A three-way stiffness customizable metamaterial vibration isolator topology optimization design method

Abstract

The application discloses a kind of three-way stiffness customizable metamaterial vibration isolator topology optimization design methods, first, based on the structural design idea of "cell-whole", the overall structure of vibration isolator is divided into multiple cell array arrangement configuration, the corresponding relationship between the overall structure three-way stiffness of vibration isolator and cell three-way stiffness is established;Then, create cell topology optimization design domain, use the structure topology optimization design method based on variable density to design and optimize cell three-way stiffness, obtain the optimal configuration of cell;Finally, according to the array arrangement mode of cell, its array modeling is carried out, and the overall structure of vibration isolator meeting the requirements of three-way stiffness is obtained.The application utilizes the characteristics of high degree of freedom of variable density topology optimization method, in the face of different application requirements, can design excellent performance new vibration isolator cell configuration, widen vibration isolator three-way stiffness design range, improve equipment vibration reduction effect.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of metamaterials, vibration and noise reduction, and structure design, and particularly relates to a topological optimization design method of a metamaterial vibration isolator with customizable three-direction stiffness. BACKGROUND

[0002] Mechanical vibration is a main noise source in large and complex equipment systems, and vibration and noise not only affect the operation state and performance of mechanical equipment, but also seriously harm the physical and mental health of workers. In particular, for important equipment such as underwater ships, vibration and noise are one of the main factors affecting the concealment performance. Therefore, vibration and noise reduction plays a crucial role in complex equipment systems.

[0003] As a common vibration isolation means, the vibration isolator can effectively reduce or eliminate the vibration transmitted by the equipment, and has a good vibration and noise reduction effect. In the design of a traditional vibration isolator, steel springs, rubber, cork, and felt are often used for vibration isolator structure design. However, the mechanical properties of these base materials are fixed, which limits the application range of the designed vibration isolator. In order to solve this problem, in the design of a traditional vibration isolator, different vibration isolator base materials need to be repeatedly selected according to different design requirements or changes in requirements. Each design needs to go through the stages of material selection, design, theoretical verification, and experimental verification. The adjustment of the design will also waste a lot of research and development resources and increase the research and development cost.

[0004] As a new vibration and noise reduction technology, the metamaterial vibration isolation technology can achieve the performance that natural materials cannot have through the design and optimization of the structure. When the application requirements change, the metamaterial only needs to be structurally adjusted to meet the new application requirements, thereby avoiding the complex research and development process of material selection, design, and verification of the traditional vibration isolator, greatly shortening the research and development cycle and reducing the research and development cost. However, the existing metamaterial vibration isolator unit cell configuration is relatively fixed and can only be optimized and adjusted through parameter optimization. The fixed unit cell configuration limits the designable range of the three-direction stiffness of the vibration isolator on the one hand, and on the other hand, in some harsh application scenarios, the internal strain may be too large, which causes the vibration isolator to be unable to be normally used. SUMMARY

[0005] Therefore, the application provides a topological optimization design method of a metamaterial vibration isolator with customizable three-direction stiffness, which utilizes the high design freedom of the variable density topological optimization method to design a new type of vibration isolator unit cell configuration with excellent performance in the face of different application requirements, widens the designable range of the three-direction stiffness of the vibration isolator, and improves the vibration and noise reduction effect of the equipment.

[0006] The application is implemented by the following technical solutions:

[0007] A three-directional stiffness customizable metamaterial vibration isolator topology optimization design method, the vibration isolator adopting metamaterial is combined by a plurality of unit cells in a set array arrangement form; the design method is as follows: firstly, based on the structure design idea of "unit cell-whole", the whole structure of the vibration isolator is divided into a plurality of unit cell array arrangement configurations, and the corresponding relationship between the whole structure three-directional stiffness of the vibration isolator and the three-directional stiffness of the unit cell is established; then, the unit cell topology optimization design domain is created, the three-directional stiffness of the unit cell is designed and optimized by using the variable density-based structure topology optimization design method, and the optimal configuration of the unit cell is obtained; finally, the array modeling is performed on the unit cell according to the array arrangement mode of the unit cell, and the whole structure of the vibration isolator with the three-directional stiffness meeting the requirements is obtained.

[0008] Further, the specific steps of the design method are as follows:

[0009] Step 1: establish an XYZ coordinate system, determine the overall design space size of the vibration isolator, and determine that the main bearing direction of the vibration isolator is Z direction, and determine the three-directional stiffness design target value K i，total of the vibration isolator, K represents stiffness, i=X, Y, Z represents three directions of the vibration isolator, and total represents the whole structure of the vibration isolator; that is, K X，total represents the X-directional stiffness design target value of the vibration isolator, K Y，total represents the Y-directional stiffness design target value of the vibration isolator, and K Z，total represents the Z-directional stiffness design target of the vibration isolator.

[0010] Step 2: set the unit cell configuration and its array arrangement form, and determine the three-directional stiffness design target value of the unit cell according to the corresponding relationship between the unit cell stiffness and the whole stiffness of the vibration isolator.

[0011] A plurality of unit cells are arranged along the X direction, the Y direction and the Z direction to form the vibration isolator, and the number of unit cells along the X direction, the Y direction and the Z direction is m, n and o respectively, so the three-directional stiffness design target value of the unit cell is:

[0012]

[0013] Wherein, cell represents the unit cell, K i，cell represents the three-directional stiffness design target value of the unit cell, when i=X, K X，cell represents the X-directional stiffness design target value of the unit cell, when i=Y, K Y，cell represents the Y-directional stiffness design target value of the unit cell, when i=Z, K Z，cell represents the Z-directional stiffness design target value of the unit cell.

[0014] Step 3: create the design domain Ω of the unit cell, divide the finite element grid, set the design variable, the objective function, the constraint condition and the convergence criterion.

[0015] Wherein, the design variable is the relative density of each grid element of the finite element grid; the objective function is the total strain energy inside the unit cell; the constraint condition is the three-dimensional stiffness design target value of the unit cell and the volume fraction or mass fraction of the material occupying the design space; therefore, the design problem of the unit cell can be mathematically described as:

[0016]

[0017] Wherein, ρ e is the design variable, e is the grid element number of the finite element grid, n is the total number of grid elements in the design domain Ω, F j represents the objective function, σ is the stress, ε is the strain, K i is the three-dimensional stiffness calculation value of the unit cell in the optimization process, K i，cell is the three-dimensional stiffness design target value of the unit cell, V f is the volume fraction of the material occupying the design domain, V0 is a constant, representing the upper limit of the material volume fraction;

[0018] The convergence criterion is:

[0019]

[0020] Wherein, k is the current iteration step number, α is a constant, and the value range is 10 -6 ~ 10 -3 ;

[0021] Step 4: Filtering the design variable ρ e to obtain the design variable After filtering the design variable , projection is carried out to obtain the design variable , and the material properties are interpolated and penalized;

[0022] Step 5: According to the projected design variable , the material properties obtained by interpolation and penalty, set the finite element simulation calculation model of the unit cell, and according to the stiffness design requirements of the main bearing direction and the non-bearing direction, the three-dimensional load F i of the unit cell is set, and the finite element simulation calculation model of the unit cell is simulated and solved by using the finite element simulation method of structural mechanics;

[0023] Step 6: According to the finite element simulation results, the three-dimensional stiffness calculation value K i of the unit cell is calculated:

[0024]

[0025] Wherein, i=X, Y, Z represents the three directions of the vibration isolator, F i represents the load size in the i direction, d irepresents the displacement size of the load-bearing boundary in the i direction, avg represents the average value;

[0026] According to the finite element simulation results, the objective function F is calculated according to formula (2) in step 3 j , and the sensitivity of the objective function with respect to the design variable

[0027] Step 7: According to the sensitivity in the design domain obtained in step 6 , the design variable p of step 3 is updated using the optimization solver e ;

[0028] Step 8: The updated design variable p obtained in step 7 e and the calculated three-dimensional stiffness value K of the unit cell obtained in step 6 i are substituted into formula (2) and formula (3) in step 3, and it is judged whether the constraint condition and the convergence criterion are met according to the constraint condition and the convergence criterion set in step 3; if yes, the topology optimization iteration is ended, and step 9 is executed; if no, steps 4-8 are repeated until the constraint condition and the convergence criterion are met, and step 9 is executed;

[0029] Step 9: The optimization result is extracted using any contour or isosurface of the projected design variable , and the optimal topology configuration of the unit cell is obtained;

[0030] Step 10: Array modeling is performed on the unit cell configuration according to the array arrangement form of the unit cell in step 2, and the three-dimensional overall structure of the vibration isolator is obtained.

[0031] Further, the three-dimensional overall structure of the vibration isolator obtained in step 10 is verified, and the verification method is as follows:

[0032] Step 11: The three-dimensional stiffness of the overall structure of the vibration isolator is analyzed again using the finite element simulation method of structural mechanics, and the three-dimensional loading force-displacement curve of the vibration isolator is drawn, and the actual value of the three-dimensional stiffness of the vibration isolator is obtained by analyzing the curve;

[0033] Step 12: The actual value of the three-dimensional stiffness of the vibration isolator obtained in step 11 is compared with the design target value K of the three-dimensional stiffness of the vibration isolator in step 1 i，total , and the deviation between the actual value of the three-dimensional stiffness of the vibration isolator and the design target value K of the three-dimensional stiffness of the vibration isolator is calculated i，total ; if the deviation is less than 1%, step 13 is executed, and if the deviation is greater than or equal to 1%, step 2 is returned to redesign the unit cell configuration and its arrangement form, until the deviation is less than 1%, and step 13 is executed;

[0034] Step 13: The final three-dimensional overall structure of the vibration isolator is output.

[0035] Further, in step 2, the single cell configuration includes but is not limited to tetrahedron, pentahedron, hexahedron and other polyhedral configurations.

[0036] Further, in step 4, the filtering method of the design variable includes but is not limited to using the Helmholtz equation:

[0037]

[0038] wherein r min is the filtering radius, and the value range is 1-3 times the grid size;

[0039] The formula for projecting the filtered design variable includes but is not limited to the hyperbolic tangent formula:

[0040]

[0041] wherein β is the control projection slope, and the value range is 6-10, γ β is the projection point, and the value range is 0.1-0.9;

[0042] The interpolation penalty of the properties of the metamaterial refers to using the projected design variable The interpolation penalty of the properties of the metamaterial refers to using the projected design variable

[0043] Further, in step 6, the method for calculating the sensitivity includes but is not limited to the adjoint method.

[0044] Further, in step 7, the available optimization solver includes but is not limited to SNOPT, GCMMA, MMA and IPOPT.

[0045] Beneficial effects:

[0046] (1) The present application provides a three-directional stiffness customizable metamaterial isolator topology optimization design method for the three-directional stiffness customized design problem of the isolator. First, based on the "single cell-whole" structure design idea, the whole structure of the isolator is divided into a plurality of single cell array arrangement configurations, and the corresponding relationship between the three-directional stiffness of the whole structure of the isolator and the three-directional stiffness of the single cell is established. Then, a single cell topology optimization design domain is created, and a variable density-based structure topology optimization design method is used to design and optimize the three-directional stiffness of the single cell to obtain the optimal configuration of the single cell. Finally, the single cell is array modeled according to the array arrangement method of the single cell to obtain the whole structure of the isolator with the three-directional stiffness meeting the requirements. The present method does not depend on the prior engineering experience, can design and obtain a new isolator configuration with excellent performance, widens the three-directional stiffness design range of the metamaterial isolator, and improves the equipment vibration isolation and noise reduction effect.

[0047] (2) the step 2 of the method determines the three-dimensional rigidity design target value of the unit cell according to the correspondence between the unit cell rigidity and the overall rigidity of the vibration isolator, converts the design problem of the overall structure of the vibration isolator into the design problem of the unit cell, reduces the design difficulty, reduces the design workload, shortens the research and development cycle, and ensures the unity and stability of the internal performance of the structure based on the "unit cell overall" design idea, thereby increasing the reliability of the structure;

[0048] Meanwhile, the updated design variable p e and the three-dimensional rigidity calculation value K i of the unit cell calculated in step 6 are substituted into the formula in step 3, and whether the constraint condition is met and the convergence criterion is reached is judged according to the constraint condition and the convergence criterion set in step 3; if yes, the topology optimization iteration is ended, and step 9 is performed; if no, steps 4-8 are repeated until the constraint condition is met and the convergence criterion is reached, and then step 9 is performed; based on the topology optimization, the design and optimization of the vibration isolator structure are carried out, which eliminates the dependence on the engineering experience of the design and research personnel compared with the traditional method, is beneficial to the development of new vibration isolator configurations with excellent performance, and solves the application demand that cannot be met by the traditional configuration.

[0049] (3) the method verifies the three-dimensional overall structure of the vibration isolator obtained in step 10, calculates the deviation between the three-dimensional rigidity actual value of the vibration isolator and the three-dimensional rigidity design target value of the vibration isolator, if the deviation is less than 1%, the final three-dimensional overall structure of the vibration isolator is output, if the deviation is greater than or equal to 1%, the unit cell configuration and its arrangement form are redesigned, and the final three-dimensional overall structure of the vibration isolator is output until the deviation is less than 1%; so that the three-dimensional overall structure of the vibration isolator obtained finally meets the three-dimensional rigidity requirement.

[0050] (4) the unit cell configuration of the method includes but is not limited to tetrahedron, pentahedron, hexahedron and other polyhedral configurations, which improves the applicability of the method. BRIEF DESCRIPTION OF DRAWINGS

[0051] Figure 1 The flowchart of the design method of the application.

[0052] Figure 2 The overall design space diagram of the vibration isolator;

[0053] Figure 3 The unit cell division and arrangement mode diagram;

[0054] Figure 4 The two-dimensional topology optimization configuration diagram of the unit cell.

[0055] Figure 5 The three-dimensional structure diagram of the unit cell.

[0056] Figure 6 Figure 1 is a schematic diagram of the overall structure of the vibration isolator.

[0057] Figure 7 Figure 2 is a schematic diagram of the force-displacement curve of the overall structure of the vibration isolator in the X direction and the Z direction.

[0058] In the figure, 1 is a vibration isolator, and 2 is a unit cell. DETAILED DESCRIPTION

[0059] The present application will be described in detail below with reference to the accompanying drawings and examples.

[0060] The present embodiment provides a topological optimization design method for a metamaterial vibration isolator with customizable three-directional stiffness. The vibration isolator using metamaterial is composed of a plurality of unit cells arranged in a set array form. The design method is as follows: first, based on the structure design idea of "unit cell-whole", the overall structure of the vibration isolator is divided into a plurality of unit cell array arrangement configurations, and the corresponding relationship between the overall structure three-directional stiffness of the vibration isolator and the unit cell three-directional stiffness is established; then, a unit cell topological optimization design domain is created, and a variable density-based structure topological optimization design method is used to design and optimize the unit cell three-directional stiffness to obtain the optimal configuration of the unit cell; finally, the unit cell is array modeled according to its array arrangement method to obtain the overall structure of the vibration isolator with three-directional stiffness meeting the requirements.

[0061] Referring to the accompanying drawings Figure 1 , the specific steps of the design method are as follows:

[0062] Step 1: Establish an XYZ coordinate system, determine the overall design space size of the vibration isolator according to the application requirements, determine the main bearing direction of the vibration isolator as the Z direction, and determine the three-directional stiffness design target value K i，total of the vibration isolator, where K represents the stiffness, i=X, Y, Z represents the three directions of the vibration isolator, and total represents the overall structure of the vibration isolator; that is, K X，total represents the X-direction stiffness design target value of the vibration isolator, K Y，total represents the Y-direction stiffness design target value of the vibration isolator, and K Z，total represents the Z-direction stiffness design target of the vibration isolator.

[0063] Step 2: Set the unit cell configuration and its array arrangement form, and determine the unit cell three-directional stiffness design target value according to the corresponding relationship between the unit cell stiffness and the overall stiffness of the vibration isolator.

[0064] Among them, the unit cell configuration includes but is not limited to the configurations of tetrahedron, pentahedron, hexahedron and other polyhedrons. For example, a cuboid, which is a hexahedron, is used as the unit cell configuration, a plurality of unit cells are arranged along the X direction, the Y direction and the Z direction to form the vibration isolator, and the number of unit cells in the X direction, the Y direction and the Z direction is m, n and o respectively. The three-directional stiffness design target value of the unit cell is:

[0065]

[0066] wherein cell represents a unit cell, K i，cell represents a design target value of the three-directional stiffness of the unit cell, when i = X, K X，cell represents a design target value of the X-directional stiffness of the unit cell, when i = Y, K Y，cell represents a design target value of the Y-directional stiffness of the unit cell, when i = Z, K Z，cell represents a design target value of the Z-directional stiffness of the unit cell;

[0067] Step 3: creating a design domain Ω of the unit cell, dividing a finite element mesh, setting a design variable, an objective function, a constraint condition and a convergence criterion;

[0068] wherein the design variable ρ e is a relative density of each mesh element of the finite element mesh, and the design variables of all mesh elements constitute a design variable field; the objective function is a total strain energy inside the unit cell; the constraint condition is a three-directional stiffness design target value of the unit cell and a volume fraction or a mass fraction of the material occupying the design space; therefore, the design problem of the unit cell can be mathematically described as:

[0069]

[0070] wherein ρ e is the design variable, e is a mesh element number of the finite element mesh, n is a total number of mesh elements in the design domain Ω, F j represents the objective function, σ is stress, ε is strain, K i is a calculated value of the three-directional stiffness of the unit cell in the optimization process, K i，cell is a design target value of the three-directional stiffness of the unit cell, V f is a volume fraction of the material occupying the design domain, and V0 is a constant representing an upper limit of the volume fraction of the material;

[0071] The convergence criterion is:

[0072]

[0073] wherein k is a current iteration step number, and α is a constant, usually taking a value in a range of 10 -6 ~ 10 -3 .

[0074] Step 4: filtering and projecting the design variable ρ e , and interpolating and penalizing the related properties of the material;

[0075] wherein the filtering manner of the design variable includes but is not limited to using a Helmholtz equation:

[0076]

[0077] Among them, r min is the filter radius, which is generally in the range of 1 to 3 times the grid size. is the design variable after filtering;

[0078] Formulas for projecting the filtered design variables include, but are not limited to, the hyperbolic tangent formula:

[0079]

[0080] in, is the design variable after projection, β is the control projection slope, and its value range is generally 6 to 10, γ β is the projection point, and its value range is generally 0.1 to 0.9;

[0081] Penalizing the interpolation of material properties means using the projected design variables Perform interpolation penalties on relevant material properties. Penalty formulas include but are not limited to SIMP formula and RAMP formula;

[0082] Step 5: Based on the projected design variables The material related properties obtained by interpolation penalty are used to set the finite element simulation calculation model of the unit cell, and a certain load size F is applied to the three directions of the unit cell according to the stiffness design requirements of the main load direction and non-load direction. i , use the structural mechanics finite element simulation method to simulate and solve the finite element simulation calculation model of the unit cell;

[0083] Step 6: Calculate the three-dimensional stiffness K of the unit cell based on the finite element simulation results i :

[0084]

[0085] Where i = X, Y, Z represents the three directions of the vibration isolator, F i Indicates the load magnitude in the i direction, d i It represents the displacement of the load-bearing boundary in the i direction, and avg represents the average value;

[0086] Further, according to the finite element simulation results, the objective function F is calculated according to formula (2) in step 3. j and calculate the sensitivity of the objective function with respect to the design variables The method for calculating the sensitivity includes but is not limited to the adjoint method.

[0087] Step 7: Based on the sensitivity within the design domain obtained in step 6 Update the design variable ρ in step 3 using the optimization solver eAmong them, the optimization solvers that can be used include but are not limited to SNOPT, GCMMA, MMA, IPOPT, etc.

[0088] Step 8: The updated design variable p obtained in step 7 e and the three-dimensional stiffness calculation value K of the unit cell calculated in step 6 i Substitute into formula (2) and formula (3) of step 3, judge whether the constraint condition is met and the convergence criterion is reached according to the constraint condition and the convergence criterion set in step 3; if yes, the topology optimization iteration is ended, and step 9 is executed; if no, steps 4-8 are repeated until the constraint condition is met and the convergence criterion is reached, and then step 9 is executed;

[0089] Step 9: Use the projected design variable Any contour or isosurface of the projected design variable is extracted to obtain the optimal topology configuration of the unit cell.

[0090] Step 10: Array modeling is performed on the unit cell configuration according to the array arrangement form of the unit cell in step 2 to obtain the three-dimensional overall structure of the vibration isolator.

[0091] Step 11: The three-dimensional stiffness of the overall structure of the vibration isolator is analyzed again using the structural mechanics finite element simulation method, and the three-dimensional loading force-displacement curve of the vibration isolator is drawn. The actual value of the three-dimensional stiffness of the vibration isolator is obtained by analyzing the curve.

[0092] Step 12: Compare the actual value of the three-dimensional stiffness of the vibration isolator obtained in step 11 with the design target value K i，total of the three-dimensional stiffness of the vibration isolator in step 1, and calculate the deviation between the actual value of the three-dimensional stiffness of the vibration isolator and the design target value K i，total of the three-dimensional stiffness of the vibration isolator. If the deviation is less than 1%, step 13 is executed, and if the deviation is greater than or equal to 1%, step 2 is returned to redesign the unit cell configuration and its arrangement form until the deviation is less than 1%, and then step 13 is executed.

[0093] Step 13: Output the final three-dimensional overall structure of the vibration isolator, and the design process is ended.

[0094] Example 2:

[0095] Based on example 1, this embodiment provides a specific implementation process of a metamaterial vibration isolator topology optimization design method with customizable three-dimensional stiffness, as follows:

[0096] In this embodiment, the metamaterial has a Young's modulus of 100 MPa and a density of 1100 kg / m 3, the X-direction stiffness of the vibration isolator designed by the metamaterial is 2500 N / mm, the Z-direction stiffness is 12500 N / mm, wherein the Z-direction is the main bearing direction of the vibration isolator, and the overall design space size of the vibration isolator is X*Y*Z=15 cm*12 cm*6 cm.

[0097] According to step 1, in this embodiment, the overall design space size of the vibration isolator 1 is 15 cm*12 cm*6 cm, and the design target value K Z，total of the Z-direction stiffness in the main bearing direction is 12500 N / mm, the design target value K X，total of the X-direction stiffness in the non-bearing direction is 2500 N / mm, and the Y-direction stiffness K Y，total is not specifically required.

[0098] According to step 2, since the Y-direction stiffness of the vibration isolator has no design requirement, the overall structure of the vibration isolator can be divided into an array arrangement form of the unit cell 2 of the hexahedral structure of 5*1*2, as shown in FIGS. 1 and 2, and the size of a single unit cell is a cuboid of 3 cm*12 cm*3 cm; according to the array arrangement form of the unit cell 2, the relationship between the design target values of the X-direction and Z-direction stiffness of the unit cell 2 and the overall stiffness of the vibration isolator 1 is as follows: Figure 2 Figure 3

[0099]

[0100] wherein K X，cell and K Z，cell are the X-direction and Z-direction stiffness values of the unit cell 2, and K X，total and K Z，total are the X-direction and Z-direction stiffness values of the overall vibration isolator 1.

[0101] According to step 3, since the vibration isolator is designed for the X-direction and Z-direction stiffness, the number of Y-direction unit cells is 1, and thus the three-dimensional structure design problem of the unit cell can be simplified to a design problem in the XZ two-dimensional plane; an XZ two-dimensional plane design domain Ω is created, the size of the design domain is 3 cm*3 cm, the grid unit size is set to 0.2 mm, the design domain is divided into a grid model of 150*150 using quadrilateral grid units;

[0102] the design variable is the relative density of each grid unit in the design domain, the objective function is to minimize the overall strain energy in the design domain, and the constraint conditions include: (1) the specified material volume fraction is not more than 60% of the design domain; (2) the X-direction stiffness value of the unit cell is not more than 1000 N / mm; (3) the Z-direction stiffness value of the unit cell is not more than 5000 N / mm. Therefore, the unit cell design problem can be mathematically described as:

[0103] ​​

[0104] where K X and K Z are the stiffness values of the unit cell in X and Z directions respectively during the optimization process.

[0105] The convergence criterion is:

[0106]

[0107] According to step 4, the design variables can be filtered using the Helmholtz equation, and the filtered design variables can be projected using the hyperbolic tangent projection formula, i.e.

[0108]

[0109]

[0110] Here, r min takes the value of 2 times the grid size, β takes the value of 8, and the projection point γ β takes the value of 0.5.

[0111] The projected design variables are used to interpolate the properties of the metamaterial, including Young's modulus, Poisson's ratio, and density. In this embodiment, the SIMP interpolation formula can be used to penalize the Young's modulus of the metamaterial in the design domain, and the calculation formula is as follows:

[0112]

[0113] where E0 is the Young's modulus of polyurethane, and P is the penalty factor, which takes the value of 6 in this embodiment. Other properties of the metamaterial in the design domain, such as density and Poisson's ratio, are consistent with polyurethane.

[0114] According to step 5, the finite element simulation model of the unit cell is set up. Since the stiffness values in X and Z directions need to be calculated simultaneously, the unit cell is subjected to X-direction load F x = 5000 N and Z-direction load F z = 5000 N. The structural mechanics finite element simulation method is used to simulate the model;

[0115] According to step 6, based on the finite element simulation results obtained in step 5, the X and Z direction stiffnesses are calculated as:

[0116]

[0117] where d X and d Z are the displacements caused by the X and Z direction load application boundaries respectively, and avg represents the average value.

[0118] Further according to the simulation results, the objective function Fj The size of the design variable is calculated by the adjoint method.

[0119] According to step 7, the design variable ρ in step 3 is updated using the optimization solver GCMMA based on the obtained sensitivity information in the design domain. e .

[0120] According to step 8, the updated design variable ρ obtained in step 7 is e The calculated value K of the three-dimensional stiffness of the unit cell obtained in step 6 i Substitute into formula (8) and formula (9) in step 3, and judge whether the constraints and convergence criteria set in step 3 are met; if so, the topology optimization iteration ends and step 9 is executed; if not, repeat steps 4 to 8 until the constraints are met and the convergence criteria are reached, and then execute step 9;

[0121] According to step 9, extract the topology optimization results and use The optimal structure is extracted by isoline extraction, that is, the optimal two-dimensional topological configuration of the unit cell is obtained, as shown in the attached figure. Figure 4 The three-dimensional structure of the unit cell is obtained by stretching it in the Y direction with a stretching length of 12 cm, as shown in the attached figure. Figure 5 shown.

[0122] According to step 10, the unit cell three-dimensional configuration is arrayed and modeled according to the unit cell array arrangement form of step 2. The number of X-direction arrays is 5, the number of Y-direction arrays is 1, and the number of Z-direction arrays is 2. The three-dimensional overall structure of the vibration isolator is obtained, as shown in the attached figure. Figure 6 shown.

[0123] According to step 11, the structural mechanics finite element simulation method is used to analyze the overall structure and X- and Z-direction stiffness of the vibration isolator, and the X- and Z-direction loading force-displacement curves of the vibration isolator are obtained as shown in the attached figure. Figure 7 As shown in the figure, by analyzing the curve, its lateral stiffness is 2496.9N / mm and its longitudinal stiffness is 12497N / mm.

[0124] According to step 12, by comparing the isolator stiffness value obtained in step 11 with the target design value of the isolator stiffness in step 1, it is calculated that the deviations between the design value and the target value of the isolator stiffness in the X-direction and Z-direction are 0.124% and 0.024% respectively, which are less than 1% and meet the design requirements.

[0125] According to step 13, the final three-dimensional overall structure of the vibration isolator is output, and the design process is completed.

[0126] To sum up, the above is only the preferred embodiment of the present application, and is not used to limit the protection scope of the present application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A method for topology optimization design of a metamaterial vibration isolator with customizable three-directional stiffness, wherein the vibration isolator using metamaterials is composed of a plurality of unit cells arranged in a set array arrangement; characterized in that, The design method is as follows: firstly, based on the structure design idea of "unit cell-whole", the overall structure of the vibration isolator is divided into multiple unit cell array configurations, and the corresponding relationship between the overall structure three-directional stiffness of the vibration isolator and the unit cell three-directional stiffness is established; Then, the unit cell topology optimization design domain is created, the variable density-based structure topology optimization design method is used to design and optimize the unit cell three-directional stiffness, and the optimal configuration of the unit cell is obtained; finally, the array modeling is performed according to the array arrangement mode of the unit cell, and the overall structure of the vibration isolator with the required three-directional stiffness is obtained; The specific steps of the design method include: Step 1: Establish an XYZ coordinate system, determine the overall design space size of the vibration isolator, and determine the main bearing direction of the vibration isolator as Z direction, and determine the three-direction stiffness design target value of the vibration isolator , represents stiffness, represents the three directions of the vibration isolator, represents the overall structure of the vibration isolator; that is represents the X-direction stiffness design target value of the vibration isolator, represents the Y-direction stiffness design target value of the vibration isolator, represents the Z-direction stiffness design target of the vibration isolator; Step 2: set the unit cell configuration and its array arrangement form, and determine the unit cell three-directional stiffness design target value according to the corresponding relationship between the unit cell stiffness and the overall stiffness of the vibration isolator; The several cells are arranged along X to, Y to and Z to constitute a vibration isolator, and X to, Y to and Z The number of cells along the three directions is respectively , , The three-way stiffness design target value of the cell is: Equation (1) wherein, represents a unit cell, represents a three-way rigidity design target value of the unit cell when represents a three-way rigidity design target value of the unit cell when represents an X-way rigidity design target value of the unit cell when represents an X-way rigidity design target value of the unit cell when represents a Y-way rigidity design target value of the unit cell when represents a Y-way rigidity design target value of the unit cell when represents a Z-way rigidity design target value of the unit cell.

2. The metamaterial isolator topology optimization design method of claim 1, wherein, The specific steps of the design method further include: Step 3: Creating the design domain of the cell Divide the finite element mesh, set the design variables, objective function, constraints and convergence criteria; Wherein, the design variable is the relative density of each grid element of the finite element grid; the objective function is the total strain energy inside the unit cell; the constraint condition is the three-directional stiffness design target value of the unit cell and the volume fraction or mass fraction of the material occupying the design space; therefore, the design problem of the unit cell can be mathematically described as: Formula (2) wherein, is a design variable, is a mesh element number of the finite element mesh, is a design domain is a total number of the mesh elements within the design domain, denotes an objective function, is a stress, is a strain, is a calculated value of the three-directional stiffness of the unit cell in the optimization process, is a design target value of the three-directional stiffness of the unit cell, is a volume fraction of the material in the design domain, is a constant, representing an upper limit of the volume fraction of the material; The convergence criterion is: Equation (3) wherein, is the current iteration step number, is a constant with a value range of 10 -6 ~10 -3 ; Step 4: Filtering the design variables to obtain the design variables , filtering the design variables to obtain the design variables and interpolating penalties on the material properties; Step 5: according to the projected design variables The material properties obtained by interpolation punishment, set the finite element simulation calculation model of the unit cell, and apply three-way load to the unit cell according to the stiffness design requirements of the main bearing direction and the non-bearing direction , use the finite element simulation method of structural mechanics to simulate and solve the finite element simulation calculation model of the unit cell; Step 6: Calculate the three-directional stiffness of the unit cell according to the finite element simulation results : Formula (4) wherein denotes the three directions of the vibration isolator, denotes the load size in the direction, denotes the displacement size of the load boundary in the direction, denotes the average value; According to the finite element simulation results, the size of the objective function is calculated according to formula (2) in step 3, and the sensitivity of the objective function with respect to the design variable is calculated . ​ Step 7: Update the design variables of Step 3 using the optimization solver based on the sensitivity within the design domain obtained in Step 6 ; and ; Step 8: the updated design variables obtained in Step 7 and the three-directional stiffness calculation value of the unit cell obtained in Step 6 Substitute into the formula (2) and formula (3) of Step 3, and judge whether the constraint condition is met and the convergence criterion is reached according to the constraint condition and the convergence criterion set in Step 3; if yes, the topology optimization iteration is ended, and Step 9 is executed; if no, Steps 4-8 are repeated until the constraint condition is met and the convergence criterion is reached, and Step 9 is executed; Step 9: Using projected design variables extracts the optimization result of any isosurface or isovolume to obtain the optimal topological configuration of the unit cell; Step 10: array modeling is performed on the unit cell configuration according to the array arrangement form of the unit cell in step 2, and the three-dimensional overall structure of the vibration isolator is obtained.

3. The metamaterial isolator topology optimization design method of claim 2, wherein, The three-dimensional overall structure of the vibration isolator obtained in step 10 is verified, and the verification method is as follows: Step 11: the three-directional stiffness of the overall structure of the vibration isolator is analyzed again using the structure mechanics finite element simulation method, the three-directional loading force-displacement curve of the vibration isolator is drawn, and the actual value of the three-directional stiffness of the vibration isolator is obtained by analyzing the curve; Step 12: comparing the actual value of the three-directional stiffness of the vibration isolator obtained in step 11 with the design target value of the three-directional stiffness of the vibration isolator in step 1 , calculating the deviation between the actual value of the three-directional stiffness of the vibration isolator and the design target value of the three-directional stiffness of the vibration isolator , if the deviation is less than 1%, executing step 13, if the deviation is greater than or equal to 1%, returning to step 2 to redesign the unit cell configuration and its arrangement until the deviation is less than 1%, and then executing step 13; Step 13: output the final three-dimensional overall structure of the vibration isolator.

4. The topology optimization design method of a metamaterial isolator with customizable three-directional stiffness according to claim 2 or 3, characterized in that, In step 2, the unit cell configuration includes but is not limited to tetrahedron, pentahedron, hexahedron and other polyhedral configurations.

5. The topology optimization design method of a metamaterial isolator with customizable three-directional stiffness according to claim 2 or 3, characterized in that, In step 4, the filtering method of the design variable includes but is not limited to using Helmholtz equation: Equation (5) wherein, is the filter radius, which is in the range of 1-3 times the mesh size; The formula for projecting the filtered design variable includes but is not limited to hyperbolic tangent formula: Formula (6) wherein, is a control projection slope, and the value range is 6-10, is a projection point, and the value range is 0.1-0.9; Interpolation penalization of properties of metamaterials refers to using projected design variables Interpolation penalization of properties of metamaterials refers to using projected design variables 6. The topology optimization design method of a metamaterial isolator with customizable three-directional stiffness according to claim 2 or 3, characterized in that, In step 6, the method for calculating the sensitivity includes but is not limited to adjoint method.

7. The topology optimization design method of a metamaterial isolator with customizable three-directional stiffness according to claim 2 or 3, characterized in that, In step 7, the available optimization solver includes but is not limited to SNOPT, GCMMA, MMA, IPOPT.

Images