the set geometric model are defined, wherein the non-design domain is a custom area and the design domain is the remaining part excluding the non-design domain.

[0017] As a preferred technical solution, an objective function for minimizing the difference in local relative displacements is constructed, specifically expressed as:

[0018]

[0019] Where J represents the objective function of local minimum average relative displacement, j and s represent the numbers corresponding to each node, M represents the total number of nodes in the specified region, and u j u represents the displacement of the j-th node in the specified region. s This represents the displacement of the s-th node in the specified region.

[0020] As a preferred technical solution, the sensitivity of solving the objective function using the adjoint vector method is specifically expressed as follows:

[0021] The Lagrangian function for the local minimum average relative displacement minimization problem is:

[0022]

[0023] Among them, J *The objective function represents the local minimum average relative displacement, where j and s represent the node numbers, M represents the total number of nodes in the specified region, and u j u represents the displacement of the j-th node in the specified region. s λ represents the displacement of the s-th node in the specified region, λ represents the Lagrange multiplier, K represents the stiffness matrix, U represents the displacement matrix, and F represents the load vector.

[0024] Differentiating the Lagrangian function with respect to the element density, we obtain the solved and optimized expression as follows:

[0025]

[0026] The transpose of the Lagrange coefficients is obtained as follows:

[0027]

[0028] λ T Substituting back into the Lagrange function, we obtain the simplified expression for the derivative of the objective function's sensitivity:

[0029]

[0030] Where ρ represents the element density.

[0031] As a preferred technical solution, the stiffness matrix K is expressed as:

[0032]

[0033] The load vector F is represented as:

[0034] F = KU

[0035] Where B is the strain matrix and D is the material elasticity matrix.

[0036] As a preferred technical solution, the volume constraint is expressed as follows:

[0037] V(ρ)≤V0f

[0038] Where V(ρ) and V0 represent the material volume and the design domain volume, respectively, and f represents the optimized volume fraction.

[0039] As a preferred technical solution, the stress constraint is expressed as follows:

[0040]

[0041] in, Let [σ] represent the Von Mises stress of the i-th element, and [σ] represent the allowable stress of the material.

[0042] The element sensitivity corresponding to stress constraints is calculated by taking the partial derivative of its design variables with respect to density ρ, and is expressed as follows:

[0043]

[0044] Where, σ PN The stress after applying the P-Normal function, cp is the correction factor, σ i For stress components, This represents the derivative of the P-Normal function with respect to von Mises stress. This represents the derivative of von Mises stress with respect to the stress components. This represents the derivative of the stress component with respect to the design variable.

[0045] As a preferred technical solution, the final topology optimization results include topology, displacement distribution map and stress distribution map. The topology is drawn using the optimized element density, where an element density of 0 represents empty material and an element density of 1 represents solid material. The optimized topology is derived from the solid material, and the optimized topology is subjected to finite element analysis to obtain the corresponding displacement distribution map and stress distribution map.

[0046] This invention provides a topology optimization system for minimizing local relative displacement differences, comprising: an input parameter setting module, a design domain setting module, a finite element solution module, an objective function construction module, an objective function sensitivity solution module, a volume constraint setting module, a stress constraint setting module, an element sensitivity calculation module, an optimization update module, a convergence condition setting module, a convergence condition judgment module, and a topology optimization result output module;

[0047] The input parameter setting module is used to preset the input parameters required for topology optimization.

[0048] The design domain setting module is used to set the design domain and non-design domain of the geometric model;

[0049] The finite element solution module is used to perform finite element solutions on the geometric model;

[0050] The objective function construction module is used to construct an objective function that minimizes the local relative displacement difference. The objective function that minimizes the local relative displacement difference means that the relative difference between the displacement of each node on the target surface and the mean value of each updated node displacement is minimized.

[0051] The objective function sensitivity solution module is used to solve the sensitivity of the objective function using the adjoint vector method;

[0052] The volume constraint setting module is used to set the volume constraints of the geometric model, and based on the volume constraints, the weighted average of the density of each element is limited to within the set volume constraints.

[0053] The stress constraint setting module is used to set the stress constraints of the geometric model, and based on the stress constraints, the stress of each element is set to be less than the allowable stress of the material.

[0054] The unit sensitivity calculation module is used to calculate the unit sensitivity corresponding to stress constraints;

[0055] The optimization and update module is used to perform optimization solutions based on the moving asymptote method and update the density variables of each element in the design domain.

[0056] The convergence condition setting module is used to preset convergence conditions;

[0057] The convergence condition judgment module is used to determine whether the current iteration satisfies the convergence condition.

[0058] The topology optimization result output module is used to output the topology optimization results;

[0059] If the convergence condition judgment module determines that the convergence condition is not met, the finite element solution module performs finite element solution on the geometric model, traversing all elements for finite element solution. If the convergence condition judgment module determines that the convergence condition is met, the topology optimization result output module outputs the final topology optimization result.

[0060] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0061] (1) This invention adopts volume constraints and stress constraints, and uses the minimum local relative displacement difference as the objective function as the topology optimization technology solution. It solves the technical problem that there is no optimal method for local flatness in current engineering practice, and achieves the technical effect of reducing the local relative displacement difference of the design object and minimizing the local relative displacement difference of the structure.

[0062] (2) The present invention adopts the technical solution of updating the design domain by moving progressive method, which solves the technical problem of multiple constraints and achieves the technical effect of clear optimization result image without obvious gray scale unit and mesh unit, and good fabrication. Attached Figure Description

[0063] Figure 1 This is a flowchart illustrating the topology optimization method for minimizing local relative displacement differences according to the present invention.

[0064] Figure 2 This is a schematic diagram of the two-dimensional cantilever beam optimization model of the present invention;

[0065] Figure 3This is a schematic diagram of the three-dimensional cantilever beam optimization model of the present invention;

[0066] Figure 4 This is a schematic diagram of the optimized two-dimensional cantilever beam model of the present invention;

[0067] Figure 5 This is a schematic diagram of the optimized three-dimensional cantilever beam model of the present invention;

[0068] Figure 6 This is a displacement distribution diagram obtained from the optimized two-dimensional cantilever beam model of this invention.

[0069] Figure 7 This is a displacement distribution diagram obtained from the optimized three-dimensional cantilever beam model of this invention.

[0070] Figure 8 This is a stress distribution diagram obtained from the optimized two-dimensional cantilever beam model of the present invention.

[0071] Figure 9 This is a stress distribution diagram obtained from the optimized three-dimensional cantilever beam model of the present invention.

[0072] Figure 10 This is a schematic diagram showing the changes in relative displacement difference, stress value, and volume fraction when compliance is the objective function.

[0073] Figure 11 This is a schematic diagram showing the changes in relative displacement difference, stress value, and volume fraction when the objective function is to minimize the local relative displacement difference. Detailed Implementation

[0074] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0075] Example 1

[0076] like Figure 1 As shown, this embodiment provides a topology optimization method for minimizing local relative displacement differences. Based on the adjoint method sensitivity derivative framework, it includes sensitivity derivatives of the objective function and stress constraints. Based on the performance index of minimizing local relative displacement differences, a corresponding objective function expression is designed. Specifically, it is optimized and solved using the moving asymptotic method, and updated in each iteration to obtain the result after each iteration optimization.

[0077] Specifically, the following steps are included:

[0078] Step 1: Preset the basic parameters required for topology optimization;

[0079] In this embodiment, the basic parameters for topology optimization include geometric parameters, material parameters, load magnitude, constraint location, and filter radius.

[0080] Step 2: Define the design domain and non-design domain of the geometric model;

[0081] In step two, the design domain refers to the area that can be optimized and updated during the calculation, while the non-design domain refers to the area that cannot be updated. In this embodiment, the design domain is the remaining part excluding the upper edge of the two-dimensional or three-dimensional cantilever beam, and the non-design domain is the upper edge of the two-dimensional or three-dimensional cantilever beam.

[0082] Step 3: Solve the geometric model using the finite element method;

[0083] Step three involves finite element solution, which means performing finite element calculations using the finite element method.

[0084] Step 4: Construct an objective function that minimizes the difference in local relative displacement. In this embodiment, a two-dimensional or three-dimensional cantilever beam is used, and the objective function is to minimize the difference in relative displacement along the upper edge of the two-dimensional or three-dimensional cantilever beam.

[0085] In this embodiment, the objective function for minimizing local relative displacement differences refers to minimizing the relative difference between the displacement of each node and the average value of each updated node displacement on the designed target surface, thereby achieving the design objective of maximizing the flatness of the target surface. The specific objective function expression for the topology optimization design that satisfies the minimization of local relative displacement differences in step four is as follows:

[0086]

[0087] Where J represents the objective function of local minimum average relative displacement, j and s represent the numbers corresponding to each node, M represents the total number of nodes in the specified region, and u j This represents the displacement of the j-th node in the specified region, while u s This represents the displacement of the s-th node in the specified region.

[0088] Step 5: Solve for the sensitivity of the objective function using the adjoint vector method;

[0089] The specific steps in step five are as follows:

[0090] (1) Solve by using the method of adjoint variables to establish the expression of the Lagrangian function and establish the local minimum mean phase.

[0091] The Lagrangian function for the displacement minimization problem is:

[0092]

[0093] Where λ represents the Lagrange multiplier, which is a function of time, K represents the overall stiffness matrix of the element, U represents the overall displacement matrix, and F represents the load vector.

[0094] The expression for the stiffness matrix K is:

[0095]

[0096] Where B is the strain matrix and D is the material elasticity matrix.

[0097] The expressions for the displacement matrix U, the load vector F, and the stiffness matrix K are as follows:

[0098] KU = F

[0099] (2) Differentiating the Lagrangian function with respect to the element density ρ, we obtain the solved and optimized expression as follows:

[0100]

[0101] Where ρ represents the element density, u j u s K and U are all expressions related to ρ.

[0102] (3) The transpose λ of the Lagrange coefficients can be obtained from the above expression. T The expression is:

[0103]

[0104] (4) λ T Substituting back into the Lagrange function, we obtain the simplified expression for the derivative of the objective function's sensitivity:

[0105]

[0106] Step Six, as Figure 2 and Figure 3 As shown, two-dimensional and three-dimensional cantilever beam structural models before optimization are established, their volume constraints and stress constraints are set, and the corresponding element sensitivities are calculated.

[0107] The specific steps in step six are as follows:

[0108] (1) Calculating volume constraints means that the weighted average density of each element is limited to the set volume constraints. The specific expression is:

[0109] V(ρ)≤V0f

[0110] Where V(ρ) and V0 represent the material volume and the design domain volume, respectively, and f represents the optimized volume fraction.

[0111] (2) Calculate stress constraints, specifically including ensuring that the stress in each element is less than the allowable stress of the material. The expression for the constraint equation can be simplified to:

[0112]

[0113] Where σ i vm Let [σ] represent the Von Mises stress of the i-th element, and [σ] represent the allowable stress of the material.

[0114] (3) Calculate the sensitivity of the stress constraint by taking the partial derivative of the design variables with respect to ρ:

[0115]

[0116] Where, σ PN The stress after applying the P-Normal function, cp is the correction factor, σ i For the stress components, the above formula includes three sensitivity analyses. This represents the derivative of the P-Normal function with respect to von Mises stress. This represents the derivative of von Mises stress with respect to the stress components. This represents the derivative of the stress component with respect to the design variable.

[0117] Step 7: Update the design domain using the moving asymptote method, specifically by using the moving asymptote method to optimize and solve the design domain elements within the design domain.

[0118] The update method in step seven uses the moving asymptote method to update the density variables of each element in the design domain, realizes the solution of 0 or 1 for the density variables, and further realizes the removal and retention of materials to obtain the optimal topology.

[0119] Step 8: Determine if convergence has occurred. If not, skip to Step 3 and perform finite element analysis on the geometric model, iterating through all elements. If convergence has occurred, output the data.

[0120] In this embodiment, the convergence criterion can be set according to actual needs. One available convergence criterion is to determine the maximum number of iterations. When the number of iterations reaches the maximum number of iterations, convergence is achieved.

[0121] The data output in step eight includes the topology, displacement distribution map, and stress distribution map. The topology is plotted using the optimized element density; an element density of 0 represents empty material, and an element density of 1 represents solid material. The optimized topology is derived from the solid material, as shown below. Figures 6-9 As shown, finite element analysis of the optimized topology can yield the corresponding displacement and stress distribution diagrams.

[0122] like Figure 4 and Figure 5 As shown, the final optimization results obtained from the initial two-dimensional and three-dimensional configurations are illustrated. The black areas represent areas with material, i.e., areas with a unit density of 1, while the others represent areas without material, i.e., areas with a unit density of 0. The optimization results show virtually no grayscale or checkerboard patterns, indicating good manufacturability. Figure 6 and Figure 7 As shown, the two-dimensional and three-dimensional optimized structural displacement distribution diagrams are presented. It can be seen that the optimized structure of this invention achieves a smaller difference in relative displacement along the upper edge of the cantilever beam, meaning the upper edge of the cantilever beam has a higher flatness. Figure 10 and Figure 11 The figures show the iterative transformation trends of the relative displacement difference, maximum stress, and volume fraction along the upper edge of the beam in the two-dimensional case, with the traditional topology optimization using flexibility as the objective function and minimizing the local phase displacement difference as the objective function. It can be seen that the final optimization result obtained by this invention is smaller than that obtained by the traditional optimization using flexibility as the objective function. The optimization result can be used in engineering practice, such as the aerodynamic shape of sports cars and civil aircraft, and has good engineering application value.

[0123] Example 2

[0124] Except for the following technical contents, the remaining technical contents of this embodiment are the same as those of Embodiment 1;

[0125] This embodiment provides a topology optimization system for minimizing local relative displacement differences, including: an input parameter setting module, a design domain setting module, a finite element solution module, an objective function construction module, an objective function sensitivity solution module, a volume constraint setting module, a stress constraint setting module, an element sensitivity calculation module, an optimization update module, a convergence condition setting module, a convergence condition judgment module, and a topology optimization result output module;

[0126] In this embodiment, the input parameter setting module is used to preset the input parameters required for topology optimization;

[0127] In this embodiment, the design domain setting module is used to set the design domain and non-design domain of the geometric model;

[0128] In this embodiment, the finite element solution module is used to perform finite element solution on the geometric model;

[0129] In this embodiment, the objective function construction module is used to construct an objective function that minimizes the local relative displacement difference. The objective function that minimizes the local relative displacement difference means that the relative difference between the displacement of each node on the target surface and the mean value of each updated node displacement is minimized.

[0130] In this embodiment, the objective function sensitivity solution module is used to solve the sensitivity of the objective function using the adjoint vector method;

[0131] In this embodiment, the volume constraint setting module is used to set the volume constraints of the geometric model, and based on the volume constraints, the weighted average of the density of each element is limited to within the set volume constraints;

[0132] In this embodiment, the stress constraint setting module is used to set the stress constraints of the geometric model, and based on the stress constraints, the stress of each element is set to be less than the allowable stress of the material;

[0133] In this embodiment, the element sensitivity calculation module is used to calculate the element sensitivity corresponding to stress constraints;

[0134] In this embodiment, the optimization update module is used to perform optimization solutions based on the moving asymptote method and update the density variables of each element in the design domain;

[0135] In this embodiment, the convergence condition setting module is used to preset convergence conditions;

[0136] In this embodiment, the convergence condition judgment module is used to determine whether the current iteration satisfies the convergence condition;

[0137] In this embodiment, the topology optimization result output module is used to output the topology optimization results;

[0138] In this embodiment, if the convergence condition judgment module determines that the convergence condition is not met, the finite element solution module performs finite element solution on the geometric model, traversing all elements for finite element solution. If the convergence condition judgment module determines that the convergence condition is met, the topology optimization result output module outputs the final topology optimization result.

[0139] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A topology optimization method for minimizing local relative displacement differences, characterized in that, Includes the following steps: Preset the input parameters required for topology optimization; Define the design domain and non-design domain of the geometric model; The geometric model is solved using the finite element method. Construct an objective function that minimizes the local relative displacement difference, whereby the objective function for minimizing the local relative displacement difference represents minimizing the relative difference between the displacement of each node on the target surface and the mean value of each updated node displacement; The sensitivity of the objective function is solved using the adjoint vector method; Set volume constraints and stress constraints for the geometric model. Based on the volume constraints, limit the weighted average density of each element to within the set volume constraints. Based on the stress constraints, set the stress of each element to be less than the allowable stress of the material. Calculate the element sensitivity corresponding to the stress constraints. The optimization solution is based on the moving asymptote method, and the density variables of each element in the design domain are updated. The system presets convergence conditions. If the convergence conditions are not met, it returns to solving the geometric model using finite element methods. It iterates through all elements to solve the finite element method. If the convergence conditions are met, it outputs the final topology optimization result.

2. The topology optimization method for minimizing local relative displacement differences according to claim 1, characterized in that, The parameters required for the preset topology optimization include geometric parameters, material parameters, load magnitude, constraint location, and filter radius.

3. The topology optimization method for minimizing local relative displacement differences according to claim 1, characterized in that, The design domain and non-design domain of the defined geometric model are defined as follows: the non-design domain is a custom area, and the design domain is the remaining part excluding the non-design domain.

4. The topology optimization method for minimizing local relative displacement differences according to claim 1, characterized in that, The objective function for minimizing the difference in local relative displacements is constructed as follows: Where J represents the objective function of local minimum average relative displacement, j and s represent the numbers corresponding to each node, M represents the total number of nodes in the specified region, and u j u represents the displacement of the j-th node in the specified region. s This represents the displacement of the s-th node in the specified region.

5. The topology optimization method for minimizing local relative displacement differences according to claim 1, characterized in that, The sensitivity of solving the objective function using the adjoint vector method is specifically expressed as follows: The Lagrangian function for the local minimum average relative displacement minimization problem is: Among them, J * The objective function represents the local minimum average relative displacement, where j and s represent the node numbers, M represents the total number of nodes in the specified region, and u j u represents the displacement of the j-th node in the specified region. s λ represents the displacement of the s-th node in the specified region, λ represents the Lagrange multiplier, K represents the stiffness matrix, U represents the displacement matrix, and F represents the load vector. Differentiating the Lagrangian function with respect to the element density, we obtain the solved and optimized expression as follows: The transpose of the Lagrange coefficients is obtained as follows: λ T Substituting back into the Lagrange function, we obtain the simplified expression for the derivative of the objective function's sensitivity: Where ρ represents the element density.

6. The topology optimization method for minimizing local relative displacement differences according to claim 5, characterized in that, The stiffness matrix K is expressed as: The load vector F is represented as: F = KU Where B is the strain matrix and D is the material elasticity matrix.

7. The topology optimization method for minimizing local relative displacement differences according to claim 1, characterized in that, Volume constraints are expressed as: V(ρ)≤V0f Where V(ρ) and V0 represent the material volume and the design domain volume, respectively, and f represents the optimized volume fraction.

8. The topology optimization method for minimizing local relative displacement differences according to claim 1, characterized in that, Stress constraints are expressed as: in, Let [σ] represent the Von Mises stress of the i-th element, and [σ] represent the allowable stress of the material. The element sensitivity corresponding to stress constraints is calculated by taking the partial derivative of its design variables with respect to density ρ, and is expressed as follows: Where, σ PN The stress after applying the P-Normal function, cp is the correction factor, σ i For stress components, This represents the derivative of the P-Normal function with respect to von Mises stress. This represents the derivative of von Mises stress with respect to the stress components. This represents the derivative of the stress component with respect to the design variable.

9. The topology optimization method for minimizing local relative displacement differences according to claim 1, characterized in that, The final topology optimization results include the topology, displacement distribution map, and stress distribution map. The topology is plotted using the optimized element density, where an element density of 0 represents empty material and an element density of 1 represents solid material. The optimized topology is derived from the solid material, and the corresponding displacement distribution map and stress distribution map are obtained by performing finite element analysis on the optimized topology.

10. A topology optimization system for minimizing local relative displacement differences, characterized in that, include: The module includes: input parameter setting module, design domain setting module, finite element solution module, objective function construction module, objective function sensitivity solution module, volume constraint setting module, stress constraint setting module, element sensitivity calculation module, optimization update module, convergence condition setting module, convergence condition judgment module, and topology optimization result output module. The input parameter setting module is used to preset the input parameters required for topology optimization. The design domain setting module is used to set the design domain and non-design domain of the geometric model; The finite element solution module is used to perform finite element solutions on the geometric model; The objective function construction module is used to construct an objective function that minimizes the local relative displacement difference. The objective function that minimizes the local relative displacement difference means that the relative difference between the displacement of each node on the target surface and the mean value of each updated node displacement is minimized. The objective function sensitivity solution module is used to solve the sensitivity of the objective function using the adjoint vector method; The volume constraint setting module is used to set the volume constraints of the geometric model, and based on the volume constraints, the weighted average of the density of each element is limited to within the set volume constraints. The stress constraint setting module is used to set the stress constraints of the geometric model, and based on the stress constraints, the stress of each element is set to be less than the allowable stress of the material. The unit sensitivity calculation module is used to calculate the unit sensitivity corresponding to stress constraints; The optimization and update module is used to perform optimization solutions based on the moving asymptote method and update the density variables of each element in the design domain. The convergence condition setting module is used to preset convergence conditions; The convergence condition judgment module is used to determine whether the current iteration satisfies the convergence condition. The topology optimization result output module is used to output the topology optimization results; If the convergence condition judgment module determines that the convergence condition is not met, the finite element solution module performs finite element solution on the geometric model, traversing all elements for finite element solution. If the convergence condition judgment module determines that the convergence condition is met, the topology optimization result output module outputs the final topology optimization result.

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