Structure dual-scale topological optimization method and device based on basic characteristic frequency

Abstract

The invention belongs to the related technical field of structural topological optimization, and discloses a structural dual-scale topological optimization method and device based on basic characteristic frequency, and the method comprises the following steps: (1) obtaining an overall stiffness matrix and an overall mass matrix based on an elastic matrix and a volume fraction of an actual microstructure at any cutting height, then obtaining a basic characteristic frequency through a topological optimization model based on the obtained overall stiffness matrix and overall mass matrix; wherein the design variable of the topological optimization model is the cutting height of each grid unit, the target function is the basic characteristic frequency maximization of the dual-scale structure, and the constraint function is the volume fraction of the overall structure; and (2) calculating the sensitivity of the basic characteristic frequency to the cutting height, and updating a design variable based on a sensitivity result obtained by calculation so as to obtain an optimal dual-scale structure. According to the invention, the efficiency of optimization iteration is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field related to structural topology optimization, and more specifically, relates to a structural dual-scale topology optimization method and device based on basic characteristic frequencies. Background Art

[0002] The optimization method of the basic characteristic frequency of the structure is to consider keeping the characteristic frequency of the mechanical structure away from the frequency of the external excitation during the structural optimization process to avoid the occurrence of resonance. For structures that are forced to vibrate, it is assumed that the spatial distribution mode of the external excitation is non-orthogonal to the vibration mode corresponding to the natural frequency of the structure. When the frequency of the external excitation is close to the characteristic frequency of the structure, the amplitude of the structure will be drastically amplified. This is the resonance phenomenon. Avoiding resonance is one of the important goals of structural vibration design. The demand for avoiding resonance when designing and optimizing structures is growing in the engineering field, especially in the fields of construction, automobiles, and electronic equipment.

[0003] There are many methods available for optimizing the basic eigenfrequency of structures, such as solid isotropic material penalty method (SIMP), evolutionary structural optimization method (ESO), level set method (LSM), mobile deformable component method (MMC), etc. Most methods focus on macro-scale design. For the optimization of the natural frequency of engineering structures, dual-scale design has been proven to have more obvious advantages such as light weight and high strength. The dual-scale topology optimization method can obtain better optimization results, but its optimization efficiency is usually not satisfactory. Summary of the invention

[0004] In view of the above defects or improvement needs of the prior art, the present invention provides a structural dual-scale topology optimization method and device based on basic characteristic frequencies, which aims to solve the problem of low optimization efficiency of the structural basic characteristic frequencies of mechanical structures.

[0005] To achieve the above object, according to one aspect of the present invention, a structural dual-scale topology optimization method based on basic eigenfrequency is provided, the method comprising the following steps:

[0006] (1) obtaining an overall stiffness matrix and an overall mass matrix based on the elastic matrix and volume fraction of the actual microstructure at any cutting height, and then obtaining a basic eigenfrequency through a topology optimization model based on the obtained overall stiffness matrix and overall mass matrix; wherein the design variable of the topology optimization model is the cutting height of each grid unit, the objective function is to maximize the basic eigenfrequency of the dual-scale structure, and the constraint function is the volume fraction of the overall structure; performing Boolean operations on multiple virtual microstructures of the grid unit to obtain the actual microstructure, wherein the virtual microstructure is obtained by cutting different level set functions at different cutting heights;

[0007] (2) Calculate the sensitivity of the basic characteristic frequency to the cutting height, update the design variables based on the calculated sensitivity results, and then obtain the optimal dual-scale structure.

[0008] Furthermore, Boolean operations are performed on multiple virtual unit microstructures corresponding to each grid unit in the design domain of the structure to be optimized to obtain the actual microstructure, and the elastic matrix and volume fraction of the actual microstructure corresponding to all grid units at equal cutting height intervals are calculated to form an offline database.

[0009] Furthermore, the elasticity matrix and volume fraction of the actual microstructure in the offline database are interpolated to obtain the elasticity matrix and volume fraction of the actual microstructure at any cutting height.

[0010] Furthermore, the compactly supported radial basis function is used to interpolate the elastic matrix and volume fraction of the actual microstructure in the offline database. The corresponding formula is:

[0011]

[0012] φ(r q )=max{0,(1-r q ) 4}(4r q +1)

[0013]

[0014] in, are the elements in the elastic matrix, m=1,2,3,n=1,2,3,V e is the unit volume fraction, q represents different center points, q = 1, ..., R, R is the number of center points selected during radial basis function interpolation; φ(r q ) is the radial basis function, β q is the radial basis function φ(r q ) weight coefficient, φ′(r q ) and β′ q are the radial basis function and weight coefficient of the interpolation function of the volume fraction, respectively; is the distance from the interpolation point to the center point, where H q is the position of the center point of the interpolation function, is the design variable, i is the number of the selected virtual microstructure prototype, i = 1, ..., N, N is the total number of selected virtual microstructure prototypes, e is the number of the structure division grid unit, e = 1, ..., L, L is the total number of structure division grid units; σ is the free shape parameter, d s A custom constant.

[0015] Furthermore, the virtual unit microstructure is obtained by first using the level set function and cutting function Cut each grid unit, and obtain the combination function corresponding to each grid unit after cutting Then calculate the combined function If the combined function is negative, it is a solid area, otherwise it is a blank area.

[0016] Furthermore, the mathematical expression of the topology optimization model is:

[0017]

[0018] In the formula, ω j is the objective function, i.e., the jth order characteristic frequency of the structure, j, k represents different vibration modes, and the characteristic value is the square of the j-th eigenfrequency of the structure, K is the overall stiffness matrix, M is the overall mass matrix, is the j-th eigenmode, δ jk is the Dirac function, h min and h max are the lower and upper limits of the cutting height, V is the actual volume fraction of the structure, is the volume constraint.

[0019] Furthermore, the moving asymptote method is used to update the design variables based on the calculated sensitivity results.

[0020] Furthermore, the sensitivity calculation formula is:

[0021]

[0022] Among them, ω j is the objective function, j represents different vibration modes and eigenvalues is the square of the j-th eigenfrequency of the structure, is the design variable, i is the number of the selected virtual microstructure prototype, i=1,…,N, N is the total number of selected virtual microstructure prototypes, e is the number of the structure division grid unit, e=1,…,L, L is the total number of the structure division grid units.

[0023] The present invention also provides a structural dual-scale topology optimization system based on basic eigenfrequency, the system comprising a memory and a processor, the memory storing a computer program, and the processor executing the structural dual-scale topology optimization method based on basic eigenfrequency as described above when executing the computer program.

[0024] 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 structural dual-scale topology optimization method based on basic characteristic frequencies as described above.

[0025] In general, compared with the prior art, the above technical solution conceived by the present invention, the structural dual-scale topology optimization method and device based on basic characteristic frequency provided by the present invention mainly have the following beneficial effects:

[0026] 1. Taking the maximization of the basic characteristic frequency of the dual-scale structure as the optimization goal, the cutting height of each grid unit as the design variable, and the volume fraction of the overall structure as the constraint condition, the elastic matrix and volume fraction of the actual microstructure are solved at the same time, and then the dual-scale topology optimization is realized through a data-driven method to significantly improve the efficiency of optimization iterations.

[0027] 2. The present invention improves the calculation formula of sensitivity, specifically: converting the characteristic frequency maximization problem into the reciprocal characteristic frequency minimization problem, so as to better adapt to the optimization of the moving asymptote method.

[0028] 3. The compactly supported radial basis function is used for interpolation, which makes the optimization result smoother and the interpolation calculation faster, resulting in better interpolation results, thereby improving the accuracy of structural optimization.

[0029] 4. The microstructure was used to replace the intermediate density unit of the traditional SIMP method. The optimized dual-scale structure has better connectivity and obtains the corresponding optimal material distribution results. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 It is a structural schematic diagram of a dual-scale structural optimization design of a cantilever beam constructed in an embodiment of the present invention;

[0031] Figure 2 (a), (b), (c), and (d) are the microstructure models of the four virtual microstructures in each unit of the dual-scale structure when the cutting height is 0;

[0032] Figure 3 is a flow chart of a structural dual-scale topology optimization method based on basic characteristic frequencies provided by an embodiment of the present invention;

[0033] Figure 4 It is an initial design diagram of the dual-scale structure optimization constructed by the preferred embodiment of the present invention;

[0034] Figure 5It is an optimized structural diagram of a dual-scale structure constructed by a preferred embodiment of the present invention with the first-order natural frequency as the objective function. DETAILED DESCRIPTION

[0035] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain 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 can be combined with each other as long as they do not conflict with each other.

[0036] The present invention provides a structural dual-scale topology optimization method based on basic eigenfrequency. The method takes maximization of the basic eigenfrequency of the dual-scale structure as the optimization goal, takes the cutting height of each unit as the design variable, takes the volume fraction of the overall structure as the constraint condition, and uses the moving asymptotic method to update the design variables until the optimization termination condition is met, thereby obtaining the optimal dual-scale structure.

[0037] The topology optimization method mainly includes the following steps:

[0038] Step 1: interpolate the elastic matrix and volume fraction of the actual microstructure in the offline database to obtain the elastic matrix and volume fraction of the actual microstructure at any cutting height; at the same time, define a structural dual-scale topology optimization problem based on the basic eigenfrequency to obtain a topology optimization model, wherein the design variable of the topology optimization model is the cutting height of each grid unit, the objective function is to maximize the basic eigenfrequency of the dual-scale structure, and the constraint function is the volume fraction of the overall structure;

[0039] Wherein, the acquisition of the offline database includes the following steps:

[0040] S1, mesh the design domain of the structure to be optimized to obtain multiple mesh units, and then use different cutting heights to cut different level set functions to obtain virtual unit microstructures.

[0041] The virtual unit microstructure is performed in the following manner: first, the level set function is used and cutting function Cut each grid unit, and obtain the combination function corresponding to each grid unit after cutting Then calculate the combined function If the combined function is negative, it is a solid area, that is, a virtual microstructure; otherwise, it is a blank area.

[0042] S2, Boolean operations are performed on multiple virtual unit microstructures corresponding to each grid unit to obtain the actual microstructure, and the elastic matrix and volume fraction of the actual microstructure corresponding to all grid units at equal cutting height spacing are calculated to form an offline database.

[0043] The actual microstructure is calculated according to the following formula:

[0044] Wherein, x is the element of the grid unit, i is the number of the selected virtual microstructure prototype, i=1,…,N, N is the total number of selected virtual microstructure prototypes, e is the number of the structure division grid unit, e=1,…,L, L is the total number of structure division grid units, is the level set function, is the cutting function, is a combination function.

[0045] In this embodiment, a compactly supported radial basis function is used to interpolate the elastic matrix and volume fraction of the actual microstructure in the offline database, and the corresponding formula is:

[0046]

[0047]

[0048] φ(r q )=max{0,(1-r q ) 4}(4r q +1)

[0049]

[0050] in, are the elements in the elastic matrix, m=1,2,3,n=1,2,3,V e is the unit volume fraction, q represents different center points, q = 1, ..., R, R is the number of center points selected during radial basis function interpolation; φ(r q ) is the radial basis function, β q is the radial basis function φ(r q ) weight coefficient, φ′(r q ) and β′ q are the radial basis function and weight coefficient of the interpolation function of the volume fraction, respectively; is the distance from the interpolation point to the center point, where H q is the position of the center point of the interpolation function, is the design variable, i is the number of the selected virtual microstructure prototype, i = 1, ..., N, N is the total number of selected virtual microstructure prototypes, e is the number of the structure division grid unit, e = 1, ..., L, L is the total number of structure division grid units; σ is the free shape parameter, d s A custom constant.

[0051] The mathematical expression of the topology optimization model is:

[0052]

[0053] In the formula, ω j is the objective function, i.e., the jth order characteristic frequency of the structure, j, k represents different vibration modes, and the characteristic value is the square of the j-th eigenfrequency of the structure, K is the overall stiffness matrix, M is the overall mass matrix, is the j-th eigenmode, δ jk is the Dirac function, h min and h max are the lower and upper limits of the cutting height, V is the actual volume fraction of the structure, is the volume constraint.

[0054] Step 2: Based on the elastic matrix and volume fraction of the actual microstructure at any cutting height, the overall stiffness matrix and the overall mass matrix are obtained through finite element analysis, and then the characteristic frequency and characteristic vector are obtained through the topology optimization model based on the obtained overall stiffness matrix and overall mass matrix.

[0055] Among them, the overall stiffness matrix K is composed of the unit stiffness matrix k of the actual microstructure e Assemble, the overall mass matrix M is composed of the unit mass matrix m of the actual microstructure e Assembled:

[0056]

[0057] Among them, D e is the unit elastic matrix, B is the unit strain matrix, V e is the unit volume fraction, is the consistent mass matrix, ρ is the material density, A is the area of ​​the unit, and t is the thickness of the unit.

[0058] Using the formula in the mathematical expression of the topology optimization model Solve for the characteristic frequency ω j and the eigenvector Using the formula The eigenvectors are orthogonally normalized to complete the solution of eigenfrequency and eigenvector.

[0059] Step three, calculate the sensitivity of the basic characteristic frequency to the cutting height, and update the design variables based on the calculated sensitivity results to obtain a new optimized structure.

[0060] In order to adapt to the optimization of the moving asymptote method, the characteristic frequency maximization problem is converted into the reciprocal characteristic frequency minimization problem, and the sensitivity is calculated according to the following formula:

[0061]

[0062] Among them, ω j is the objective function, j represents different vibration modes and eigenvalues is the square of the j-th eigenfrequency of the structure, is the design variable, i is the number of the selected virtual microstructure prototype, i=1,…,N, N is the total number of selected virtual microstructure prototypes, e is the number of the structure division grid unit, e=1,…,L, L is the total number of the structure division grid units.

[0063] The eigenvalue λ j For design variables The derivative of is calculated according to the following formula:

[0064]

[0065] Among them, the eigenvalue is the square of the j-th eigenfrequency of the structure, is the j-th eigenmode, where j represents different vibration modes; is the design variable, i is the number of the selected virtual microstructure prototype, i = 1, ..., N, N is the total number of selected virtual microstructure prototypes, e is the number of the structure division grid unit, e = 1, ..., L, L is the total number of structure division grid units; K is the overall stiffness matrix, M is the overall mass matrix, k e is the element stiffness matrix, m e is the unit mass matrix, which can be expressed as:

[0066] k e =B T D e Bt e

[0067]

[0068] Among them, D e is the unit elastic matrix, B is the unit strain matrix, V e is the unit volume fraction, is the consistent mass matrix, ρ is the material density, A is the area of ​​the unit, and t is the thickness of the unit. e Only D eRelated to the design variables, m e Only V e The key to the sensitivity of the objective function to the design variables is the elasticity matrix D e and volume fraction V e For design variables The derivative of .

[0069] The elasticity matrix D e and volume fraction V e The derivatives with respect to the design variables can be calculated as follows:

[0070] Elasticity Matrix D e For the design variables The partial derivative of can be expressed as:

[0071]

[0072] Volume fraction V e For the design variables The partial derivative of can be expressed as:

[0073]

[0074] Among them, φ is the radial basis function, β q is the weight coefficient of the radial basis function φ, which is independent of the design variables. φ′ and β q ′ are radial basis functions and weight coefficients of the interpolation function of the volume fraction respectively; q represents different center points, q=1,…,R, R is the number of center points selected by the radial basis interpolation function.

[0075] Radial basis function φ for design variables The partial derivatives of can be transformed into the derivatives of the radial basis function φ with respect to the distance function r and the derivatives of the distance function r with respect to the design variables Combinations of derivatives:

[0076]

[0077] Distance function for design variables The partial derivative of can be expressed as:

[0078]

[0079] Among them, H q is the position of the center point of the interpolation function, is the design variable, i is the number of the selected virtual microstructure prototype, i = 1, ..., N, N is the total number of selected virtual microstructure prototypes, e is the number of the structural division unit, e = 1, ..., L, L is the total number of structural division units; σ is the free shape parameter, d s A custom constant.

[0080] Step 4: Determine whether the characteristic frequency error of the new optimized structure meets the preset termination condition. If yes, the optimization is completed and a dual-scale structure that meets the requirements is obtained; otherwise, go to step 1.

[0081] The characteristic frequency error is calculated according to the following formula:

[0082]

[0083] Among them, ω err is the error of the objective function, ω k-p+1 It represents the value of the objective function at the k-p+1th iteration, where k is the current iteration number. To obtain the values ​​of the objective function for the last ten times, set p=1, 2, 3, 4, 5.

[0084] The present invention also provides a structural dual-scale topology optimization system based on basic eigenfrequency, the system comprising a memory and a processor, the memory storing a computer program, and the processor executing the structural dual-scale topology optimization method based on basic eigenfrequency as described above when executing the computer program.

[0085] 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 structural dual-scale topology optimization method based on basic characteristic frequencies as described above.

[0086] The present invention is further described in detail below with reference to specific embodiments.

[0087] like Figure 1 As shown, this embodiment takes the basic characteristic frequency optimization problem of a cantilever beam structure with concentrated mass as an example to explain the present invention. An initial dual-scale structure is set in a given 2m×1m rectangular design domain D, with a thickness of 0.01m, the horizontal and vertical degrees of freedom in the left area are fixed, and a concentrated mass is applied at the midpoint of the right end, and the Young's modulus E of the solid material is set s =200GPa, mass density ρ s =7.8×10 3 Kg / m 3 , Poisson's ratio ν = 0.3; Young's modulus E of weak material w =200×10 -3 GPa, mass density ρ w =7.8×10 -2 Kg / m 3 , Poisson's ratio ν = 0.3; concentrated mass density is 100×ρ s .

[0088] like Figure 3 As shown, the present embodiment provides a structural dual-scale topology optimization method based on basic characteristic frequencies, comprising the following steps:

[0089] S1: The design domain of the structure to be optimized is meshed to obtain multiple mesh cells, and the virtual unit microstructure corresponding to each mesh cell under different level set functions and cutting height divisions is calculated.

[0090] Specifically, the design domain Ω is divided into 80×40 square unit grids, and the corresponding unit number n is 3200; each grid unit is divided into 40×40 grids, and the level set function is established. The four virtual microstructure prototypes selected in this embodiment are as follows: Figure 2 shown.

[0091] S2: Boolean operations are performed on the obtained multiple virtual unit microstructures to obtain the actual microstructure. Based on the unit density, the actual microstructure corresponding to all grid units at equal cutting height spacing and their corresponding elastic matrix and volume fraction are calculated by a homogenization method.

[0092] Specifically, the cutting height is defined as 0.1, the cutting range is from -1.2 to 1.2, and 25 × 25 × 25 × 25 groups of actual microstructures and their corresponding elastic matrices and volume fractions are obtained through the homogenization method to form an offline database.

[0093] S3: Interpolate the elastic matrix and volume fraction in the offline database to obtain the elastic matrix and volume fraction corresponding to the actual microstructure at any cutting height.

[0094] Specifically, the number of radial basis function interpolation center points is selected to be 81, the free shape parameter σ in the radial basis function is set to 0.05, and the custom constant d s It is set to 2, and the interpolation function is used to obtain the elastic matrix and volume fraction corresponding to the actual microstructure at any cutting height.

[0095] S4: Define the optimization design problem of the basic characteristic frequency of the dual-scale structure to obtain an optimization model: the design variable of the optimization model is the cutting height of each unit, the objective function is to maximize the basic characteristic frequency of the dual-scale structure, and the constraint function is the volume fraction of the overall structure.

[0096] Specifically, the basic characteristic frequency under volume constraint is solved with cutting height as the design variable, and the mathematical expression of the optimization model is:

[0097]

[0098] Where i represents the number of virtual microstructure prototypes. In this example, four virtual microstructure prototypes are selected. j represents different vibration modes. In this example, the first-order natural frequency is selected. hmin and h max are the lower and upper limits of the cutting height, which are -1.5 and 1.5 respectively; is the volume constraint, set to 0.5; the initial design of the structure is as follows Figure 4 shown.

[0099] S5: obtaining an overall stiffness matrix and an overall mass matrix through finite element analysis, and then obtaining the characteristic frequency and characteristic vector of the structure based on the optimization model.

[0100] Specifically, the unit stiffness matrix and unit mass matrix are calculated by combining the elastic matrix and volume fraction of the unit at any cutting height with the unit strain matrix and consistent mass matrix, and then the overall stiffness matrix and the overall mass matrix are obtained by assembling the unit stiffness matrix and the unit mass matrix. Considering the basic eigenfrequency problem under the volume constraint in S4, the eigenfrequency and eigenvector of the entire structure are calculated and normalized.

[0101] S6: Calculate the sensitivity of the objective function, i.e., the characteristic frequency, to the cutting height of the design variable, and then update the design variables based on the moving asymptote algorithm to obtain a new optimized structure.

[0102] Specifically, the design variables are updated by combining the moving asymptote method The moving asymptote method is a commonly used nonlinear optimization algorithm that uses updated design variables to represent the distribution of the actual microstructure of each unit.

[0103] S7: When the updated characteristic frequency error meets the preset termination condition, the optimization analysis is completed and a dual-scale structure that meets the requirements is obtained; otherwise, go to step S3.

[0104] Specifically, the optimization termination condition is set as:

[0105]

[0106] Among them, ω err is the error of the objective function, ω k-p+1 It represents the value of the objective function at the k-p+1th iteration, k is the current iteration number, and ε is the threshold of the objective function error, which is 0.5% here.

[0107] The initial design of the dual-scale structure is as follows Figure 4 As shown in Figure 2, the first-order natural frequency value is 89.2347 Hz; the optimized dual-scale structure is shown in Figure 2 Figure 5 As shown in Figure 1, the first-order natural frequency is 166.8620 Hz. By comparison, the first-order natural frequency of the dual-scale structure after optimization increases by about 86.70%. Figure 5The optimization results shown have clear and definite structural boundaries. At the same time, the microstructure replaces the intermediate density unit of the traditional SIMP method. The optimized dual-scale structure has good connectivity and obtains the corresponding optimal material distribution results.

[0108] It will be easily understood by those skilled in the art that the above description is only 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 protection scope of the present invention.

Claims

1. A structural dual-scale topology optimization method based on basic eigenfrequency, characterized by: The method comprises the following steps: (1) obtaining an overall stiffness matrix and an overall mass matrix based on the elastic matrix and volume fraction of the actual microstructure at any cutting height, and then obtaining a basic eigenfrequency through a topology optimization model based on the obtained overall stiffness matrix and overall mass matrix; wherein the design variable of the topology optimization model is the cutting height of each grid unit of the structure to be optimized, the objective function is to maximize the basic eigenfrequency of the dual-scale structure, and the constraint function is the volume fraction of the overall structure; performing Boolean operations on multiple virtual microstructures of the grid unit to obtain the actual microstructure, wherein the virtual microstructure is obtained by cutting different level set functions of the grid unit at different cutting heights; (2) Calculate the sensitivity of the basic characteristic frequency to the cutting height, update the design variables based on the calculated sensitivity results, and then obtain the optimal dual-scale structure.

2. The structural dual-scale topology optimization method based on basic characteristic frequency according to claim 1, characterized in that: Boolean operations are performed on multiple virtual unit microstructures corresponding to each grid unit in the design domain of the structure to be optimized to obtain the actual microstructure, and the elastic matrix and volume fraction of the actual microstructure corresponding to all grid units at equal cutting height spacing are calculated to form an offline database.

3. The structural dual-scale topology optimization method based on basic characteristic frequency according to claim 2, characterized in that: The elastic matrix and volume fraction of the actual microstructure in the offline database are interpolated to obtain the elastic matrix and volume fraction of the actual microstructure at any cutting height.

4. The structural dual-scale topology optimization method based on basic characteristic frequency according to claim 3, characterized in that: The compactly supported radial basis function is used to interpolate the elastic matrix and volume fraction of the actual microstructure in the offline database. The corresponding formula is: in, are the elements in the elastic matrix, m=1,2,3,n=1,2,3,V e is the unit volume fraction, q represents different center points, q = 1, ..., R, R is the number of center points selected during radial basis function interpolation; φ(r q ) is the radial basis function, βq is the radial basis function φ(r q ) weight coefficient, φ′(r q ) and βq′ are the radial basis function and weight coefficient of the interpolation function of the volume fraction, respectively; is the distance from the interpolation point to the center point, where H q is the position of the center point of the interpolation function, is the design variable, i is the number of the selected virtual microstructure prototype, i = 1, ..., N, N is the total number of selected virtual microstructure prototypes, e is the number of the structure division grid unit, e = 1, ..., L, L is the total number of structure division grid units; σ is the free shape parameter, d s A custom constant.

5. The structural dual-scale topology optimization method based on basic characteristic frequency according to claim 2, characterized in that: The virtual unit microstructure is obtained by first using the level set function and cutting function Cut each grid unit, and obtain the combination function corresponding to each grid unit after cutting Then calculate the combined function If the combined function is negative, it is a solid area, that is, a virtual microstructure; otherwise, it is a blank area.

6. The structural dual-scale topology optimization method based on basic characteristic frequency according to any one of claims 1 to 5, characterized in that: The mathematical expression of the topology optimization model is: In the formula, ω j is the objective function, i.e., the jth order characteristic frequency of the structure, j, k represents different vibration modes, and the characteristic value is the square of the j-th eigenfrequency of the structure, K is the overall stiffness matrix, M is the overall mass matrix, is the j-th eigenmode, δ jk is the Dirac function, h min and h max are the lower and upper limits of the cutting height, V is the actual volume fraction of the structure, and V is the volume constraint.

7. The structural dual-scale topology optimization method based on basic characteristic frequency according to any one of claims 1 to 5, characterized in that: The moving asymptote method is used to update the design variables based on the calculated sensitivity results.

8. The structural dual-scale topology optimization method based on basic eigenfrequency according to any one of claims 1 to 5, characterized in that: The sensitivity calculation formula is: Among them, ω j is the objective function, j represents different vibration modes and eigenvalues is the square of the j-th eigenfrequency of the structure, is the design variable, i is the number of the selected virtual microstructure prototype, i=1,…,N, N is the total number of selected virtual microstructure prototypes, e is the number of the structure division grid unit, e=1,…,L, L is the total number of the structure division grid units.

9. A structural dual-scale topology optimization system based on basic eigenfrequency, characterized by: The system includes a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, the structural dual-scale topology optimization method based on basic eigenfrequency according to any one of claims 1 to 8 is executed.

10. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores machine-executable instructions. When the machine-executable instructions are called and executed by the processor, the machine-executable instructions prompt the processor to implement the structural dual-scale topology optimization method based on basic characteristic frequency as described in any one of claims 1-8.

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