A topology optimization method for applying a structure and rounding a minimum size control thereof

Abstract

The present application belongs to the technical field of structure optimization design, and discloses a topology optimization method of a structure and its rounded minimum size control, which comprises the following steps: (1) associating the pseudo density in the design domain with the corresponding grid element as a design variable, and sequentially performing first density filtering and step projection on the design variable to obtain a physical variable field vector, and then calculating a density indication vector reflecting the presence or absence of solid or empty phase materials; (2) sequentially performing second density filtering and second step projection on the relative volume fraction coefficient corresponding to the grid element after updating the relative volume fraction coefficient to obtain a solid or empty phase minimum size satisfaction rate vector, and then calculating an additional element density vector; (3) constructing a solid or empty phase and its rounded minimum size constraint function, integrating the obtained minimum size constraint function and its sensitivity into a topology optimization model, and iteratively solving the topology optimization model by using an MMA algorithm to obtain an optimal topology configuration. The present application improves the control effect of the minimum size.

Description

Technical Field

[0001] This invention belongs to the technical field of structural optimization design, and more specifically, relates to a topology optimization method that applies minimum size control to the structure and its rounding. Background Technology

[0002] Topology optimization aims to obtain novel configurations with optimal material distribution under specific objectives and constraints. It offers high design freedom and is independent of human experience, making it a promising structural optimization design technique widely used in aerospace, shipbuilding, vehicle, and weaponry. However, existing methods primarily focus on physical properties, neglecting manufacturing processes. This leads to issues such as small or thin-walled components and local hinges in topology optimization, which are detrimental to manufacturing and affect structural performance. Furthermore, topology optimization generally cannot automatically round corners, easily generating sharp angles that lead to stress concentration, accelerate fatigue failure, and fail to meet processing and usage requirements. Therefore, effectively controlling the minimum dimensions of the structure and its rounding is crucial for structural optimization design in engineering applications.

[0003] In the field of structural topology optimization, previously developed topology optimization methods considering the minimum size of structures or rounded corners are mainly divided into two categories: implicit and explicit. Early on, Guest et al. proposed an implicit control method for the minimum size of the solid phase (solid) and void phase (hole) based on filtered projection. This method uses a circular (spherical) filter and a Heaviside projection function to achieve minimum size control of the solid phase (void) and its rounded corners. However, this method easily generates gray-scale elements at the interface between the solid and void phases and struggles to solve the local hinge problem in flexible mechanisms, resulting in imprecise minimum size control. Furthermore, this method requires defining design variables separately for the void and solid phases, increasing computational complexity. Sigmund et al. proposed an implicit minimum size control method based on the robust formula of "corrosion, intermediate, and expansion," which has good applicability and stability. However, the minimum size control is not strict, requiring three finite element analyses per optimization iteration, leading to a significant increase in computational complexity. Moreover, it cannot guarantee a consistent topological configuration across various physical problems, limiting its applicability. Zhang Weisheng, Li Hao, and others introduced a minimum size control method based on skeleton extraction into SIMP or level set topology optimization methods. This method can effectively control the minimum size of the solid phase, but it struggles to simultaneously control the minimum size of the solid phase rounding, the empty phase, and its rounding. Furthermore, this method converges slowly in SIMP topology optimization. In summary, existing methods have certain shortcomings in terms of applicability, minimum size control effectiveness, and computational efficiency. There is an urgent need to develop a minimum size control method for structures and their rounding that is theoretically feasible, easy to implement, versatile, adaptable, computationally efficient, and effective. Summary of the Invention

[0004] In view of the above-mentioned 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 rounding, which aims to solve the problem of poor minimum size control effect of the existing structure optimization method.

[0005] To achieve the above objectives, according to one aspect of the present invention, a topology optimization method for applying minimum control over the structure and its rounding dimensions 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 then filtered by the first density and the step projection to obtain the physical variable field vector, and then the density indicator vector reflecting the presence or absence of solid or empty phase material is calculated. (2) After updating the relative volume fraction coefficient of the mesh element based on the density indicator vector and the current topological gray value, perform secondary density filtering and secondary step projection in sequence to obtain the minimum size satisfaction rate vector of the solid phase or the empty phase, and then calculate the additional element density vector. (3) Based on the additional unit density vector, the minimum size constraint function of the solid phase or empty phase and its rounding is constructed by adopting the principle of additional volume constraint. 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. The MMA algorithm is used to iteratively solve the topology optimization model to obtain the optimal topology configuration.

[0006] Furthermore, when the topological grayscale value of the design domain reaches a predetermined value... Modify the relative volume fraction coefficients corresponding to the mesh elements. The formula for calculating the topological grayscale value is:

[0007] In the formula, For physical variable field vectors; This indicates the total number of grid cells within the design domain.

[0008] Furthermore, the formula used to identify the boundary region between the solid and air phases near the design domain boundary is:

[0009] In the formula, This represents the set of all element numbers within the design domain. Represents the unit on the boundary Density indication, Represents the unit on the boundary The coordinates of the center point, This indicates that the distance from the boundary is not greater than the minimum size. The coordinates of the center point of element k within the cell. Representation unitk volume, Representation unit k Acting on unit b The weighting coefficients.

[0010] 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 that element, and the product of the two is used as the element. i Additional unit density , .

[0011] Furthermore, 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 or empty phase, so that all elements satisfy the minimum size constraint.

[0012] Furthermore, the constraint function is The corresponding formula is:

[0013] In the formula, The penalty term is an integer not less than 1; Represents the total volume of the solid or open phase; This indicates the error term representing the size constraint of the solid or air phase.

[0014] Furthermore, the formula for calculating the sensitivity of the constraint function relative to the design variables is: .

[0015] Furthermore, the steepness of the first step projection Second step projection truncation threshold Minimum size constraint error term for solid phase or open phase Perform parameter extension.

[0016] Furthermore, during the iteration process, the steepness of the first step projection is... Second step projection truncation threshold Solid phase or open phase size constraint error item and As the number of iterations increases With the increase of , the parameter extension is performed in the form of a linear or exponential function, and the corresponding formula is:

[0017]

[0018]

[0019] in, Indicates to The number of steps between updates; Indicates to The number of steps between updates; Indicates to The number of steps between updates; Indicates the second step projection cutoff threshold initial value, Indicates the size constraint error term of the solid phase or the unused phase. The minimum value.

[0020] Furthermore, the objective function and its sensitivity of the topology optimization model, as well as all constraint functions and their sensitivities, are substituted into the MMA algorithm to solve the optimization problem, thereby achieving iterative updating of design variables.

[0021] In summary, compared with the prior art, the topology optimization method for controlling the minimum size of the applied structure and its rounding provided by the present invention has the following advantages: 1. Based on the additional element density vector, the minimum size constraint function of the solid phase or empty phase and its rounded shape is constructed using the principle of additional volume constraint. The sensitivity of the minimum size constraint function with respect to the design variables is solved. 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 stricter minimum size control and improving the minimum size control effect. At the same time, no complex p-norm aggregation and other calculations are required. Only one finite element analysis is needed in one iteration, resulting in low overall computational cost.

[0022] 2. The minimum size constraint introduced in this invention is a geometric constraint with good versatility and adaptability. It can be easily combined with various topology optimization methods such as variable density and level set, and does not depend on specific physical field problems.

[0023] 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 various gradient-type optimization algorithms to solve the problem, which is conducive to widespread software integration. Moreover, the topology optimization method has diverse functions, which can not only achieve effective control of the minimum size of solid and empty phases, but also produce good control effect on the minimum size of solid or empty phase rounding.

[0024] 4. The topology optimization method can also be used as a post-processing technique to add an optimization process when the gray level of the topology graph is less than a certain value. It is especially suitable for some topology optimization problems with slow convergence speed and unclear early optimization results.

[0025] 5. Steepness of the first step projection Second step projection truncation threshold Minimum size constraint error term for solid phase or open phase Parameter extension is performed to ensure the stability of the MMA algorithm iteration process and the reasonable generation of topological configurations. Attached Figure Description

[0026] Figure 1 This is a flowchart of a topology optimization method for applying structural and rounding minimum size control provided by the present invention; Figure 2 This is a schematic diagram illustrating the principle of determining whether the minimum size constraint is met through secondary density filtering. Figure 3 This is a schematic diagram of the design domain model of the MBB beam (right half) provided in Embodiment 1 of the present invention; Figure 4 This is a schematic diagram of the topology optimization results of Embodiment 1 of the present invention without applying structural and rounding minimum size control; Figure 5 This is a schematic diagram of the topology optimization results of Embodiment 1 of the present invention, which only applies the solid phase and its minimum rounding size control; Figure 6 This is a schematic diagram of the topology optimization results of simultaneously applying solid phase, empty phase and their minimum rounding size control in Embodiment 1 of the present invention; Figure 7 This is a schematic diagram of the design domain model of the flexible inverter (lower half) according to Embodiment 2 of the present invention; Figure 8 This is a schematic diagram of the topology optimization results of Embodiment 2 of the present invention without applying structural and rounding minimum size control; Figure 9 This is a schematic diagram of the topology optimization results of Embodiment 2 of the present invention, which only applies solid phase and its minimum rounding size control; Figure 10 This is a schematic diagram of the topology optimization results of simultaneously applying solid phase, empty phase and their minimum rounding size control in Embodiment 2 of the present invention. Detailed Implementation

[0027] 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. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0028] This invention provides a topology optimization method for applying minimum size control to a structure and its rounded corners. The method first uses a pseudo-density associated with the finite element mesh as a design variable. After two density filtering and projection operations, an additional element density characterizing the minimum size of the structure and its rounded corners is obtained. Then, based on the principle of additional volume constraints, an explicit, differentiable minimum size constraint function is established, its sensitivity is derived, and this function and its sensitivity are integrated into the topology optimization model. The design variable is updated iteratively using the MMA algorithm, and parameter extension ensures the stability of the iteration process and the rationality of the topology configuration generation, ultimately yielding an optimization result with good minimum size control.

[0029] Please see Figure 1 and Figure 2 The method mainly includes the following main steps: Step 1: Perform finite element mesh generation on the geometric model of the structure to be optimized. Use the pseudo-density associated with the mesh cells within the design domain of the structure to be optimized as the design variable. The design variable is then subjected to initial density filtering and step projection to obtain the physical variable field vector. Finally, the density indicator vector reflecting the presence or absence of solid or empty phase material is calculated.

[0030] Finite element meshing is performed on the geometric model of the structure to be optimized, and pseudo-density associated with mesh elements within the design domain is defined. ( ) are the design variables (in pseudo-density vector form) After initial filtering and step projection, the physical variable field vector is obtained. This allows for the calculation of a density indicator vector that reflects the presence or absence of a solid or empty phase material. .

[0031] Specifically, density filtering is performed on the design variable vector to obtain the filter vector. The formula for density filtering is: (1) In the formula, This represents the set of all element numbers in the structural design domain. and These represent the coordinates of the center points of element i and element j, respectively. express and The distance between them This represents the volume of element j. This represents the weighting coefficient of unit j on unit i (a real number between 0 and 1). Represents the total number of grid cells in the design domain. For the first density filtration Weight matrix, for Summing by column vector, The filtration radius is typically equal to the solid phase (empty phase) and its minimum size. .

[0032] The formula corresponding to step projection is: (2) In the formula, and Let represent the cutoff threshold and steepness of the first step projection function, respectively. Through projection operations, the physical variable field vector is made... The elements in the grid gradually approach 1 or 0, thus better reflecting the presence or absence of solid material corresponding to the grid cells.

[0033] When minimum size control is applied to a solid material, the density indicator vector This can be expressed by formula (3): (3) When minimum size control is applied to an empty phase material, the density indicator vector This can be expressed by formula (4): (4) Step 2: Based on the density indicator vector and the current topological gray value, update the relative volume fraction coefficient of the mesh element, and then perform secondary density filtering and secondary step projection to obtain the minimum size satisfaction rate vector of the solid phase or the empty phase. Then, calculate the additional element density vector that further reflects the minimum size satisfaction.

[0034] Based on the current topology grayscale value Update the relative volume fraction coefficients corresponding to the mesh elements. Then, secondary filtering and projection are performed to obtain the minimum size satisfaction rate vector of the solid or empty phase. This allows for the calculation of additional element density vectors that better reflect the minimum size requirement. .

[0035] like Figure 2 As shown, topology optimization typically yields truss-like or plate-beam-like structures, provided that the element center is the center of the circle (sphere) and the minimum size is achieved. Within a circle (sphere) of radius , when the area occupied by solid phase (empty phase) material reaches a certain proportion, the minimum size requirement is easily met. The proportion of solid phase (empty phase) material within the filter circle (sphere) that satisfies the minimum size requirement is defined as the relative volume fraction coefficient, and is used as... express.

[0036] When the filter circle (sphere) of the cell is located inside the design domain, This represents the ratio of the area of ​​a semi-circular (spherical) shape to the area of ​​the entire filter circle (sphere). Here, we use... This indicates that for a two-dimensional topology optimization problem, Calculated using formula (5): (5) When the filter circle (sphere) of the cell is located near a corner of the design domain, This represents the ratio of the area of ​​the corner region to the area of ​​the entire filter circle (sphere), expressed here as... This means that the calculation is performed according to formula (6): (6) in, This represents the angle of the corner openings of the design domain. For a rectangular design domain, .

[0037] During topology optimization, the solid-vacuum boundary region near the design domain boundary may exhibit similar characteristics to that near the design corner. In such cases, it is necessary to dynamically identify and correct this region. This operation only occurs when the topological grayscale value in the design domain reaches a certain value. The topological grayscale value is calculated according to formula (7) during the process: (7) The identification of the boundary region between the solid and air phases near the design domain boundary is performed using formula (8): (8) In the formula, This represents the set of all element numbers within the design domain. Represents the unit on the boundary Density indication, Represents the unit on the boundary The coordinates of the center point, This indicates that the distance from the boundary is not greater than the minimum size. The coordinates of the center point of element k within the cell. Representation unit k volume, Representation unit k Acting on unit b The weighting coefficients (which are Boolean variables with values ​​of 0 or 1).

[0038] The relative volume fraction of the solid-void boundary region near the design domain boundary use This indicates that after identifying the region, a relaxation process is applied, and the calculation is performed according to formula (9): (9) Next, with the smallest size Using a radius of [radius], a secondary density filtration was performed, and the volume fraction of the solid or free-floating material within the filter circle (sphere) relative to the filter circle (sphere) was statistically determined. As shown in formula (10): (10) in, This represents the set of all element numbers in the structural design domain. Represents the total number of grid cells (design variables) in the design domain. and Representing units respectively i and unit j The coordinates of the center point, express and The distance between them Representation unit j volume, Representation unit j The weight coefficient applied to unit i (is a Boolean variable with a value of 0 or 1). Indicates the second filtering Weight matrix, for Summing by column vector.

[0039] Next, a second step projection is performed, and the minimum dimensional satisfaction rate of the solid phase (empty phase) and its rounding is calculated according to formula (11). : (11) in, and These represent the cutoff threshold and steepness of the second-order step projection function, respectively.

[0040] When unit i Density indication and minimum size satisfaction rate When both are equal to 1, the minimum size constraint is considered to be fully satisfied at that element location. Therefore, the product of the two is used as the element size. i Additional unit density As shown in formula (12): (12) Step 3: Based on the additional element density vector, construct the minimum size constraint function of the solid phase or empty phase and its rounded shape using the principle of additional volume constraint. Then, solve the sensitivity of the minimum size constraint function with respect to the design variables and integrate the obtained minimum size constraint function and sensitivity into the topology optimization model.

[0041] Using the principle of additional volume constraints, a minimum size constraint function for the solid phase or empty phase and its rounding is established. Then, based on the chain rule, its sensitivity relative to design variables is derived. Then, they are integrated into the topology optimization model to be solved.

[0042] Specifically, if all elements satisfy the minimum size constraint, and the sum of the volumes of all elements satisfying the minimum size constraint equals the total volume constraint value of the corresponding solid or empty phase, then the minimum size constraint function for the solid phase (empty phase) and its rounded corners is defined as follows: It must satisfy formula (13): (13) in, The penalty term is an integer not less than 1 (here, we take...). The larger the value, the stricter the additional volume constraints; It represents the total volume of the solid or empty phase. It is a very small positive number that represents the error term for the size constraint of the solid or open phase.

[0043] Furthermore, according to the chain rule, the sensitivity of the solid phase (empty phase) and its minimum rounding size constraint function to the design variables is calculated, as shown in formula (14): (14) Specifically, a topology optimization model is established with constraints such as the total volume of the solid phase, the solid phase (empty phase), and the minimum size of its rounding. (15) in, This represents the design variable (pseudo-density) associated with the finite element mesh element. Indicates the total number of design variables. , These represent the design variable vector and the physical variable vector, respectively. Represents the volume vector of a mesh element in the design domain; Represent the objective function; Indicates the total volume limit of the solid phase; Indicates solid volume constraint; This represents the minimum dimensional constraint of the solid phase and its rounding. This represents the minimum size constraint for the empty phase and its rounding. Indicates the upper and lower limits of the design variable.

[0044] Step 4: Use the MMA algorithm to iteratively solve the topology optimization model to obtain the optimal topology configuration.

[0045] The optimization problem is solved using the MMA algorithm; to ensure the stability of the algorithm's iterative process and the reasonableness of the generated topology, the steepness of the first step projection is adjusted. Second step projection truncation threshold Minimum size constraint error term for solid phase or open phase Perform parameter extension; after the convergence condition or the maximum number of iterations is reached, output the optimal topology configuration.

[0046] For the topology optimization model in step three, combined with finite element analysis, the objective function and its sensitivity of the topology optimization model, as well as all constraint functions and their sensitivities, are substituted into the MMA algorithm to solve the optimization problem, thereby realizing the iterative update of design variables.

[0047] During the iteration process, the steepness of the first step projection will be... Second step projection truncation threshold Solid phase or open phase size constraint error item and (Uniform usage) (represented by) the number of iterations The increase of is extended by parameters in the form of linear or exponential functions, as shown in the following equation: (16) (17) (18) in, Indicates to The number of steps between updates; Indicates to The number of steps between updates; Indicates to The number of steps between updates; Indicates the second step projection cutoff threshold initial value, Indicates the size constraint error term of the solid phase or the unused phase. The minimum value.

[0048] Iteration stops when certain convergence conditions are met (e.g., the maximum change in the design variable is less than a specific value) or the maximum number of iterations is reached. For the design variables of the last iteration... The physical variable field vector is obtained after density filtering and step projection. After necessary post-processing, the final topology configuration is output.

[0049] The present invention will be further described in detail below with reference to specific embodiments.

[0050] Example 1 Figure 3The diagram shows an MBB beam (right half) with length L=40cm and width H=10cm. A concentrated load F=1kN is located at the upper left corner in the negative Y-axis direction. The left boundary X-axis direction and the lower right corner Y-axis direction are fixed constraints. The beam is divided into 400×100=400000 regular quadrilateral elements. The volume constraint is 0.4 times the total volume of the design domain. The minimum dimensions of the initial density filter radius, the solid phase and the empty phase, and their rounding radii are all 4 element lengths. The elastic modulus of the solid phase material is E=10MPa. The objective is to minimize structural flexibility, with a maximum of 200 iterations. The steepness of the first step projection, the truncation threshold of the second step projection, and the parameter extension methods for the minimum size constraint error term are shown below: (19) (20) (twenty one) 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 rounding minimum size control. It can be seen that without structural and rounding minimum size control, there are many places in the topology optimization results that do not meet the constraints of the solid phase (empty phase) and its rounding minimum size. With only solid phase and its rounding minimum size control applied, the solid phase region structure is more robust, and the constraints of the empty phase, solid phase and its minimum size are basically satisfied. However, with the topology optimization results that apply both solid phase and its rounding and empty phase and its rounding minimum size control, the rounding area is more open and round (spherical), and can better satisfy the constraints of the solid phase and its rounding minimum size and the empty phase and its rounding minimum size.

[0051] Example 2 Figure 7 The image shows a flexible inverter (lower half) with a length L = 15cm and a width H = 4.5cm. There is a [something] in the upper left corner. x The concentrated load in the positive direction of the axis is F=1kN. A spring with a stiffness coefficient of 10kN / m is added to the upper left and upper right corners respectively. The upper boundary Y-axis direction and the lower left corner are fixed and constrained. It is divided into 150×45=6750 regular quadrilateral elements. The volume constraint is 0.3 times the total volume of the design domain. The minimum size of the initial density filtering radius, solid phase and empty phase and their rounding radius are all 3 element lengths. The elastic modulus of the solid phase material is E=1MPa. The goal is to maximize the output displacement of the structure. The maximum number of iterations is 200. The parameters are still extended according to formula (19), formula (20) and formula (21) (the same as the MBB beam example).

[0052] Figure 8 , Figure 9 and Figure 10The diagram shows a comparison of topology optimization results for the flexible inverter with and without structural and rounding minimum size control. It can be seen that without minimum size control, the topology optimization results show several instances where the solid phase (empty phase) and its rounding minimum size constraints are not met. With only solid phase and rounding minimum size control applied, the solid phase and minimum size constraints are satisfied, although some areas do not meet the rounding minimum size constraint. However, with both solid phase (empty phase) and rounding minimum size control applied, the solid phase region is more robust, the rounding is more open and rounded, and the hinge phenomenon of the flexible inverter is significantly alleviated, fully satisfying the solid phase (empty phase) and its minimum size constraints.

[0053] The present invention provides a topology optimization design method for structural solid phase (empty phase) and its minimum rounding size control. It is a general, efficient and multifunctional system design method that solves the problems of poor universality, limited applicability, non-strict minimum size control, single function and high computational cost of most existing topology optimization methods.

[0054] The present invention also provides a topology optimization system for applying structural and rounding minimum size control, the system including a memory and a processor, the memory storing a computer program, and the processor executing the computer program to perform the topology optimization method for applying structural and rounding minimum size control as described above.

[0055] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the topology optimization method described above for applying a structure and controlling its minimum rounding size.

[0056] Those skilled in the art will readily understand 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 within the scope of protection of the present invention.

Claims

1. A topology optimization method for applying minimum size control to the structure and its rounding, characterized in that, The method includes 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 then filtered by the first density and the step projection to obtain the physical variable field vector, and then the density indicator vector reflecting the presence or absence of solid or empty phase material is calculated. (2) After updating the relative volume fraction coefficient of the mesh element based on the density indicator vector and the current topological gray value, perform secondary density filtering and secondary step projection in sequence to obtain the minimum size satisfaction rate vector of the solid phase or the empty phase, and then calculate the additional element density vector. (3) Based on the additional unit density vector, the minimum size constraint function of the solid phase and its rounded or empty phase and its rounded is constructed by using the principle of additional volume constraint. 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. The MMA algorithm is used to iteratively solve the topology optimization model to obtain the optimal topology configuration. When the topological grayscale value of the design domain reaches a predetermined value Modify the relative volume fraction coefficients corresponding to the mesh elements. The formula for calculating the topological grayscale value is: In the formula, For physical variable field vectors; Indicates the total number of grid cells within the design domain; When unit i Density indication and minimum size satisfaction rate When both are equal to 1, the minimum size constraint is fully satisfied at that element, and the product of the two is used as the element. i Additional unit density , ; The constraint function is The corresponding formula is: In the formula, The penalty term is an integer not less than 1; Represents the total volume of the solid or open phase; This represents the error term indicating the size constraint of the solid or open phase. Let i be the volume of unit i.

2. The topology optimization method for applying structure and controlling the minimum size of its rounding as described 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 or empty phase, so that all elements satisfy the minimum size constraint.

3. The topology optimization method for applying structure and controlling the minimum size of its rounding as described in claim 1, characterized in that: The formula for calculating the sensitivity of the constraint function relative to the design variables is: In the formula, For design variables, Density indicator vector, The minimum size satisfaction rate vector for solid or unused phases. For the additional unit density vector, This is the filter vector after the initial density filtering. The volume vector represents the mesh element volume within the design domain; the proportion of solid material within the filter circle that satisfies the minimum size requirement is the relative volume fraction coefficient, expressed as... express, , These are the initial step projection steepness and the truncation threshold, respectively. , These are the second step projection steepness and the truncation threshold, respectively. For the first density filtration Weight matrix, for Summing by column vector; Indicates the second filtering Weight matrix, for Summing by column vector.

4. The topology optimization method for applying structure and controlling the minimum size of its rounding as described in claim 1, characterized in that: Steepness of the first step projection Second step projection truncation threshold Minimum size constraint error term for solid phase or open phase Perform parameter extension.

5. The topology optimization method for applying structure and controlling the minimum size of its rounding as described in claim 4, characterized in that: During the iteration process, the steepness of the first step projection will be... Second step projection truncation threshold Solid phase or open phase size constraint error item and As the number of iterations increases With the increase of , the parameter extension is performed in the form of a linear or exponential function, and the corresponding formula is: in, Indicates to The number of steps between updates; Indicates to The number of steps between updates; Indicates to The number of steps between updates; Indicates the second step projection cutoff threshold initial value, Indicates the size constraint error term of the solid phase or the unused phase. and The minimum value.

6. The topology optimization method for applying structural and rounding minimum size control as described in claim 1, characterized in that: The objective function and its sensitivity of the topology optimization model, as well as all constraint functions and their sensitivities, are substituted into the MMA algorithm to solve the optimization problem, thereby achieving iterative updating of design variables.

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