Topological optimization method for applying structure and rounding minimum size control thereof

Abstract

The invention belongs to the technical field related to structure optimization design, and discloses a topological optimization method for applying structure and rounding minimum size control thereof, which comprises the following steps of: (1) taking pseudo density associated with a corresponding grid unit in a design domain as a design variable, and sequentially performing first density filtering and step projection on the design variable to obtain a physical variable field vector, calculating to obtain a density indication vector reflecting whether a solid-phase or empty-phase material exists or not; (2) updating relative volume fraction coefficients corresponding to the grid units, then sequentially carrying out secondary density filtering and secondary step projection to obtain a solid-phase or empty-phase minimum size satisfaction rate vector, and further calculating to obtain an additional unit density vector; and (3) constructing a solid phase or an empty phase and a rounding minimum size constraint function thereof, integrating the obtained minimum size constraint function and the sensitivity thereof into the topological optimization model, and carrying out iterative solution on the topological optimization model by adopting an MMA algorithm to obtain an optimal topological configuration. According to the invention, the control effect of the minimum size is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field related to structural optimization design, and more specifically, relates to a topology optimization method for applying minimum size control of a structure and its fillet. Background Art

[0002] Topology optimization aims to obtain novel configurations with optimal material distribution under specific goals and constraints. It has a high degree of design freedom and does not rely on human experience. It is a very promising structural optimization design technology and is widely used in aerospace, shipbuilding, vehicles, weapons and other fields. However, existing methods focus more on physical properties and lack consideration of manufacturing processes. As a result, small or thin-walled components and local hinges are prone to appear in topology optimization, which is not conducive to manufacturing and affects structural performance. At the same time, topology optimization generally cannot automatically fillet, which easily produces sharp corners, leading to stress concentration, accelerated fatigue failure, and does not meet processing and use requirements. Therefore, effectively controlling the structure and its minimum fillet size is of great significance to the structural optimization design of engineering applications.

[0003] In the field of structural topology optimization, previously developed topology optimization methods that consider the minimum size of structures or fillets are primarily categorized as implicit and explicit. Earlier, Guest et al. proposed an implicit minimum size control method for solid (solid) and void (hole) phases based on filtered projection. This method employed a circular (spherical) filter and the Heaviside projection function to achieve minimum size control for the solid (void) phase and its fillets. However, this method was prone to generating grayscale units at the interface between the solid and void phases, struggled to resolve local hinges in flexible mechanisms, and lacked strict minimum size control. Furthermore, this method required separate design variables for the void and solid phases, increasing the computational effort. Sigmund et al. proposed an implicit minimum size control method based on the robust "erosion, intermediate, and expansion" formulation. While this method offers good applicability and stability, it suffers from lax minimum size control, requiring three finite element analyses for each optimization iteration, significantly increasing the computational effort. Furthermore, this method cannot guarantee consistent topological configurations across multiple physical problems, limiting its applicability. Zhang Weisheng, Li Hao, and others introduced a skeleton-extraction-based minimum size control method for structures within SIMP or level set topology optimization methods. This method effectively controls the minimum size of the solid phase, but struggles to simultaneously address the minimum size control of solid-phase fillets, the void phase, and its fillets. Furthermore, this method exhibits slow convergence in SIMP topology optimization. In summary, existing methods have limitations in terms of applicability, minimum size control effectiveness, and computational efficiency. There is an urgent need to develop a method for minimum size control of structures and their fillets that is both feasible in principle, easy to implement, versatile, adaptable, computationally efficient, and effective. Summary of the Invention

[0004] In response to the above defects or improvement needs of the prior art, the present invention provides a topology optimization method for applying minimum size control of the structure and its fillet, which aims to solve the problem of poor minimum size control effect of the existing structure optimization method.

[0005] To achieve the above object, according to one aspect of the present invention, a topology optimization method for applying minimum size control of a structure and its fillet is provided, the method comprising the following steps: (1) The pseudo-density associated with the corresponding grid cell in the design domain of the structure to be optimized is used as the design variable. The design variable is sequentially subjected to the first density filtering and step projection to obtain the physical variable field vector, and then the density indicator vector reflecting the presence or absence of solid or void phase material is calculated; (2) Based on the density indicator vector and the current topological grayscale value, the relative volume ratio coefficient corresponding to the grid unit is updated, and then secondary density filtering and secondary step projection are performed in sequence to obtain the minimum size satisfaction rate vector of the solid phase or the empty phase, and then the additional unit density vector is calculated; (3) Based on the additional unit density vector, the additional volume constraint principle is used to construct the minimum size constraint function of the solid phase or empty phase and its inverted rounding, and the sensitivity of the minimum size constraint function relative to the design variables is solved. The obtained minimum size constraint function and sensitivity are integrated into the topology optimization model, and the MMA algorithm is used to iteratively solve the topology optimization model to obtain the optimal topological configuration.

[0006] Furthermore, when the topological gray value of the design domain reaches a predetermined value, the relative volume ratio coefficient corresponding to the grid unit is modified. , the calculation formula of topological gray value is: Where, is the physical variable field vector; Represents the total number of mesh elements in the design domain.

[0007] Furthermore, the formula used to identify the interface between the solid phase and the void phase near the boundary of the design domain is: Where, represents the set of all unit numbers in the design domain, Represents the unit on the boundary b The density indication, Represents the unit on the boundary b The center point coordinates, Indicates that the distance from the boundary is no greater than the minimum size d The coordinates of the center point of cell k in represents the volume of unit k, Indicates that unit k acts on unit b The weight coefficient of .

[0008] Furthermore, when the unit i Density indication and minimum size satisfaction rate When both are equal to 1, the minimum size constraint is fully satisfied at the unit, and the product of the two is used as the unit i Additional cell density , .

[0009] Furthermore, the sum of the volumes of all cells that satisfy the minimum size constraint is equal to the total volume constraint value of the corresponding solid phase or void phase, so that all cells satisfy the minimum size constraint.

[0010] Furthermore, the constraint function is , and the corresponding formula is: Where, is a penalty term, which is an integer not less than 1; Indicates the total volume of the solid phase or void phase; The error term represents the size constraint of the solid or void phase.

[0011] Furthermore, the calculation formula of the sensitivity of the constraint function relative to the design variable is: .

[0012] Furthermore, the first step projection steepness , the second step projection truncation threshold , solid or empty phase minimum size constraint error term Perform parameter extension.

[0013] Furthermore, in the iterative process, the first step projection steepness is , the second step projection truncation threshold , solid or void phase size constraint error term and , as the number of iterations The increase is carried out by extending the parameters in the form of linear or exponential function, and the corresponding formula is: in, Express The number of steps between updates; Express The number of steps between updates; Express The number of steps between updates; Represents the second step projection truncation threshold The initial value of Represents the minimum value of the sum of the solid phase or void phase size constraint error terms.

[0014] Furthermore, the objective function and sensitivity of the topology optimization model, all constraint functions and their sensitivities are substituted into the MMA algorithm to solve the optimization problem and realize the update and iteration of the design variables.

[0015] In general, the above technical solutions conceived by the present invention, compared with the prior art, provide a topology optimization method for applying a structure and controlling the minimum size of filleting thereof, which has the following beneficial effects: 1. Based on the additional unit density vector, the principle of additional volume constraint is adopted to construct the minimum size constraint function of the solid or void phase and its fillet. The sensitivity of the minimum size constraint function with respect to the design variables is solved, and the obtained minimum size constraint function and sensitivity are integrated into the topology optimization model. The MMA algorithm is used to iteratively solve the topology optimization model to obtain the optimal topology configuration. This ensures that the minimum size constraint conditions are fully satisfied, thereby achieving more stringent minimum size control and improving the minimum size control effect. At the same time, no complex operations such as p-norm aggregation are required. Only one finite element analysis is required in one iteration, which reduces the overall computational cost.

[0016] 2. The minimum size constraint introduced in this invention is a geometric constraint with good versatility and strong adaptability. It can be easily combined with various topology optimization methods such as variable density and level set, and is not dependent on specific physical field problems.

[0017] 3. The minimum size constraint function introduced by the topology optimization method is explicit and differentiable. By calculating this function and its sensitivity value, it is convenient to use a variety of gradient optimization algorithms to solve the problem, which is conducive to extensive software integration. In addition, the topology optimization method is versatile and can not only effectively control the minimum size of the solid phase and the void phase, but also have a good control effect on the minimum size of the solid phase or void phase fillet.

[0018] 4. The topology optimization method can also be used as a post-processing technique. When the grayscale of the topology map is less than a certain value, the optimization process is added. This method is particularly suitable for topology optimization problems with slow convergence speed and unclear early optimization results.

[0019] 5. Projection steepness of the first step , the second step projection truncation threshold , solid or empty phase minimum size constraint error term Parameter extension is performed to ensure the stability of the MMA algorithm iteration process and the reasonableness of the generated topology configuration. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 This is a flow chart of a topology optimization method for applying structure and minimum size control of filleting thereof provided by the present invention; Figure 2 This is a schematic diagram of the principle of determining whether the minimum size constraint is satisfied through secondary density filtering; Figure 3 Schematic diagram of the design domain model of the MBB beam (right half) provided in Example 1 of the present invention; Figure 4 1 is a schematic diagram of the topology optimization results without applying the structure and minimum fillet size control of Example 1 of the present invention; Figure 5 Schematic diagram of the topology optimization result of Example 1 of the present invention in which only the solid phase and its minimum fillet size control are applied; Figure 6 1 is a schematic diagram of the topology optimization results of Example 1 of the present invention in which the solid phase, the void phase, and the minimum size of the fillet are controlled simultaneously; Figure 7 Schematic diagram of the design domain model of the flexible inverter (lower half) according to embodiment 2 of the present invention; Figure 8 1 is a schematic diagram of the topology optimization results of Example 2 of the present invention without applying the structure and the minimum size control of the fillet; Figure 9 Schematic diagram of the topology optimization result of Example 2 of the present invention in which only the solid phase and its minimum fillet size control are applied; Figure 10 Schematic diagram of the topology optimization results of Example 2 of the present invention in which the solid phase, the void phase and the minimum size control of the fillet are simultaneously applied. DETAILED DESCRIPTION

[0021] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is 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 for the purpose of explaining the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0022] The present invention provides a topology optimization method for applying minimum size control of a structure and its fillet. The method first adopts a pseudo-density associated with a finite element mesh unit as a design variable, and obtains an additional unit density characterizing the minimum size of the structure and its fillet after two density filtering and projections. Then, based on the principle of additional volume constraints, an explicit and differentiable minimum size constraint function is established, and its sensitivity is derived. The function and its sensitivity are integrated into a topology optimization model. The design variables are updated and iterated with the help of an MMA algorithm. The stability of the iterative process and the rationality of the topological configuration generation are ensured through parameter extension, and finally an optimization result with a good minimum size control effect is obtained.

[0023] See also Figure 1 and Figure 2 , the method mainly includes the following main steps: Step 1: The geometric model of the structure to be optimized is divided into finite element meshes, and the pseudo-density associated with the divided mesh units in the design domain of the structure to be optimized is used as the design variable. The design variables are sequentially subjected to the first density filtering and step projection to obtain the physical variable field vector, and then the density indicator vector reflecting the presence or absence of solid or void phase material is calculated.

[0024] Finite element meshing is performed on the geometric model of the structure to be optimized, and the pseudo density associated with the mesh elements in the design domain is defined. is the design variable (the pseudo density vector form is ), after the first filtering and step projection, the physical variable field vector is obtained , and then calculate the density indicator vector reflecting the presence or absence of solid or empty phase material .

[0025] Specifically, density filtering is performed on the design variable vector to obtain the filtered vector , the formula corresponding to density filtering is: (1) Where, represents the set of all element numbers in the structural design domain, and Represent the center point coordinates of unit i and unit j respectively, express and The distance between represents the volume of unit j, represents the weight coefficient of unit j acting on unit i (a real number between 0 and 1), represents the total number of mesh elements in the design domain, For the first density filter weight matrix, for Sum by column Vector, R is the filter radius, usually equal to the solid phase (empty phase) and its minimum size d .

[0026] The formula corresponding to the step projection is: (2) Where, and Represent the cutoff threshold and steepness of the first step projection function respectively. Through the projection operation, the physical variable field vector The elements in gradually approach 1 or 0, thereby better reflecting the presence or absence of solid phase material corresponding to the grid unit.

[0027] When minimum size control is applied to a solid material, the density indicator vector It is expressed by formula (3): (3) When minimum size control is applied to the void phase material, the density indicator vector It is expressed by formula (4): (4) Step 2: Based on the density indicator vector and the current topological grayscale value, the relative volume ratio coefficient corresponding to the grid unit is updated, and then secondary density filtering and secondary step projection are performed in sequence to obtain the minimum size satisfaction rate vector of the solid phase or the empty phase, and then the additional unit density vector that further reflects the minimum size satisfaction is calculated.

[0028] According to the current topology gray value , update the relative volume ratio coefficient corresponding to the grid cell , and then perform secondary filtering and projection to obtain the minimum size satisfaction rate vector of solid or empty phase , and then calculate the additional unit density vector that can better reflect the minimum size satisfaction .

[0029] like Figure 2 As shown in the figure, topology optimization usually obtains a truss or plate beam structure, as long as the center of the element is the center of the circle (sphere) and the minimum size is d When the area occupied by the solid phase (empty phase) material in a circle (sphere) with a radius of , reaches a certain proportion, it is easy to meet the minimum size requirement. The proportion of the solid phase (empty phase) material in the filter circle (sphere) when the minimum size requirement is met is defined as the relative volume fraction coefficient, and is expressed as express.

[0030] When the filter circle (sphere) of the element is located inside the design domain, It represents the ratio of the area of ​​a semicircular (spherical) shape to the area of ​​the entire filter circle (sphere). For two-dimensional topology optimization problems, Calculated by formula (5): (5) When the filter circle (sphere) of the element is located near the corner of the design domain, Indicates the ratio of the area of ​​the corner area to the area of ​​the entire filter circle (sphere), here we use It is expressed as follows: (6) in, Represents the opening angle of the design domain corners. For a rectangular design domain, .

[0031] During the topology optimization process, the solid-air interface area near the design domain boundary will also have similar situations as those near the design corners. In this case, it is necessary to dynamically identify this area and correct it. This operation is only performed when the topological grayscale value of the design domain reaches a certain value. The topological grayscale value is calculated according to formula (7): (7) The identification of the interface between the solid phase and the void phase near the boundary of the design domain is performed using formula (8): (8) Where, represents the set of all unit numbers in the design domain, Represents the unit on the boundary b The density indication, Represents the unit on the boundary b The center point coordinates, Indicates that the distance from the boundary is no greater than the minimum size d The coordinates of the center point of cell k in Display unit k The volume, Display unit k Acting on the unit b The weight coefficient of (It is a Boolean variable with a value of 0 or 1).

[0032] Relative volume fraction coefficient of the interface between solid and void phase near the boundary of the design domain use It means that after the area is identified, it is relaxed and calculated according to formula (9): (9) Next, with the smallest size d For the radius, perform secondary density filtration and calculate the volume ratio of solid or empty phase materials in the filter circle (sphere) relative to the filter circle (sphere). , as shown in formula (10): (10) in, represents the set of all element numbers in the structural design domain, represents the total number of mesh elements (design variables) in the design domain, Represents units i and unit j The center point coordinates, express The distance between Display unit j The volume, Display unit j The weight coefficient acting on unit i (a Boolean variable with a value of 0 or 1), Indicates the second filtering weight matrix, for Sum by column vector.

[0033] Then, a secondary step projection is performed and the minimum size satisfaction rate of the solid phase (empty phase) and its fillet is calculated according to formula (11): : (11) in, and Represent the cutoff threshold and steepness of the quadratic step projection function, respectively.

[0034] When the unit i Density indication and minimum size satisfaction rate When both are equal to 1, it is considered that the unit fully meets the minimum size constraint. Therefore, the product of the two is used as the unit i Additional cell density , as expressed in formula (12): (12) Step three: Based on the additional unit density vector, the additional volume constraint principle is used to construct the minimum size constraint function of the solid phase or void phase and its fillet, and then the sensitivity of the minimum size constraint function relative to the design variables is solved. The obtained minimum size constraint function and sensitivity are integrated into the topology optimization model.

[0035] Adopting the principle of additional volume constraint, the minimum size constraint function of solid phase or void phase and its inverted rounding is established , and then derive its sensitivity relative to the design variables according to the chain rule , and then integrate them into the topology optimization model to be solved.

[0036] Specifically, if all units satisfy the minimum size constraint, the sum of the volumes of all units that satisfy the minimum size constraint is equal to the total volume constraint value of the corresponding solid phase or void phase. The minimum size constraint function of the solid phase (void phase) and its fillet is defined as , must satisfy formula (13): (13) in, p is the penalty term, which is an integer not less than 1 (here p =2), the larger the value, the stricter the additional volume constraint; Represents the total volume of the solid phase or void phase. is a very small positive number, representing the error term due to the size constraint of the solid phase or the void phase.

[0037] Then, according to the chain rule, the sensitivity of the solid phase (void phase) and its minimum inverted size constraint function relative to the design variables is calculated, as shown in formula (14): (14) Specifically, a topology optimization model is established with the total volume of the solid phase, the minimum size of the solid phase (empty phase) and its fillet as constraints: (15) in, represents the design variable (pseudo-density) associated with the finite element mesh element, represents the total number of design variables, Represent the design variable vector and physical variable vector respectively, represents the volume vector of the mesh element in the design domain; represents the objective function; represents the total volume limit of the solid phase; represents the solid phase volume constraint; Represents the minimum size constraint of the solid phase and its fillet; Indicates the minimum size constraint of the void phase and its fillet; Indicates the upper and lower limits of the design variables.

[0038] Step 4: Use the MMA algorithm to iteratively solve the topology optimization model to obtain the optimal topology configuration.

[0039] The MMA algorithm is used to solve the optimization problem. To ensure the stability of the algorithm iteration process and the reasonableness of the generated topological configuration, the first step projection steepness is , the second step projection truncation threshold , solid or empty phase minimum size constraint error term Perform parameter extension; after reaching the convergence condition or the maximum number of iterations, output the optimal topology configuration.

[0040] For the topology optimization model in step three, combined with finite element analysis calculation, the objective function and sensitivity of the topology optimization model, all constraint functions and their sensitivities are substituted into the MMA algorithm to solve the optimization problem and realize the update and iteration of design variables.

[0041] During the iteration process, the first step projection steepness , the second step projection truncation threshold , solid or void phase size constraint error term and (Used uniformly ), as the number of iterations loop The increase of is carried out by extending the parameters in the form of linear or exponential function, as shown in the following formula: in, Express The number of steps between updates; Express The number of steps between updates; Express The number of steps between updates; Represents the second step projection truncation threshold The initial value of Represents the solid phase or void phase size constraint error term The minimum value of .

[0042] When a certain convergence condition is met (such as the maximum change of the design variable is less than a specific value), or the maximum number of iterations is reached, the iteration is stopped. , after density filtering and step projection, the physical variable field vector is obtained , and then after necessary post-processing, the final topological configuration is output.

[0043] The present invention is further described in detail below with reference to specific embodiments.

[0044] Example 1 Figure 3The figure shows an MBB beam (right half) with a length of L = 40 cm and a width of H = 10 cm. A concentrated load F = 1 kN is applied in the upper left corner in the negative Y-axis direction. The left boundary is fixed in the X-axis direction and the lower right corner is fixed in the Y-axis direction. The beam is divided into 400 × 100 = 400,000 regular quadrilateral elements. The volume constraint is 0.4 times the total volume of the design domain. The minimum size of the first density filter radius, the solid phase, the void phase, and their fillet radii are all 4 element lengths. The elastic modulus of the solid phase material is E = 10 MPa. Minimizing structural flexibility is the goal, and the maximum number of iterations is 200. The parameter extension method for the steepness of the first step projection, the cutoff threshold of the second step projection, and the minimum size constraint error term is shown below: Figure 4 、 Figure 5 and Figure 6 The figure shows a comparison of the topology optimization results of the MBB beam with and without structural and fillet minimum size control. It can be seen that when structural and fillet minimum size control are not applied, there are many cases in the topology optimization results that do not meet the minimum size constraints of the solid phase (void phase) and its fillet. The topology optimization results with only the solid phase and its fillet minimum size control show a more robust structure in the solid phase region, and the void phase, solid phase and their minimum size constraints are basically satisfied. The topology optimization results with the simultaneous application of solid phase and fillet, void phase and their minimum fillet size controls show that the fillets are more open and round (spherical), which can better meet the minimum size constraints of the solid phase and its fillet, and the void phase and their minimum fillet size constraints.

[0045] Example 2 Figure 7 The figure shows a flexible inverter (lower half) with a length of L = 15 cm and a width of H = 4.5 cm. The concentrated load F = 1 kN in the positive direction of the axis, a spring with a stiffness coefficient of 10 kN / m is added to the upper left corner and the upper right corner respectively, the upper boundary Y-axis direction and the lower left corner are fixed constraints, divided into 150 × 45 = 6750 regular quadrilateral units, the volume constraint is 0.3 times the total volume of the design domain, the first density filtering radius, the minimum size of the solid phase and the void phase and their fillet radius are all 3 unit lengths, the elastic modulus of the solid phase material is E = 1 MPa, the goal is to maximize the output displacement of the structure, the maximum number of iterations is 200 times, and the parameter extension is still carried out according to formula (19), formula (20), and formula (21) (the same as the MBB beam case).

[0046] Figure 8 、 Figure 9 and Figure 10The figure shows a comparison of the topology optimization results of the flexible inverter with and without applying structural and fillet minimum size control. It can be seen that when no minimum size control is applied, many parts of the topology optimization results do not meet the minimum size constraints of the solid phase (empty phase) and its fillet; in the topology optimization results when only the solid phase and its fillet minimum size control are applied, the solid phase and its minimum size constraints are met, and some parts do not meet the minimum size constraints of the fillet; in the topology optimization results when the solid phase (empty phase) and its fillet minimum size control are applied at the same time, the solid phase area is thicker, the fillet is more open and round (spherical), and the hinge phenomenon of the flexible inverter is significantly alleviated, which can fully meet the solid phase (empty phase) and its minimum size constraints.

[0047] The present invention provides a topology optimization design method for controlling the minimum size of a structural solid phase (empty phase) and its fillet. The method is a universal, efficient, and multifunctional system design method that solves the problems of most existing topology optimization methods, such as poor versatility, limited scope of application, lax minimum size control, single function, and high computational cost.

[0048] The present invention also provides a topology optimization system for applying structure and minimum size control of filleting thereof, the system comprising a memory and a processor, the memory storing a computer program, and the processor executing the topology optimization method for applying structure and minimum size control of filleting thereof as described above when executing the computer program.

[0049] The present invention also provides a computer-readable storage medium, which stores machine-executable instructions. When the machine-executable instructions are called and executed by a processor, the machine-executable instructions prompt the processor to implement the topology optimization method for applying a structure and controlling the minimum size of its fillet as described above.

[0050] It will be easily understood by those skilled in the art that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A topology optimization method for applying minimum size control of a structure and its fillet, characterized in that: The method comprises the following steps: (1) The pseudo-density associated with the corresponding grid cell in the design domain of the structure to be optimized is used as the design variable. The design variable is sequentially subjected to the first density filtering and step projection to obtain the physical variable field vector, and then the density indicator vector reflecting the presence or absence of solid or void phase material is calculated; (2) Based on the density indicator vector and the current topological grayscale value, the relative volume ratio coefficient corresponding to the grid unit is updated, and then secondary density filtering and secondary step projection are performed in sequence to obtain the minimum size satisfaction rate vector of the solid phase or the empty phase, and then the additional unit density vector is calculated; (3) Based on the additional unit density vector, the additional volume constraint principle is used to construct the minimum size constraint function of the solid phase or empty phase and its inverted rounding, and the sensitivity of the minimum size constraint function relative to the design variables is solved. The obtained minimum size constraint function and sensitivity are integrated into the topology optimization model, and the MMA algorithm is used to iteratively solve the topology optimization model to obtain the optimal topological configuration.

2. The topology optimization method for applying a structure and controlling the minimum size of fillet fillets according to claim 1, characterized in that: When the topological gray value of the design domain reaches a predetermined value, the relative volume ratio coefficient corresponding to the grid unit is modified , the calculation formula of topological gray value is: Where, is the physical variable field vector; Represents the total number of mesh elements in the design domain.

3. The topology optimization method for applying a structure and controlling the minimum size of fillet fillets according to claim 2, characterized in that: The formula used to identify the interface between the solid phase and the void phase near the boundary of the design domain is: Where, represents the set of all unit numbers in the design domain, Represents the unit on the boundary b The density indication, Represents the unit on the boundary b The center point coordinates, Indicates that the distance from the boundary is no greater than the minimum size d The coordinates of the center point of cell k in represents the volume of unit k, Indicates that unit k acts on unit b The weight coefficient of .

4. The topology optimization method for applying a structure and controlling the minimum size of fillet fillets as claimed in claim 1, characterized in that: When the unit i Density indication and minimum size satisfaction rate When both are equal to 1, the minimum size constraint is fully satisfied at the unit, and the product of the two is used as the unit i Additional cell density , .

5. The topology optimization method for applying a structure and controlling the minimum size of fillet fillets as claimed in claim 1, characterized in that: The sum of the volumes of all elements that satisfy the minimum size constraint is equal to the total volume constraint value of the corresponding solid phase or void phase, so that all elements satisfy the minimum size constraint.

6. The topology optimization method for applying a structure and controlling the minimum size of fillet fillets as claimed in claim 5, characterized in that: The constraint function is , and the corresponding formula is: Where, is a penalty term, which is an integer not less than 1; Indicates the total volume of the solid phase or void phase; The error term represents the size constraint of the solid or void phase.

7. The topology optimization method for applying a structure and controlling the minimum size of fillet fillets as claimed in claim 6, characterized in that: The sensitivity of the constraint function to the design variable is calculated as: 。 8. The topology optimization method for applying a structure and controlling the minimum size of fillet fillets as claimed in claim 6, characterized in that: The first step projection steepness , the second step projection truncation threshold , solid or empty phase minimum size constraint error term Perform parameter extension.

9. The topology optimization method for applying a structure and controlling the minimum size of fillet fillets thereof according to claim 8, characterized in that: During the iteration process, the first step projection steepness , the second step projection truncation threshold , solid or void phase size constraint error term and , as the number of iterations The increase is carried out by extending the parameters in the form of linear or exponential function, and the corresponding formula is: in, Express The number of steps between updates; Express The number of steps between updates; Express The number of steps between updates; Represents the second step projection truncation threshold The initial value of Represents the minimum value of the sum of the solid phase or void phase size constraint error terms.

10. The topology optimization method for applying a structure and controlling the minimum size of fillet fillet as claimed in claim 6, characterized in that: The objective function and sensitivity of the topology optimization model, all constraint functions and their sensitivities are substituted into the MMA algorithm to solve the optimization problem and realize the update and iteration of the design variables.

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