Continuum structure topological optimization method and system based on variable density method theory

Abstract

The invention discloses a continuum structure topological optimization method and system based on a variable density method theory, and the method comprises the steps: carrying out the discretization of a continuum structure optimization region, and forming a plurality of middle density units; based on a variable density method theory, constructing a topological optimization model which takes the unit density of each intermediate density unit as a design variable, takes a preset volume fraction as a constraint condition and takes minimization of structural flexibility as a target; and adopting an interpolation model which punishes the unit density by an exponential function, and carrying out iterative optimization solution of eliminating gray units on the topological optimization model to obtain a topological optimization scheme. The method is higher in convergence speed and higher in capability of searching the optimal distribution of the material.

Description

Technical Field

[0001] The present invention relates to the field of structural optimization design, and in particular to a continuum structure topology optimization method and system based on variable density method theory. Background Art

[0002] With the increasing demand for modern engineering design, structural optimization design has become a core technology for improving material efficiency and reducing costs. In the field of structural optimization design, it mainly includes size optimization, shape optimization and topology optimization. Among them, topology optimization refers to finding the optimal distribution of structural materials in a specified optimization area through optimization algorithms under the premise that the relevant parameters of the structural materials are known and the boundary conditions and load conditions meet the given requirements.

[0003] The research field of topology optimization is mainly divided into continuum topology optimization and discrete structure topology optimization. Among them, continuum topology optimization is to discretize the material of the optimization space into a finite number of units (shell units or volume units), and then determine the retention of the units in the design space according to the algorithm. The retained units constitute the final topological structure, thereby achieving topological optimization. The main continuum topology optimization methods include homogenization method, variable density method, progressive structural optimization method, level set method and independent mapping method.

[0004] Among them, the variable density method, as a mainstream method in the field of topology optimization, is widely favored because of its advantages such as simple optimization process and easy implementation. This method is essentially a combinatorial optimization problem of 0-1 discrete variables. In order to effectively solve the combinatorial explosion problem caused by discrete design variables, the SIMP model in the variable density method uses a power function to construct the corresponding relationship between unit density and material elastic modulus, and introduces a penalty factor to force the unit density to approach 0 or 1. In addition, the traditional variable density method often uses the SIMP model combined with the Sigmund sensitivity filtering method to eliminate the checkerboard and grid dependency phenomena, and uses the OC method as a numerical solution algorithm to update the design variables.

[0005] However, the traditional variable density method does have obvious limitations in topology optimization. First, the SIMP model can effectively force the unit density to approach 0 or 1 by penalizing the unit density, but it still has the problems of slow convergence speed and insufficient ability to find the optimal material distribution. Secondly, as a numerical solution algorithm, the OC method can effectively update the design variables, but in practical applications it is often difficult to completely eliminate the grayscale units, resulting in blurred boundaries of the optimized structure. Summary of the invention

[0006] In order to solve the problems existing in the prior art, the present invention proposes a continuum structure topology optimization method and system based on the variable density method theory.

[0007] The technical solution of the present invention is as follows:

[0008] A continuum structure topology optimization method based on variable density method theory, comprising:

[0009] Discretize the optimization area of ​​the continuum structure to form several intermediate density units; based on the variable density method theory, construct a topology optimization model with the unit density of each intermediate density unit as the design variable, the preset volume fraction as the constraint condition, and the goal of minimizing the structural flexibility;

[0010] An interpolation model that penalizes unit density with an exponential function is used to iteratively optimize and solve the topology optimization model to eliminate gray units, thereby obtaining a topology optimization solution.

[0011] Furthermore, the expression of the topology optimization model is:

[0012]

[0013] Where x = [x1, x2, ..., x n ] T is a vector composed of unit density, representing the design variable, whose value can change continuously in [0,1];

[0014] c (x) is the structural flexibility, representing the objective function; U is the structural displacement matrix, F is the load vector, representing the boundary condition; K is the structural stiffness matrix; is the elastic modulus of the intermediate density unit e after interpolation; N is the total number of intermediate density units; u e is the unit displacement matrix; k e is the unit stiffness matrix; k0 is the entity unit stiffness matrix;

[0015] V0 is the initial structure volume; V (x) is the optimized structural volume; v e is the unit volume; f is the preset volume fraction, representing the constraint condition; x min is the minimum value of the intermediate density;

[0016] Furthermore, the expression of the interpolation model that penalizes the cell density with an exponential function is:

[0017]

[0018] In the formula, E min is the elastic modulus of the void material, E0 is the elastic modulus of the solid material; p is the penalty factor.

[0019] Furthermore, the interpolation model that uses an exponential function to penalize the unit density is used to iteratively optimize and solve the topology optimization model to eliminate gray units. The specific steps of obtaining the topology optimization solution include:

[0020] With the initial values ​​of the design variables, the stiffness matrix of each intermediate density unit is calculated, and the structural stiffness matrix is ​​constructed accordingly;

[0021] Under given boundary conditions, combined with the structural stiffness matrix, finite element analysis is performed on the continuum structure to calculate the structural displacement matrix;

[0022] Based on the stiffness matrix and the unit displacement matrix of each intermediate density unit, the interpolation model is used to calculate and accumulate the flexibility of each intermediate density unit to obtain the objective function of the topology optimization model;

[0023] Calculate the sensitivity of constraints and objective functions, and perform sensitivity filtering to eliminate checkerboard and grid dependency phenomena;

[0024] According to the result of sensitivity filtering, the design variables for grayscale suppression are updated;

[0025] An iteration is performed to obtain the sensitivity of a new objective function from the updated design variables and to perform new filtering, and the iteration is terminated when a convergence condition is satisfied.

[0026] Furthermore, the specific steps of performing finite element analysis on the continuum structure under given boundary conditions and combining the structural stiffness matrix to calculate the structural displacement matrix include: determining the boundary conditions applied to the continuum structure, and calculating the structural displacement matrix based on the structural stiffness matrix and the boundary conditions.

[0027] Furthermore, the specific method of obtaining the objective function of the topology optimization model by using the interpolation model to calculate and accumulate the flexibility of each intermediate density unit based on the stiffness matrix and the unit displacement matrix of each intermediate density unit includes:

[0028] Initialize the objective function and calculate the initial value of the objective function according to the initial structural design and material distribution;

[0029] Then, based on the stiffness matrix and the unit displacement matrix of each intermediate density unit, the interpolation model is adopted to calculate the flexibility of each unit respectively and accumulate it into the objective function.

[0030] Furthermore, the specific steps of calculating the sensitivity of the constraint condition and the objective function and performing sensitivity filtering for eliminating the checkerboard and grid dependency phenomena include:

[0031] The sensitivity calculation of the objective function and constraints is performed as follows:

[0032]

[0033] in, is the sensitivity of the objective function; is the sensitivity of the constraint;

[0034] Sigmund sensitivity filtering method is used for filtering, and the specific steps include:

[0035] Determine the filter radius;

[0036] Define the filtering area. For each intermediate density unit, draw a circle with its center as the center and the filtering radius as the radius. The intermediate density unit is defined as the central unit, and the area within this circle is defined as the filtering area. The units in the filtering area except the central unit are nearby units.

[0037] Calculate the sensitivity after filtering. In the filtering area, perform weighted average on the sensitivity of the central unit. The weight of the weighted average is the geometric center distance between the central unit and the nearby units. Replace the original sensitivity value with the calculated weighted average sensitivity value.

[0038] Furthermore, the specific steps of updating the design variables for grayscale suppression include:

[0039]

[0040] in, represents the cell density at the kth iteration, represents the cell density of the k+1th iteration; m is the positive movement limit, 0<m<1; η is the numerical damping coefficient; is the iteration coefficient,

[0041] q is the grayscale suppression operator, and its mathematical expression is:

[0042]

[0043] Where I is the number of iterations; α q is the step length of q;

[0044] Furthermore, the convergence conditions include stability of design variables, satisfaction of constraint conditions and convergence of objective function.

[0045] A continuum structure topology optimization system based on variable density method theory, including a model building module and a model solving module

[0046] The model building module is used to discretize the optimization area of ​​the continuum structure to form a plurality of intermediate density units; based on the variable density method theory, a topology optimization model is constructed with the unit density of each intermediate density unit as a design variable, the set volume fraction as a constraint condition, and the goal of minimizing structural flexibility;

[0047] The model solving module is used to adopt an interpolation model that penalizes unit density with an exponential function to perform iterative optimization and solve the topology optimization model to eliminate gray units, so as to obtain a topology optimization solution.

[0048] Compared with the prior art, the present invention has the following beneficial effects:

[0049] The present invention proposes a continuum structure topology optimization method and system based on the variable density method theory. The method adopts an interpolation model that penalizes the unit density with an exponential function. Compared with the traditional SIMP model, the convergence speed is faster. By accelerating the convergence speed, the present invention significantly shortens the optimization time, greatly improves the design efficiency, and effectively realizes resource conservation, thereby accelerating the product development cycle and reducing the computing cost.

[0050] The method of the present invention designs the interpolation model to ensure that E(x e ) It is always greater than 0, which improves the optimization utilization of the unit and avoids the failure of some units due to Compared with the SIMP model, the interpolation model has greater optimization potential and a stronger ability to find the best distribution of materials.

[0051] The method of the present invention uses the OC method after the integrated grayscale suppression operator to update the design variables, completely eliminating the grayscale unit, making the material distribution clear and continuous, and more in line with the requirements of actual manufacturing. Therefore, the design makes the obtained optimized structure have a higher design reference value. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is a flow chart of the continuum structure topology optimization method based on the variable density method theory;

[0053] Figure 2 It is the penalty curve diagram of SIMP model and improved SIMP model;

[0054] Figure 3 The sensitivity change curves of SIMP model and improved SIMP model: 3(a) is SIMP model, 3(b) is improved SIMP model;

[0055] Figure 4 This is the numerical case diagram of MBB beam;

[0056] Figure 5 It is the optimized structure solved by the traditional variable density method;

[0057] Figure 6 Iterative curve of structural flexibility (objective function) solved by traditional variable density method;

[0058] Figure 7 The optimized structure solved by the method of the present invention;

[0059] Figure 8 It is an iterative curve for solving the structural flexibility (objective function) of the method of the present invention. DETAILED DESCRIPTION

[0060] The present invention is described in detail below in conjunction with the accompanying drawings and examples. It should be understood that the specific examples described herein are only used to explain the present invention and are not used to limit the present invention.

[0061] Embodiment 1:

[0062] A continuum structure topology optimization method based on variable density method theory of the present invention is as follows: Figure 1 As shown, the following steps are included:

[0063] S1. Discretize the optimization area of ​​the continuum structure to form several intermediate density units; based on the variable density method theory, construct a topology optimization model with the unit density of each intermediate density unit as the design variable, the preset volume fraction as the constraint condition, and the minimization of structural flexibility as the objective function;

[0064] S2. An interpolation model that penalizes unit density with an exponential function is used to iteratively optimize the topology optimization model to eliminate gray units and obtain a topology optimization solution.

[0065] Furthermore, the expression of the topology optimization model is:

[0066]

[0067] Where x = [x1, x2, ..., x n ] T is a vector composed of unit density, representing the design variable, whose value can change continuously in [0,1]; min is the minimum value of the intermediate density. To avoid singular matrices in calculations, x min Usually set to 0.01;

[0068] c (x) is the structural flexibility, representing the objective function; U is the structural displacement matrix, F is the load vector, representing the boundary condition; K is the structural stiffness matrix; is the elastic modulus after interpolation of the intermediate density unit e, calculated by the interpolation model; N is the total number of intermediate density units; u e is the unit displacement matrix; k e is the unit stiffness matrix. According to the entity unit (unit density value is 1) stiffness matrix k0 and Calculated; k0 is the stiffness matrix of the solid unit (intermediate density value is 1);

[0069] V0 is the initial structure volume; V (x) is the optimized structural volume; v e is the unit volume; f is the preset volume fraction, representing the constraint condition. If f is set to 0.5, it means V (x) 0.5 times of V0;

[0070] Furthermore, the expression of the interpolation model that penalizes the cell density with an exponential function is:

[0071]

[0072] After simplification:

[0073]

[0074] In the formula, E min is the elastic modulus of the void material, E0 is the elastic modulus of the solid material; p is the penalty factor. Construct an exponential function Alternative power function Penalizes cell density and adds an exist Multiply the previous item The purpose is to adjust the slope of the penalty curve and control the values ​​of the penalty curve at both end points.

[0075] Figure 2 The penalty curves of the SIMP model (traditional model) and the improved SIMP model (interpolation model designed by the present invention) are shown in FIG. Figure 2 It can be clearly seen that:

[0076] When p = 1, the penalty curves of both interpolation models are approximately straight lines, which means that the penalty on unit density is negligible.

[0077] When p>1, the penalty factor comes into play.

[0078] In the range of 2≤p≤5, the penalty curve of the improved SIMP model shows a steeper characteristic. This phenomenon shows that the improved SIMP model can make the unit density approach 0 or 1 more quickly during the iteration process, thereby accelerating the convergence speed.

[0079] In order to further explore the difference in penalty performance between the two interpolation models, the elastic modulus Sensitivity analysis was performed.

[0080] SIMP model The sensitivity expression is:

[0081]

[0082] Improved SIMP model The sensitivity expression is:

[0083]

[0084] Figure 3 The sensitivity change curves of SIMP model and improved SIMP model are shown in Figure 3(a) for SIMP model and Figure 3(b) for improved SIMP model. It can be clearly seen that:

[0085] In the SIMP model, as p increases, When it approaches 0, the corresponding unit density value range will also expand. For example, when p = 3, When p approaches 0, the corresponding cell density value range is 0 to 0.1; when p = 5, this range is expanded to 0 to 0.3. It is close to 0, which means that these units are deleted soon after the iteration starts, thus limiting the optimization potential and not conducive to searching for the best distribution of materials.

[0086] Compared with the SIMP model, the improved SIMP model can ensure is always greater than 0, thus effectively avoiding the This improvement improves the optimization utilization of density units to a certain extent and effectively improves the ability to search for the best distribution of materials.

[0087] Furthermore, an interpolation model that penalizes the unit density with an exponential function is used to iteratively optimize and solve the topology optimization model to eliminate gray units. The specific steps for obtaining the topology optimization solution include:

[0088] With the initial values ​​of the design variables, according to Calculate the stiffness matrix k for each intermediate density element with k0 e , and construct the structural stiffness matrix K;

[0089] Under the given boundary condition F, combined with the structural stiffness matrix K, the continuum structure is subjected to finite element analysis to calculate the structural displacement matrix U (including the unit displacement matrix u e );

[0090] Based on the stiffness matrix k of each intermediate density element e , unit displacement matrix u e , using the interpolation model, calculate the flexibility of each intermediate density unit and accumulate it to obtain the objective function c of the topology optimization model (x) ;

[0091] Calculate the sensitivity of constraints and objective functions, and perform sensitivity filtering to eliminate checkerboard and grid dependency phenomena;

[0092] According to the result of sensitivity filtering, the design variables for grayscale suppression are updated;

[0093] The sensitivity of the new objective function is obtained from the updated design variables, and the new filtering is iterated, and the iteration is terminated when the convergence condition is met. Specifically, it is determined whether the convergence condition is met; if it is met, the iteration is terminated and the optimized structure is obtained; if it is not met, the stiffness matrix is ​​reconstructed according to the new design variables, finite element analysis and sensitivity filtering are performed, and the design variables are updated again, and the cycle is repeated until the convergence condition is met.

[0094] Furthermore, under given boundary conditions, in combination with the structural stiffness matrix, the continuum structure is subjected to finite element analysis, and the specific steps of calculating the structural displacement matrix include: determining the boundary condition F applied to the continuum structure, and calculating the structural displacement matrix U according to the structural stiffness matrix K and the boundary condition F.

[0095] Furthermore, based on the stiffness matrix k of each intermediate density unit e , unit displacement matrix u e , using the interpolation model, calculating the flexibility of each intermediate density unit and accumulating it, the specific method of obtaining the objective function of the topology optimization model includes:

[0096] Initialize the objective function and calculate the initial value of the objective function according to the initial structural design and material distribution;

[0097] Then based on the stiffness matrix (k e ), unit displacement matrix (u e ), the interpolation model is used to calculate the flexibility of each unit and add it to the objective function.

[0098] Furthermore, the specific steps of calculating the sensitivity of the constraint conditions and the objective function and performing sensitivity filtering for eliminating the checkerboard and grid dependency phenomena include:

[0099] The sensitivity calculation of the objective function and constraints is performed as follows:

[0100]

[0101] in, is the sensitivity of the objective function (structural flexibility); is the sensitivity of the constraint (volume constraint);

[0102] Sigmund sensitivity filtering method is used for filtering, and the specific steps include:

[0103] Determine the filter radius. The filter radius should not be too large or too small. It should be determined according to the specific requirements of the optimization problem and the division of the grid.

[0104] Define the filtering area. For each intermediate density unit, draw a circle with its center as the center and the filtering radius as the radius. The intermediate density unit is defined as the central unit, and the area within this circle is defined as the filtering area. The units in the filtering area except the central unit are nearby units.

[0105] Calculate the sensitivity after filtering. In the filtering area, perform weighted average on the sensitivity of the central unit. The weight of the weighted average is the geometric center distance between the central unit and the nearby units. Use the calculated weighted average sensitivity value to replace the original sensitivity value for subsequent numerical solution.

[0106] Furthermore, the specific steps of updating the design variables by the original OC method include:

[0107]

[0108] In the formula, represents the cell density at the kth iteration, represents the cell density of the k+1th iteration; m (0<m<1) is the positive movement limit, which is used to limit the x before and after the iteration e to avoid sudden changes; η is the numerical damping coefficient, which is used to ensure the stability of the iterative process;

[0109] is the iteration coefficient, which contains the sensitivity filtering information at the kth iteration. The mathematical expression is:

[0110]

[0111] The specific steps of updating the design variables for grayscale suppression in the present invention include:

[0112]

[0113] Among them, q is the grayscale suppression operator, and its mathematical expression is:

[0114]

[0115] Where I is the number of iterations; α qis the step size of q, usually 0.05. The upper limit of q is set to 5 to avoid non-convergence of optimization iterations.

[0116] In the iterative format of the OC method after integrating the grayscale suppression operator, the iteration factor changes from Transformed into During the first 20 iterations, q is always 1, which means that it has no effect at this stage. However, once the first 20 iterations are completed, the value of q is gradually increased, which will accelerate the iteration factor Polarization towards 0 or 1. In the topology optimization process, this change is manifested as the original gray cells will accelerate to transform into cells with density values ​​of 0 or 1 after the initial 20 iterations, thereby eliminating the gray cells.

[0117] Furthermore, the convergence conditions include the stability of design variables, the satisfaction of constraints, and the convergence of the objective function.

[0118] Design variable stability: As iterations proceed, the design variables should gradually become stable and no longer change significantly, which can be measured by comparing the difference between the design variables in adjacent iterations. Specifically, the difference between the design variables in adjacent iterations is less than or equal to the preset design variable stability threshold.

[0119] Constraints are satisfied: The constraints during the iteration must be satisfied. This means that after the iteration, the optimization result is within the predetermined range.

[0120] Objective function convergence: In the early stages of iteration, the objective function changes significantly, but as the iteration proceeds, the objective function should gradually stabilize, and the iteration ends when the preset convergence threshold is reached. This usually means that the change in the objective function in multiple consecutive iterations is less than the preset objective function stability threshold.

[0121] Embodiment 2:

[0122] The present invention provides a continuum structure topology optimization system based on the variable density method theory, including a model building module and a model solving module.

[0123] The model building module is used to discretize the optimization area of ​​the continuum structure to form several intermediate density units. Based on the variable density method theory, a topology optimization model is constructed with the unit density of each intermediate density unit as the design variable, the set volume fraction as the constraint condition, and the goal of minimizing the structural flexibility.

[0124] The model solving module is used to adopt an interpolation model that penalizes the unit density with an exponential function, perform iterative optimization and solve the topology optimization model to eliminate gray units, and obtain a topology optimization solution.

[0125] Application examples:

[0126] In this example, a continuum structure topology optimization method of the present invention is implemented on a continuum structure simulation, such as Figure 4 As shown, the continuum structure is a MBB (Messerschmitt– –Blohm) beam, MBB beam is one of the most classic examples in the topology optimization of continuum structures. It has a simple and symmetrical geometry and clear boundary conditions, and provides a reliable benchmark platform for studying the performance of different optimization algorithms and evaluating the lightweight design of structures. The optimization area of ​​the structure is discretized into 300×50, with both ends fixed and the top midpoint subjected to a vertical downward concentrated force F=1. The material elastic modulus E=1, Poisson's ratio ν=0.3, the preset volume fraction f is 0.5, the filter radius is set to 0.04 times the length of the structure, and the penalty factor p=3. Considering the symmetry of the model, half of the structure of the model can be used for optimization calculation, and the objective function is to minimize the structural flexibility.

[0127] As a comparison, this example also uses the traditional variable density method to perform topology optimization of the continuum structure. Figure 5 This is the optimized structure solved by the traditional variable density method. The white area in the figure represents the unit with a density of 0, and the black area represents the unit with a density of 1. Figure 5 It can also be found that there are obvious gray units at the boundaries of the optimized structure, which are intermediate density units. However, in the actual manufacturing process, such intermediate density units are not allowed. Therefore, it is necessary to take measures to eliminate these gray units to ensure the consistency and manufacturability of the structure.

[0128] Figure 6 Iterative curve of structural flexibility (objective function) solved by traditional variable density method. Figure 6 It can be found that in the initial stage, the structural flexibility dropped sharply. After 20 iterations, the change in structural flexibility began to stabilize, showing that the difference in structural flexibility before and after iteration was small. This stable state lasted until 95 iterations, when the curve finally converged, indicating that the optimization process reached a stable state. Overall, the iteration process showed a slow convergence speed, suggesting that there may be room for improvement in the optimization algorithm or parameter setting.

[0129] Figure 7 This is the optimized structure solved by the method of the present invention. Figure 7 It can also be found that the optimized structure is clearly black and white, in a completely discrete state, and does not have any grayscale units, and has strong manufacturability.

[0130] Figure 8 The iterative curve of the structural flexibility (objective function) is solved for the present invention. Figure 8It can be found that in the initial stage, the structural flexibility also drops sharply. However, after 20 iterations, the grayscale suppression operator q begins to play a role, prompting the grayscale units to be quickly converted into units with density values ​​of 0 or 1. Comparing the optimized structures of the 20th iteration and the 30th iteration, it can be found that at the 20th iteration, there are a large number of grayscale units in the optimized structure, and no obvious outline has been formed. At the 30th iteration, the optimized structure is already very clear. By the 40th iteration, there are almost no grayscale units in the optimized structure, which lasts until the 52nd iteration, and the curve finally converges. Overall, the topology optimization method under the present invention not only has a faster convergence speed and better penalty efficiency, but also can completely eliminate grayscale units to obtain an ideal optimized structure, showing its significant advantages in the field of continuum structure optimization.

Claims

1. A continuum structure topology optimization method based on variable density method theory, characterized in that: include: Discretize the optimized region of the continuum structure to form several intermediate density units; Based on the theory of variable density method, a topology optimization model is constructed with the unit density of each intermediate density unit as the design variable, the preset volume fraction as the constraint condition, and the goal of minimizing structural flexibility; An interpolation model that penalizes unit density with an exponential function is used to iteratively optimize and solve the topology optimization model to eliminate gray units, thereby obtaining a topology optimization solution.

2. The method for topology optimization of continuum structure based on variable density method theory according to claim 1, characterized in that: The expression of the topology optimization model is: Where x = [x1, x2, ..., x n ] T It is a vector composed of unit density, representing the design variable, and its value can change continuously in [0, 1]; c (x) is the structural flexibility, representing the objective function; U is the structural displacement matrix, F is the load vector, representing the boundary condition; K is the structural stiffness matrix; is the elastic modulus of the intermediate density unit e after interpolation; N is the total number of intermediate density units; u e is the unit displacement matrix; k e is the unit stiffness matrix; k0 is the entity unit stiffness matrix; V0 is the initial structure volume; V (x) is the optimized structural volume; v e is the unit volume; f is the preset volume fraction, representing the constraint condition; x min is the minimum value of the intermediate density.

3. The method for topology optimization of continuum structure based on variable density method theory according to claim 2, characterized in that: The expression of the interpolation model that penalizes the cell density with an exponential function is: In the formula, E min is the elastic modulus of the void material, E0 is the elastic modulus of the solid material; p is the penalty factor.

4. The method for topology optimization of continuum structure based on variable density method theory according to claim 3, characterized in that: The specific steps of adopting the interpolation model that penalizes the unit density with an exponential function and performing iterative optimization and solving the topology optimization model to eliminate gray units and obtaining the topology optimization solution include: With the initial values ​​of the design variables, the stiffness matrix of each intermediate density unit is calculated, and the structural stiffness matrix is ​​constructed accordingly; Under given boundary conditions, combined with the structural stiffness matrix, finite element analysis is performed on the continuum structure to calculate the structural displacement matrix; Based on the stiffness matrix and the unit displacement matrix of each intermediate density unit, the interpolation model is used to calculate and accumulate the flexibility of each intermediate density unit to obtain the objective function of the topology optimization model; Calculate the sensitivity of constraints and objective functions, and perform sensitivity filtering to eliminate checkerboard and grid dependency phenomena; According to the result of sensitivity filtering, the design variables for grayscale suppression are updated; An iteration is performed to obtain the sensitivity of a new objective function from the updated design variables and to perform new filtering, and the iteration is terminated when a convergence condition is satisfied.

5. The method for topology optimization of continuum structure based on variable density method theory according to claim 4, characterized in that: The specific steps of performing finite element analysis on the continuum structure under given boundary conditions and combining the structural stiffness matrix to calculate the structural displacement matrix include: determining the boundary conditions applied to the continuum structure, and calculating the structural displacement matrix according to the structural stiffness matrix and the boundary conditions.

6. The method for topology optimization of continuum structure based on variable density method theory according to claim 5, characterized in that: The specific method of obtaining the objective function of the topology optimization model by using the interpolation model to calculate and accumulate the flexibility of each intermediate density unit based on the stiffness matrix and the unit displacement matrix of each intermediate density unit includes: Initialize the objective function and calculate the initial value of the objective function according to the initial structural design and material distribution; Then, based on the stiffness matrix and the unit displacement matrix of each intermediate density unit, the interpolation model is adopted to calculate the flexibility of each unit respectively and accumulate it into the objective function.

7. The method for topology optimization of continuum structure based on variable density method theory according to claim 6, characterized in that: The specific steps of calculating the sensitivity of the constraint condition and the objective function and performing sensitivity filtering for eliminating the checkerboard and grid dependency phenomena include: The sensitivity calculation of the objective function and constraints is performed as follows: in, is the sensitivity of the objective function; is the sensitivity of the constraint; Sigmund sensitivity filtering method is used for filtering, and the specific steps include: Determine the filter radius; Define the filtering area. For each intermediate density unit, draw a circle with its center as the center and the filtering radius as the radius. The intermediate density unit is defined as the central unit, and the area within this circle is defined as the filtering area. The units in the filtering area except the central unit are nearby units. Calculate the sensitivity after filtering. In the filtering area, perform weighted average on the sensitivity of the central unit. The weight of the weighted average is the geometric center distance between the central unit and the nearby units. Replace the original sensitivity value with the calculated weighted average sensitivity value.

8. The method for topology optimization of continuum structure based on variable density method theory according to claim 7, characterized in that: The specific steps of updating the design variables for grayscale suppression include: in, represents the cell density at the kth iteration, represents the cell density of the k+1th iteration; m is the positive movement limit, 0<m<1; η is the numerical damping coefficient; is the iteration coefficient, q is the grayscale suppression operator, and its mathematical expression is: Where I is the number of iterations; α q is the step size of q.

9. The method for topology optimization of continuum structure based on variable density method theory according to claim 8, characterized in that: The convergence conditions include that the design variables are stable, the constraints are satisfied, and the objective function converges.

10. A continuum structure topology optimization system based on variable density method theory, characterized in that: Includes model building module and model solving module The model building module is used to discretize the continuum structure optimization region to form a plurality of intermediate density units; Based on the theory of variable density method, a topology optimization model is constructed with the unit density of each intermediate density unit as the design variable, the preset volume fraction as the constraint condition, and the goal of minimizing structural flexibility; The model solving module is used to adopt an interpolation model that penalizes unit density with an exponential function to perform iterative optimization and solve the topology optimization model to eliminate gray units, so as to obtain a topology optimization solution.

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