Topological optimization method and system for high-damping high-rigidity free damping plate

Abstract

The invention discloses a high-damping high-rigidity free damping plate topological optimization method and system, and the method employs a self-adaptive material field series expansion method to reduce the scale of a uniform multi-material interpolation model, employs a CRITIC method to distribute the weight coefficient of an objective function, and builds a comprehensive objective function to reduce the calculation amount of multi-objective topological optimization. And a low-dimensional and efficient multi-material multi-target topological optimization method is established. Taking a four-side clamped free damping plate as an example, two damping materials are adopted to carry out first six-order modal loss factor maximization topological optimization. Compared with a single-material free damping plate, the double-material multi-target topological optimization free damping plate realizes high-damping and high-rigidity design in a wide frequency band, and meanwhile, the effectiveness of the method in the multi-material multi-target topological optimization problem is verified.

Description

Technical Field

[0001] The present invention relates to the technical field of topology optimization, and particularly to a topology optimization method and system for a high-damping and high-stiffness free damping plate. Background Art

[0002] The thin plate structure is one of the common structures in engineering and is widely used in objects such as airplanes, ships, and automobiles that require lightweighting. However, the thin plate structure has a relatively large span and a small vertical stiffness, making it more likely to generate vibration and noise problems. The free damping plate lays damping materials on the surface of the thin plate to improve its damping characteristics and thus suppress the vibration of the thin plate. It has the characteristics of convenient operation, high economy, and good effect, and is one of the commonly used measures in engineering to suppress the vibration of thin plates.

[0003] In order to improve the utilization rate of damping materials and at the same time reduce the increase in the mass of the thin plate structure and the cost of damping materials caused by laying damping materials, scholars have carried out a large amount of research work on the topology optimization of damping material layout for different objective functions. Zheng et al. took the maximization of the multi-order modal damping ratio as the optimization objective, used the SIMP method to carry out the modeling topology optimization of the constrained damping plate, and discussed the optimization results under different amounts of damping materials. Kim et al. took the maximization of the modal loss factor as the optimization objective to carry out the topology optimization of the free damping material of the cylindrical plate structure, and compared the damping layout forms optimized according to the modal vibration mode and the strain energy of the thin plate, proving that the topology-optimized damping material distribution has a better effect. Kang et al. took the displacement amplitude of the free damping plate under harmonic excitation as the optimization objective and improved the damping characteristics of the structure through topology optimization. Yan et al. took the residual vibration response after the impact load as the optimization objective to carry out the topology optimization of the free damping plate structure, and proposed a complex transient dynamics objective topology optimization method based on the Lyapunov equation. Xu Wei et al. took the minimization of the sound pressure of the radiated sound field of the free damping plate as the optimization objective to carry out the topology optimization of the damping material for the acoustic response.

[0004] From different perspectives, the above research work has achieved the improvement of the damping characteristics of thin plate structures by using damping materials. However, the results of some studies have shown that the modal frequencies of thin plate structures optimized by using a single damping material topology are lower than those of undamped plates, and there are problems of high damping and low stiffness in the optimization results. To avoid the above problems of high damping and low stiffness, He Honglin et al. added a modal frequency constraint function during the topology optimization of constrained damping plates to ensure that the modal frequencies of the optimization results meet the requirements. In addition, multi-material topology optimization is also an effective method to solve the problem of high damping and low stiffness. Zhang et al. used soft and hard damping materials to conduct topology optimization of free damping plates at the micro and macro scales, and achieved the design of thin plate structures with high damping and high stiffness. Ni Weiyu et al. directly conducted topology optimization of dual-material free damping plates macroscopically and achieved the design of free damping plates with high damping and high stiffness. The topology optimization of multi-material free damping plates at the macro scale is simpler to implement and easier to apply in engineering.

[0005] When using multi-materials for joint optimization, it is necessary to establish a multi-material topology optimization model. The commonly used homogeneous multi-material interpolation model is simple to implement, but the model scale increases significantly and the optimization cost increases. For the problem of multi-material topology optimization, Wang et al. adopted the material field series expansion method, which greatly reduced the dimension of the multi-material topology optimization problem. On the basis of the material field series expansion method, Fan et al. proposed the adaptive material field series expansion method, which improved the optimization efficiency while reducing the scale of the multi-material topology optimization model, and achieved the efficient topology optimization of single-material free damping plates. Therefore, using the adaptive material field series expansion method for multi-material free damping plate topology optimization helps to reduce the scale of the multi-material topology optimization model.

[0006] The modal loss factor is a commonly used optimization objective for free damping plates. To improve the damping characteristics of thin plates in a wide frequency band, a multi-objective optimization method of weighted sum is often used to optimize multiple-order modal loss factors simultaneously, and the computational amount of multi-objective optimization is reduced. Therefore, the weight coefficient is an important factor affecting the multi-objective optimization results. In the problem of structural multi-objective topology optimization, scholars have studied different methods for allocating objective weight coefficients. Among them, the method for allocating weight coefficients in multi-objective topology optimization based on the CRITIC method

[18] can take into account the implicit correlation relationship between optimization objectives, and the allocated weight coefficients are more reasonable. There is also an implicit correlation relationship between the modal loss factors of each order of free damping plates. Using the CRITIC method to allocate the weight coefficients of each order of modal loss factors is beneficial to improving the high stiffness and high damping performance of thin plate structures in a wide frequency band.

[0007] Multi-material multi-objective free damping plate topology optimization is beneficial to further improving the performance of thin plate structures, but the increase in the scale of the optimization model and the increase in the number of optimal solutions have increased the computational cost of multi-material multi-objective topology optimization. Summary of the Invention

[0008] The present invention aims to solve at least one of the technical problems in the related art to a certain extent.

[0009] The present invention provides a topology optimization method for a high-damping and high-stiffness free damping plate, which uses the adaptive material field series expansion method to reduce the scale of the uniform multi-material interpolation model, uses the CRITIC method to allocate the weight coefficients of the objective function and establish a comprehensive objective function to reduce the computational complexity of the multi-objective topology optimization, and establishes a low-dimensional and efficient multi-material multi-objective topology optimization method.

[0010] Another object of the present invention is to provide a topology optimization system for a high-damping and high-stiffness free damping plate.

[0011] To achieve the above object, on the one hand, the present invention provides a topology optimization method for a high-damping and high-stiffness free damping plate, including:

[0012] Characterize the mechanical characteristics of the damping material by the complex stiffness method, and construct the total stiffness matrix of the free damping plate based on the complex stiffness method to solve the modal loss factor of the free damping plate through the characteristic equation;

[0013] Based on the uniform multi-material interpolation model, establish a double-loop topology optimization system of the adaptive material field series expansion method, and output the topology calculation result through topology optimization; wherein, the double-loop topology optimization system includes an inner loop that uses the moving asymptote method to optimize and update the design variables of the currently expanded material field, and an outer loop that updates the adaptive correlation function and the optimization model;

[0014] Perform multi-material multi-objective topology optimization of the free damping plate with the first N-order modal loss factors as the optimization objectives, and construct a comprehensive objective function according to the topology calculation result;

[0015] Obtain the optimization results of the single-objective topology optimization of each order of the modal loss factor, calculate the modal loss factors of each order of the optimization results to obtain an evaluation matrix, analyze the standard deviation of each index, the conflict between indexes, and the information content included in the indexes in the evaluation matrix to obtain the weight coefficients of the indexes, so as to establish a multi-material multi-objective topology optimization model of the free damping plate according to the comprehensive objective function.

[0016] To achieve the above object, on the other hand, the present invention provides a topology optimization system for a high-damping and high-stiffness free damping plate, including:

[0017] A modal loss factor calculation module, which is used to characterize the mechanical characteristics of the damping material by the complex stiffness method, and construct the total stiffness matrix of the free damping plate based on the complex stiffness method to solve the modal loss factor of the free damping plate through the characteristic equation;

[0018] The topology optimization model construction module is used to establish a double-loop topology optimization system of the adaptive material field series expansion method based on the uniform multi-material interpolation model, and output the topology calculation result through topology optimization. Among them, the double-loop topology optimization system includes an inner loop that uses the moving asymptote method to optimize and update the design variables of the currently expanded material field, and an outer loop that updates the adaptive correlation function and the optimization model.

[0019] The objective function construction module is used to perform multi-material multi-objective free damping plate topology optimization with the modal loss factors of the first N orders as the optimization objectives, and construct a comprehensive objective function according to the topology calculation result.

[0020] The multi-objective topology optimization module is used to obtain the optimization results of the single-objective topology optimization of each order of modal loss factors, calculate the modal loss factors of each order of the optimization results to obtain an evaluation matrix, analyze the standard deviation of each index, the conflict between indexes, and the information content contained in the indexes in the evaluation matrix, and obtain the weight coefficients of the indexes, so as to establish a multi-material multi-objective topology optimization model of the free damping plate according to the comprehensive objective function.

[0021] In the high-damping and high-stiffness free damping plate topology optimization method and system of the embodiments of the present invention, taking the free damping plate with four sides fixed as an example, two damping materials are used to perform topology optimization to maximize the modal loss factors of the first six orders. Compared with the free damping plate using a single material, the free damping plate with dual-material multi-objective topology optimization realizes the design of high damping and high stiffness in a wide frequency band, and at the same time verifies the effectiveness of the proposed method in the multi-material multi-objective topology optimization problem.

[0022] Additional aspects and advantages of the present invention will be given in part in the following description, become apparent in part from the following description, or be understood through the practice of the present invention. Description of the Drawings

[0023] The above and / or additional aspects and advantages of the present invention will become apparent and be readily understood from the following description of the embodiments in conjunction with the drawings, where:

[0024] Figure 1 is a flowchart of the high-damping and high-stiffness free damping plate topology optimization method according to an embodiment of the present invention;

[0025] Figure 2 is a schematic diagram of the projection function under different projection parameters according to an embodiment of the present invention;

[0026] Figure 3 is a schematic diagram of the topology optimization of the double-loop adaptive material field series expansion method according to an embodiment of the present invention;

[0027] Figure 4 is a schematic diagram of the free damping plate with four sides fixed according to an embodiment of the present invention;

[0028] Figure 5 Schematic diagram of the single-material single-objective topology optimization result according to an embodiment of the present invention;

[0029] Figure 6 Schematic diagram of the dual-material multi-objective topology optimization result according to an embodiment of the present invention;

[0030] Figure 7 Comparison diagram of the amplitude-frequency functions of the single-material and dual-material free damping plates according to an embodiment of the present invention;

[0031] Figure 8 Structural diagram of the topology optimization system of the high-damping and high-stiffness free damping plate according to an embodiment of the present invention. Detailed implementation manners

[0032] It should be noted that, without conflict, the embodiments in the present invention and the features in the embodiments may be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and in conjunction with the embodiments.

[0033] In order to enable those skilled in the art to better understand the solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments in the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0034] The topology optimization method and system of the high-damping and high-stiffness free damping plate according to an embodiment of the present invention will be described below with reference to the accompanying drawings.

[0035] Figure 1 Is the flowchart of the topology optimization method of the high-damping and high-stiffness free damping plate according to an embodiment of the present invention, as Figure 1 shown, the method includes:

[0036] S1, characterizing the mechanical characteristics of the damping material by using the complex stiffness method, and constructing the total stiffness matrix of the free damping plate based on the complex stiffness method to solve the modal loss factor of the free damping plate through the characteristic equation;

[0037] S2, establishing a double-loop topology optimization system of the adaptive material field series expansion method based on the uniform multi-material interpolation model, and outputting the topology calculation result through topology optimization; wherein, the double-loop topology optimization system includes an inner loop that uses the moving asymptote method to optimize and update the design variables for the currently expanded material field, and an outer loop that updates the adaptive correlation function and the optimization model;

[0038] S3. Take the modal loss factors of the first N orders as the optimization objectives for multi-material and multi-objective topology optimization of the free damping plate, and construct a comprehensive objective function according to the topological calculation results;

[0039] S4. Obtain the optimization results of the single-objective topology optimization of the modal loss factors of each order, calculate the modal loss factors of the optimization results to obtain an evaluation matrix, analyze the standard deviation of each index, the conflict between indexes, and the information content contained in the indexes in the evaluation matrix, and obtain the weight coefficients of the indexes, so as to establish a multi-material and multi-objective topology optimization model of the free damping plate according to the comprehensive objective function.

[0040] It can be understood that in order to describe the mechanical properties of the viscoelastic damping material, the complex stiffness method is used to characterize the mechanical properties of the damping material.

[0041]

[0042] In the formula, E v is the elastic modulus of the damping material, α is the loss factor of the damping material, and i im is the imaginary unit. Therefore, is the complex elastic modulus.

[0043] Based on the characterization method of the complex stiffness method, the stiffness matrix of the free damping material is also a complex matrix. The total stiffness matrix K of the free damping plate structure is

[0044] K = K R + K I (31)

[0045] In the formula, K R and K I are the real part and the imaginary part of the total stiffness matrix, respectively.

[0046] The characteristic equation of the motion differential equation of the free damping plate is

[0047] (K - λ l M)φ l = 0 (32)

[0048] In the formula, M is the total mass matrix of the damped structure; λ l and φ l are the l-th order complex eigenvalue and eigenvector of the characteristic equation, respectively.

[0049] The modal loss factor can reflect the vibration energy dissipation degree of the damped structure and is a common objective function for the structural damping topology optimization. The l-th order modal loss factor ζ l of the damped structure is the ratio of the modal strain energy dissipated by the damping material in the l-th order modal vibration mode to the modal strain energy stored in the structure.

[0050]

[0051] According to the modal strain energy method, ignoring the influence of the damping material on the structural modal vibration mode, the real modal vibration mode is used to replace the complex modal vibration mode for calculation, reducing the computational complexity of modal analysis. Therefore, the l-th order modal loss factor ζ of the damped structure l is expressed as

[0052]

[0053] Furthermore, the uniform multi-material interpolation model continues the idea of element property interpolation of the single-material SIMP method. Multiple design variables corresponding to different materials are set within the element, and the mechanical indices of the element are obtained by interpolating all the design variables together, thereby forming a multi-material topology optimization mathematical model. For a multi-material topology optimization problem involving m m materials and N e elements, the uniform multi-material interpolation model is

[0054]

[0055] where A (j) is the mechanical index of material j (material elastic modulus E v (j) , loss factor α (j) , density ρ (j) ), and A i is the corresponding mechanical index of element i after interpolation. p is the penalty factor, taking 3 for elastic modulus interpolation and 1 for loss factor and density interpolation.

[0056] In Equation (6), when and only when x ij = 1 and x iξ = 0 (ξ≠j), w ij = 1, and element i only contains material j after interpolation; in other cases, element i does not contain any material after interpolation.

[0057] The sensitivity of the mechanical index A (j) of the interpolated element with respect to the design variable x ij is

[0058]

[0059] Furthermore, the scale of the multi-material structure topology optimization model established using the uniform multi-material interpolation model is N e ×m m . Compared with the number N e of finite element units, the model scale increases significantly as the number of material types increases. To reduce the scale of the multi-material topology optimization model based on the uniform multi-material interpolation model and reduce the computational amount of multi-material topology optimization, the adaptive material field series expansion method is used to reduce the dimension of the required design variables.

[0060] The topology optimization based on the material field assumes that each finite element is a field point z i , and a material field is established for each material to characterize the structural topology. The distribution probability information of the material field is unknown, but the design variables characterizing the structural topology are bounded between [0, 1]. Therefore, a bounded material field can be established, and its uncertainty can be described based on the correlation of the bounded uncertain field. For a structural topology optimization problem involving m m materials, the material field functions corresponding to each material are constructed respectively field point The mapping relationship between the structural topology at the location and the value of the material field function of the j-th material is defined as

[0061]

[0062] where Ω void and Ω solid respectively represent the solid material elements and empty material elements in the design domain Ω des .

[0063] To obtain a structural topology with clear boundaries, the material field function needs to be projected before interpolating the relative density of the elements. The projection function is

[0064]

[0065] where β is the projection parameter, and the projection functions under different projection parameters are as shown in Figure 2 .

[0066] The interpolation relationship between the projected material field and the relative density of the elements is

[0067]

[0068] For the material field characterizing the structural topology, from the perspective of the continuous distribution of materials, there is a spatial correlation based on distance between its field points; from the perspective of the load transfer of the material field, the correlation between solid field points is greater. Therefore, the following adaptive correlation function is established to describe the correlation between field points

[0069]

[0070] where l c is defined as the correlation length between the field points of the material field, and its magnitude controls the minimum size in the optimization result.

[0071] Based on the adaptive correlation function, the correlation matrix of the material field is

[0072]

[0073] Similar to the K-L series expansion in the random field model, the series expansion can be performed on the non-probabilistic bounded field model. For the material field of material j Its series expansion form is

[0074]

[0075] In the formula, is the expansion coefficient of the material field corresponding to material j, and are the eigenvalues and eigenvectors of the correlation matrix C j , which can be obtained from the following formula

[0076]

[0077] Equation (13) transforms the uncertainty of the point value of the material field into the uncertainty of the expansion coefficient , that is, the design variable is transformed into However The dimension of still remains the same as that of x ij , and all the information of the material field after expansion is retained. The terms with smaller eigenvalues in the expansion formula (13) contribute less to the material field. Therefore, truncating the terms with smaller eigenvalues can reduce the dimension of the expansion coefficient. By defining the truncation criterion as the percentage of the sum of the retained eigenvalues in the sum of all eigenvalues

[0078]

[0079] In the formula, ε t is the truncation error, and a relatively small value is generally taken to ensure the expansion accuracy. The truncated material field reduced-dimension series expansion formula is

[0080]

[0081] Usually, K is much smaller than N e , thus realizing the reduction of the dimension of the design variables required to characterize the material field.

[0082] Since the adaptive correlation function changes following the evolution of the material field, the correlation matrix and its eigenvalues and eigenvectors also change accordingly. Therefore, an adaptive material field series expansion method topology optimization with a double loop as shown in Figure 3 is established. The inner loop uses the moving asymptote (MMA) method to optimize and update the design variables of the currently expanded material field, and the outer loop updates the adaptive correlation function and the optimization model to maintain the expansion accuracy and improve the optimization efficiency during the evolution of the material field.

[0083] From equations (9), (10), and (16), it can be obtained that the relative density x of the element ij with respect to the expansion coefficient The sensitivity is

[0084]

[0085] Furthermore, to improve the damping effect of the free damping plate within a wide frequency band, the first m o order modal loss factors are selected as the common optimization objectives, and multi-material and multi-objective topology optimization of the free damping plate is carried out. Based on the weighted sum method, the comprehensive objective function is constructed as

[0086]

[0087] In Equation (18), the weight coefficient w l of each order modal loss factor is a direct factor affecting the multi-objective optimization result. There are often implicit correlation relationships among the modal loss factors of each order of the same structure. Therefore, the CRITIC method that simultaneously considers the comparison intensity and relevance of the objective function is used to allocate the weight coefficient w l . First, single-objective topology optimization of each order modal loss factor is carried out, and the modal loss factors of each order of each optimization result are calculated to obtain the evaluation matrix The evaluation matrix is normalized:

[0088]

[0089] The evaluation matrix in subsequent calculations refers to the normalized evaluation matrix.

[0090] To calculate the weight coefficient, first analyze the standard deviation σ l2 of each index (column of the evaluation matrix). It characterizes the comparison intensity of each order modal loss factor itself, and the greater the comparison intensity, the greater the weight.

[0091]

[0092] In the formula, is the average value of the l2th column of the evaluation matrix.

[0093] Secondly, analyze the conflict between indicators. The greater the conflict between an indicator and other indicators, the greater the weight should be. Using the Pearson or Spearman correlation coefficient, the correlation coefficient between indicators l1 and l2 is denoted as Then the conflict between indicator l2 and other indicators is

[0094]

[0095] Based on the comparison intensity of the indicator itself and its conflict with other indicators, evaluate the information content of the indicator

[0096]

[0097] The information content of different indicators is different, and the indicator with a larger information content is more important in decision-making. By normalizing with the total information content of the indicators, the weight coefficient of indicator l2 can be obtained.

[0098]

[0099] Based on the multi-material topology optimization using the adaptive material field series expansion method and the multi-objective topology optimization of allocating the target weight coefficient by the CRITIC method, a multi-material multi-objective topology optimization model of a free damping plate is established as

[0100]

[0101] In the above model, the sensitivity of the objective function is

[0102]

[0103] In the formula, is

[0104]

[0105] For the multi-material free damping plate, the real part K R and the imaginary part K I of the total stiffness matrix are

[0106]

[0107] In the formula, is the stiffness matrix of the base plate in element i, and are respectively the real part and the imaginary part of the stiffness matrix of the damping layer in element i, is the element stiffness matrix corresponding to the unit elastic modulus. Therefore

[0108]

[0109] In the formula, and can be obtained from equations (7) and (17) respectively.

[0110] The sensitivity of the constraint function is

[0111]

[0112] In the formula, v i is the volume of element i.

[0113] In summary, the numerical example of the present invention is as follows:

[0114] In traditional topology optimization of damped thin plates, high-damping materials such as rubber are mostly used. However, rubber damping materials have a relatively large density and a small elastic modulus. After optimization, the modal frequency of the thin plate decreases, showing the characteristics of high damping and low stiffness. To improve the damping and stiffness characteristics of the thin plate structure simultaneously, materials with a smaller loss factor but a larger elastic modulus, such as plastic, are added, and a multi-material multi-objective topology optimization method is used for the topology optimization of free damping plates to achieve the design of thin plates with high damping and high stiffness in a wide frequency band.

[0115] Taking Figure 4 a simply supported free damping plate with four edges fixed as an example, the first six modal loss factors (ζ1 to ζ6) are respectively selected as the optimization objectives for multi-material multi-objective topology optimization. The size of the thin plate is 0.5×0.4m, quadrilateral elements are used, and 50×40 elements are divided for modeling.

[0116] The base plate is made of steel with a thickness of 1.5mm; the damping materials are rubber (blue) and plastic (red) with a thickness of 3mm. The parameters of the three materials are shown in Table 1. In the adaptive material field series expansion method, the truncation error is taken as ε t = 0.01, and the correlation length l c of the material field is taken as 0.1 times the short side of the thin plate. The initial projection parameter is taken as 0.1 and gradually increases at a rate of 1.2 times with iteration. The convergence accuracies of the inner and outer loops are 0.01 and 0.001 respectively.

[0117] Table 1

[0118] Elastic modulus (GPa) <![CDATA[Density (kg·m -3 )]]> Poisson's ratio Material loss factor Steel plate 210 7860 0.29 0 Rubber (blue) 0.05 1500 0.4 1 Plastic (red) 2 1000 0.4 0.05

[0119] Only using rubber damping material, the first six modal loss factors (ζ1 to ζ6) are respectively selected as the optimization objectives for single-objective topology optimization, and the volume fraction constraint of the rubber material is taken as 0.5. The optimization results are as Figure 5 shown. The distribution of the rubber damping material basically corresponds to the modal vibration mode of the simply supported base plate with four edges fixed. The damping material is mainly concentrated at the place with larger modal strain energy, which conforms to the principle of damping vibration reduction.

[0120] Analysis Figure 5The modal loss factors and modal frequencies of the single-material single-objective topology optimization results are shown in Tables 2 and 3. The results show that the damping characteristics of the four-edge clamped free damping plate are significantly enhanced, but the modal frequencies decrease significantly. Taking the optimization result of maximizing ζ1 as an example, compared with the first six-order modal frequencies of the base plate, which are 69.50 Hz, 122.06 Hz, 159.29 Hz, 207.46 Hz, 207.54 Hz, and 288.59 Hz, the first six-order modal frequencies of the optimized four-edge clamped free damping plate are reduced by 8.11 Hz, 10.19 Hz, 13.56 Hz, 12.68 Hz, 11.97 Hz, and 13.48 Hz respectively. Therefore, only using rubber damping material for topology optimization produces a high-damping and low-stiffness structure.

[0121] Table 2

[0122] <![CDATA[ζ1]]> <![CDATA[ζ1]]> <![CDATA[ζ1]]> <![CDATA[ζ1]]> <![CDATA[ζ1]]> <![CDATA[ζ1]]> <![CDATA[ζ1]]> 1.31 1.10 1.14 0.90 0.88 0.77 <![CDATA[ζ2]]> 0.98 1.29 1.03 1.14 1.04 0.86 <![CDATA[ζ3]]> 1.02 0.89 1.38 1.14 0.80 0.99 <![CDATA[ζ4]]> 0.80 1.04 0.84 1.34 0.92 1.11 <![CDATA[ζ s > 0.82 1.05 0.93 1.07 1.31 1.00 <![CDATA[ζ6]]> 0.83 0.94 0.95 1.00 0.93 1.21

[0123] Table 3

[0124] <![CDATA[f1]]> <![CDATA[f2]]> <![CDATA[f3]]> <![CDATA[f4]]> <![CDATA[f5]]> <![CDATA[f6]]> <![CDATA[ζ1]]> 61.39 111.87 145.73 194.78 195.57 275.11 <![CDATA[ζ2]]> 62.39 106.78 145.31 186.18 189.54 269.91 <![CDATA[ζ3]]> 61.97 110.5 138.33 183.87 190.97 259.83 <![CDATA[ζ4]]> 61.46 108.42 142.90 179.42 187.23 254.17 <![CDATA[ζ5]]> 61.00 107.40 141.15 181.81 181.86 259.00 <![CDATA[ζ6]]> 62.54 109.97 143.48 186.51 187.45 252.92

[0125] In the single-material single-objective topology optimization, the dimensions of the design variables at each outer loop step are shown in Table 4. Compared with the scale of the uniform multi-material interpolation model (2000), the model scale in the adaptive material field series expansion method is reduced by more than 85%, verifying the effect of the adaptive material field series expansion method in reducing the model dimension.

[0126] Table 4

[0127] Outer loop 1 Outer loop 2 Outer loop 3 <![CDATA[ζ1]]> 207 253 253 <![CDATA[ζ2]]> 207 265 265 <![CDATA[ζ3]]> 207 244 244 <![CDATA[ζ4]]> 207 246 246 <![CDATA[ζ5]]> 207 252 252 <![CDATA[ζ6]]> 207 279 -

[0128] To achieve a high-damping and high-stiffness thin plate in a wide frequency band, both rubber and plastic materials are used simultaneously, and ζ1 to ζ6 are selected as the optimization objectives together. The proposed multi-material multi-objective topology optimization method is used for optimization. The volume fraction constraint of the rubber material is kept at 0.5, and the volume fraction constraint of the plastic material is taken as 0.2. To obtain the weight coefficients of each optimization objective, first, single-objective topology optimization under two materials is carried out, and the evaluation matrix shown in Table 5 is obtained.

[0129] Table 5

[0130] <![CDATA[ζ1]]> <![CDATA[ζ1]]> <![CDATA[ζ1]]> <![CDATA[ζ1]]> <![CDATA[ζ1]]> <![CDATA[ζ1]]> <![CDATA[ζ1]]> 1.59 1.33 1.38 1.27 1.25 1.23 <![CDATA[ζ2]]> 1.24 1.49 1.28 1.28 1.15 1.14 <![CDATA[ζ3]]> 1.21 1.07 1.52 1.27 1.19 1.25 <![CDATA[ζ4]]> 1.02 1.18 0.99 1.53 1.17 1.34 <![CDATA[ζ5]]> 0.93 1.04 0.92 1.05 1.40 1.09 <![CDATA[ζ6]]> 1.08 1.28 1.22 1.32 1.28 1.49

[0131] From the evaluation matrix in Table 5, using the CRITIC method, the weight coefficients of the optimization objective of the sixth-order modal loss factor are calculated as shown in Table 6. Based on the weight coefficients assigned to each order of modal loss factors in, the topology optimization model of the two-material multi-objective of the free damping plate can be established by the weighted sum method. Using MMA to solve the optimization problem, the two-material layout scheme obtained is as Figure 6 shown. Comparing Figure 5For each free damping plate with a single material and a single objective in topology optimization, the material distribution of the free damping plate obtained by the multi-objective optimization of two materials is more complex. The result of the multi-objective topology optimization of two materials cannot be directly obtained by referring to the modal vibration mode of the thin plate.

[0132] Table 6

[0133]

[0134] In the process of multi-objective topology optimization of two materials, the dimension of the optimization model is shown in Table 7. Compared with the traditional uniform multi-material interpolation model, the multi-material multi-objective topology optimization method using the adaptive material field series expansion method effectively reduces the problem of the increase in the model scale caused by the increase in the number of material types.

[0135] Table 7

[0136] Material field Outer loop 1 Outer loop 2 Outer loop 3 Rubber 207 261 261 Plastic 207 236 236

[0137] Figure 6 The modal loss factor and frequency of the results in the middle are shown in Table 8. Compared with the results of each single-material single-objective optimization in Table 2 and Table 3, the modal loss factor and frequency of the multi-objective optimization results of two materials are significantly improved.

[0138] Table 8

[0139]

[0140] To verify the improvement of the performance of the free damping plate after the multi-objective topology optimization of two materials, harmonic response analysis is carried out on the Figure 5 and Figure 6 results. The excitation points and response points of the harmonic response analysis are as shown in Figure 4 , which are at (0.13, 0.1) m and (0.39, 0.3) m from the upper left vertex of the thin plate respectively. Calculate the amplitude-frequency function of the Z-direction displacement of the response point within 0 - 300 Hz, and the results are as shown in Figure 7 . It can be seen from the figure that within 0 - 300 Hz, the amplitude-frequency function of the displacement of the multi-objective topology optimization results of two materials has a significant decrease in the curve peak compared with the results of each single-material topology optimization, and the peak frequency is generally increased. Therefore, the simply supported free damping plate with multi-objective topology optimization of two materials realizes the design of high damping and high stiffness in a wide frequency band.

[0141] In summary, for the free damping plate structure that only uses rubber damping materials with a higher loss factor and a lower elastic modulus for topology optimization, the modal frequency decreases significantly compared to the undamped thin plate, resulting in a structure with high damping and low stiffness. To achieve the design of a thin plate structure with high damping and high stiffness in a wide frequency band, a multi-material multi-objective topology optimization method is established. In the multi-material topology optimization model, the adaptive material field series expansion method is used to reduce the scale, and in the multi-objective topology optimization, the CRITIC method is used to allocate weight coefficients for optimization. Based on the proposed multi-material multi-objective topology optimization method, rubber damping materials with a higher loss factor and a lower elastic modulus and plastic damping materials with a lower loss factor and a higher elastic modulus are used to jointly perform topology optimization on the first six modal loss factors of the free damping plate. The optimization results show that compared with the free damping plate of single-material single-object topology optimization, the free damping plate of dual-material multi-objective obtains higher modal loss factors and modal frequencies, realizing the design of a thin plate with high damping and high stiffness in a wide frequency band.

[0142] According to the high-damping and high-stiffness free damping plate topology optimization method of the embodiments of the present invention, a multi-material multi-objective topology optimization method based on the adaptive material field series expansion method and the CRITIC method is used to reduce the scale of the multi-material topology optimization model and reduce the computational cost of the multi-objective topology optimization. Taking the free damping plate as the object, multi-material multi-objective topology optimization is carried out to verify the effectiveness of the proposed method and at the same time achieve the optimization effect of high damping and high stiffness in a wide frequency band.

[0143] To implement the above embodiments, as Figure 8 shown, the high-damping and high-stiffness free damping plate topology optimization system 10 is further provided in this embodiment, including:

[0144] A modal loss factor calculation module 100, configured to characterize the mechanical characteristics of the damping material by using the complex stiffness method, and construct the total stiffness matrix of the free damping plate based on the complex stiffness method, so as to solve the modal loss factor of the free damping plate through the characteristic equation;

[0145] A topology optimization model construction module 200, configured to establish a double-loop topology optimization system of the adaptive material field series expansion method based on the uniform multi-material interpolation model, and output the topology calculation result through topology optimization; wherein, the double-loop topology optimization system includes an inner loop that uses the moving asymptote method to optimize and update the design variables of the currently expanded material field, and an outer loop that updates the adaptive correlation function and the optimization model;

[0146] A target function construction module 300, configured to perform multi-material multi-objective free damping plate topology optimization by using the first N modal loss factors as optimization objectives, and construct a comprehensive target function according to the topology calculation result;

[0147] The multi-objective topology optimization module 400 is used to obtain the optimization results of single-objective topology optimization of the modal loss factors of each order, calculate the modal loss factors of each order of the optimization results to obtain an evaluation matrix, analyze the standard deviation of each index, the conflict between indexes, and the information content contained in the indexes in the evaluation matrix, and obtain the weight coefficients of the indexes, so as to establish a multi-material multi-objective topology optimization model of a free damping plate according to the comprehensive objective function.

[0148] According to the high-damping and high-stiffness free damping plate topology optimization system of the embodiments of the present invention, a multi-material multi-objective topology optimization method based on the adaptive material field series expansion method and the CRITIC method is used to reduce the scale of the multi-material topology optimization model and reduce the computational cost of multi-objective topology optimization. Taking the free damping plate as the object, multi-material multi-objective topology optimization is carried out to verify the effectiveness of the proposed method, and at the same time, the optimization effect of high damping and high stiffness in a wide frequency band is achieved.

[0149] In the description of this specification, the description with reference to terms such as "one embodiment", "some embodiments", "example", "specific example", or "some examples" means that the specific features, structures, materials, or characteristics described in connection with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials, or characteristics described can be combined in any one or more embodiments or examples in a suitable manner. In addition, without contradiction, those skilled in the art can combine and combine the different embodiments or examples described in this specification and the features of different embodiments or examples.

[0150] In addition, the terms "first" and "second" are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly indicating the quantity of the indicated technical features. Thus, the features defined with "first" and "second" may explicitly or implicitly include at least one of the features. In the description of the present invention, "a plurality" means at least two, such as two, three, etc., unless otherwise specifically defined.

Claims

1. A topology optimization method for a high-damping and high-stiffness free damping plate, characterized in that, Including: Characterize the mechanical properties of damping materials using the complex stiffness method, and construct the total stiffness matrix of the free damping plate based on the complex stiffness method to solve the modal loss factor of the free damping plate through the characteristic equation; Establish a double-loop topology optimization system for the adaptive material field series expansion method based on the uniform multi-material interpolation model, and output the topology calculation results through topology optimization; wherein, the double-loop topology optimization system includes an inner loop that uses the moving asymptote method to optimize and update the design variables of the currently expanded material field, and an outer loop that updates the adaptive correlation function and the optimization model; Perform multi-material multi-objective topology optimization of the free damping plate with the first N modal loss factors as the optimization objectives, and construct a comprehensive objective function based on the topology calculation results; Obtain the optimization results of the single-objective topology optimization of each order of modal loss factors, calculate the modal loss factors of each order of the optimization results to obtain an evaluation matrix, analyze the standard deviation of each index, the conflict between indexes, and the information content contained in the indexes in the evaluation matrix to obtain the weight coefficients of the indexes, so as to establish a multi-material multi-objective topology optimization model of the free damping plate based on the comprehensive objective function.

2. The method according to claim 1, characterized in that Characterize the mechanical properties of damping materials using the complex stiffness method: where E v is the elastic modulus of the damping material, α is the loss factor of the damping material, and i im is the imaginary unit, is the complex elastic modulus; The total stiffness matrix K of the free damping plate structure is: K = K R + K I (2) where K R and K I are the real and imaginary parts of the total stiffness matrix, respectively; The characteristic equation of the motion differential equation of the free damping plate is: (K-λ l M)φ l =0 (3) where M is the total mass matrix of the damped structure; λ l and φ l are the l-th order complex eigenvalue and eigenvector of the characteristic equation, respectively; The modal loss factor ζ of the l-th order containing the damping structure l is the ratio of the modal strain energy dissipated by the damping material in the l-th order modal vibration mode to the modal strain energy stored in the structure: The l-th order modal loss factor ζ of the damping structure l It is expressed as:

3. The method according to claim 1, characterized in that, For the multi-material topology optimization problem involving m m materials and N e units, the uniform multi-material interpolation model is as follows: Where, A (j) is the mechanical index of material j, the elastic modulus E of the material v (j) , the loss factor α (j) , the density ρ (j) , A i is the corresponding mechanical index of element i after interpolation, and p is the penalty factor; The mechanical index A of the interpolated element (j) with respect to the design variable x ij is:

4. The method according to claim 1, characterized in that The topology optimization based on the material field assumes that each finite element is a field point z i , and a material field is established for each material characterizing the structural topology, including the structural topology optimization problem of m m types of materials. The material field functions corresponding to each material are constructed respectively field point The mapping relationship between the structural topology at and the material field function value of the j-th material is defined as: where, Ω void and Ω solid represent the solid material elements and void material elements in the design domain Ω des respectively; The material field function needs to be projected before interpolating the relative density of the unit, and the projection function is: In the formula, β is the projection parameter; Projected material field The interpolation relationship with the relative density of the element is as follows: Establish an adaptive correlation function to describe the correlation between field points: where, l c is defined as the correlation length between the field points of the material field, and its magnitude controls the minimum size in the optimization result; Based on the adaptive correlation function, the correlation matrix of the material field is: For the material field of material j Its series expansion form is as follows: In the formula, is the expansion coefficient of the material field corresponding to material j, and are the eigenvalues and eigenvectors of the correlation matrix C j , which are obtained from the following formula: Define the truncation criterion by the percentage of the sum of the retained eigenvalues in the sum of the total eigenvalues: where ε t is the truncation error, and the truncated dimensionality reduction series expansion of the material field is:

5. The method according to claim 4, wherein From equations (9), (10), and (16), the relative density x of the unit ij with respect to the expansion coefficient is:

6. The method according to claim 1, characterized in that Select the first m o The modal loss factors of the first m orders are jointly used as the optimization objectives to perform multi-material and multi-objective topology optimization of the free damping plate; Based on the weighted sum method, the comprehensive objective function is constructed as follows:

7. The method according to claim 6, wherein Perform single-objective topology optimization of the modal loss factors of each order, calculate the modal loss factors of each order of each optimization result, and obtain the evaluation matrix 8. The method according to claim 7, wherein Standard deviation of each index in the analysis and evaluation matrix In the formula, is the average value of the l2-th column of the evaluation matrix; Analyze the conflict between indicators. Using Pearson or Spearman correlation coefficient, the correlation coefficient between indicator l1 and l2 is denoted as Then the conflict between indicator l2 and other indicators is as follows: Based on the magnitude of the comparison intensity of the indicator itself and its conflict with other indicators, evaluate the amount of information contained in the indicator Normalize using the total information content of the index to obtain the weight coefficient of index l2:

9. The method according to claim 8, characterized in that, Based on the multi-material topology optimization of the adaptive material field series expansion method and the multi-objective topology optimization of allocating target weight coefficients by the CRITIC method, establish a multi-material multi-objective topology optimization model of the free damping plate as: The sensitivity of the objective function is: In the formula, is: For a multi-material free damping plate, the real part K R and the imaginary part K I are as follows: In the formula, is the stiffness matrix of the base plate in element i, and are respectively the real part and the imaginary part of the stiffness matrix of the damping layer in element i, is the element stiffness matrix corresponding to the unit elastic modulus; then: wherein, and are obtained from formulas (7) and (17) respectively; The sensitivity of the constraint function is: where v i is the volume of unit i.

10. A topology optimization system for a high-damping and high-stiffness free damping plate, characterized in that, Including: A modal loss factor calculation module, which is used to characterize the mechanical properties of damping materials using the complex stiffness method, and construct the total stiffness matrix of the free damping plate based on the complex stiffness method to solve the modal loss factor of the free damping plate through the characteristic equation; A topology optimization model construction module, which is used to establish a double-loop topology optimization system for the adaptive material field series expansion method based on the uniform multi-material interpolation model, and output the topology calculation results through topology optimization; wherein, the double-loop topology optimization system includes an inner loop that uses the moving asymptote method to optimize and update the design variables of the currently expanded material field, and an outer loop that updates the adaptive correlation function and the optimization model; An objective function construction module, which is used to perform multi-material multi-objective topology optimization of the free damping plate with the first N modal loss factors as the optimization objectives, and construct a comprehensive objective function based on the topology calculation results; The multi-objective topology optimization module is used to obtain the optimization results of single-objective topology optimization of modal loss factors of each order, calculate the modal loss factors of each order of the optimization results to obtain an evaluation matrix, analyze the standard deviation of each index, the conflict between indexes, and the information content contained in the indexes in the evaluation matrix, obtain the weight coefficients of the indexes, and establish a multi-material multi-objective topology optimization model of a free damping plate according to the comprehensive objective function.

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