Cross-scale topological optimization method for auxetic metamaterial with specified deformation mode

Abstract

The invention belongs to the technical field related to structure optimization, and discloses a cross-scale topological optimization method of an auxetic metamaterial with a specified deformation mode, which comprises the following steps: (1) according to a given metamaterial function target, introducing an elastic tensor component constraint condition to establish a negative Poisson's ratio microstructure topological optimization model; (2) gradually changing the volume rate constraint of the initial microstructure to obtain a series of negative Poisson's ratio gradient microstructures with different equivalent densities; (3) constructing a multi-phase material interpolation model based on a Kriging model, constructing an auxetic metamaterial cross-scale topological optimization model with a specified deformation mode, and deducing the sensitivity of a target function to design variables; and (4) calculating and updating the design variable until the output meets the convergence condition, and outputting the auxetic metamaterial with the target deformation mode. According to the method, the limitation that the deformation mode of the traditional auxetic metamaterial cannot be locally regulated and controlled is overcome, and the design efficiency of the cross-scale design problem of the two-dimensional auxetic metamaterial under the complex condition is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field related to structural optimization, and more specifically, relates to a cross-scale topology optimization method for auxetic metamaterials with a specified deformation mode. Background Technique

[0002] Auxetic metamaterials with negative Poisson's ratio characteristics are a class of multi-scale functional materials that can expand laterally during axial tension. Their unique deformation behavior stems from the strong coupling between the microstructure topology and the macroscopic mechanical response, and has important application values in fields such as impact energy absorption, flexible sensing devices, and adaptive deformation structures. The design method of auxetic metamaterials with a specified deformation mode provides a theoretical framework for the directional regulation of a specified deformation mode under target loads by establishing the mapping relationship between the microscopic microstructure configuration and the macroscopic equivalent mechanical parameters. Traditional multi-scale optimization methods combine the homogenization theory and the gradient optimization algorithm to realize the basic design and performance prediction of auxetic metamaterials through the iteration of the geometric parameters of the microscopic microstructure and the matching of the macroscopic equivalent performance.

[0003] For the cross-scale design of auxetic metamaterials, existing methods still face technical challenges. For example, in Reference 1: "Wu, J., Sigmund, O. & Groen, J. P. Topology optimization of multi-scale structures: a review. Struct Multidisc Optim 63, 1455–1480 (2021)", it is proposed that although the parametric level set method can synchronously optimize the macroscopic material distribution and microscopic unit cell topology through geometric description, it relies on a preset geometric parameterization model, making it difficult to flexibly adapt to the anisotropic deformation requirements under complex loads, and the connection compatibility problem between microscopic configurations has not been completely solved. In Reference 2: "Wang Y, Sigmund O, Chen JS. Multiscale topology optimization for graded auxetic metamaterials with non-periodic microstructures[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 372:113370.", a multi-scale topology optimization framework for non-periodic microstructures is proposed. By introducing a gradient penalty function, the correlation between macroscopic performance and microscopic configuration is realized, but the non-linear influence of microscopic topology evolution on the macroscopic boundary layer stress distribution is not considered, resulting in insufficient design accuracy in local stress concentration regions. Moreover, existing studies mostly adopt a separated optimization strategy in micro-macro coupling modeling, making it difficult to achieve the synchronous coordination of structural topology evolution and macroscopic deformation behavior, and the design flexibility for non-uniform boundary conditions and anisotropic deformation requirements is insufficient. Summary of the Invention

[0004] In view of the above defects or improvement requirements of the prior art, the present invention provides a cross-scale topology optimization method for auxetic metamaterials with a specified deformation mode, aiming to solve the problem that the structure obtained by directly interpolating using an interpolation model in the existing topology optimization method is unreasonable, resulting in low accuracy of topology optimization.

[0005] To achieve the above object, according to one aspect of the present invention, there is provided a cross-scale topology optimization method for auxetic metamaterials with a specified deformation mode, the method comprising the following steps:

[0006] (1) Set the objective function to minimize the combination of elastic tensor components of the initial microstructure of the auxetic metamaterial to be optimized, and introduce a constraint function to construct a topology optimization model of the initial microstructure;

[0007] (2) Gradually change the volume fraction of the topology optimization model to obtain a series of negative Poisson's ratio microstructures with different equivalent densities;

[0008] (3) After describing the negative Poisson's ratio microstructure using the level set function, the homogenization method is used to calculate the equivalent elastic tensor of the negative Poisson's ratio microstructure, and a cross-scale material interpolation model of the auxetic metamaterial to be optimized is constructed based on the equivalent elastic tensor;

[0009] (4) Based on the target deformation mode and the cross-scale material interpolation model, a cross-scale topology optimization model of the auxetic metamaterial is constructed;

[0010] (5) Calculate the sensitivities of the objective function and constraint function of the cross-scale topology optimization model of the auxetic metamaterial with respect to the design variables. Based on the obtained sensitivity results, update the design variables using the gradient-based moving asymptote method until the convergence condition is satisfied. The cross-scale topology optimization model of the auxetic metamaterial outputs the optimal negative Poisson's ratio gradient microstructure and the auxetic metamaterial with the target deformation mode based on the current design variables.

[0011] Furthermore, the mathematical expression of the cross-scale material interpolation model is:

[0012]

[0013] where represents the elastic tensor of the base microstructure, m = 1, 2, …, M, M is the total number of prototype level set functions, n = 1, 2, …, N, N is the total number of finite element cells in the design domain, represents the elastic tensor of the base microstructure, i, j, k, l = 1, 2, …, d, d is the spatial dimension, is the continuous design variable ρ mn after density filtering, ρ mn is the continuous design variable representing the equivalent density of the base microstructure, δ is a positive number used to avoid matrix singularity, is the discrete design variable representing the configuration of the base microstructure and satisfies p represents the penalty parameter.

[0014] Furthermore, the mathematical expression of the cross-scale topology optimization model of the auxetic metamaterial is:

[0015] Find: s mn , ρ mn

[0016]

[0017] Subject to: K(s mn , ρ mn )U = F

[0018] K0U0 = F

[0019] 0 ≤ smn ≤ 1,

[0020]

[0021] Among them, J represents the objective function, which is used to measure the relative difference between the structural displacement field U at the current iteration step and the desired auxetic metamaterial displacement field U0 under the action of the external load.

[0022] Furthermore, the mathematical expression of the topology optimization model is:

[0023]

[0024] K mic U A(kl) = F (kl) , k, l = 1, ..., d

[0025]

[0026] In the formula, γ is a fixed weighting factor, set to 0.8, iter represents the current number of iteration steps; v e represents the area / volume of the elements within the design domain; V mic represents the maximum volume fraction of the solid material within the microstructure design domain; K mic is the global stiffness matrix assembled from ; U A(kl) and F (kl) respectively represent the global displacement vector and its corresponding load vector under the unit test strain (kl) condition during the application of the periodic boundary conditions.

[0027] Furthermore, perform a morphological dilation operation on the element density distribution of the obtained initial microstructure. Based on the dilated element density distribution, increase the volume ratio constraint of the microstructure. The increase value of the volume ratio constraint for the new prototype negative Poisson's ratio microstructure is y, and repeat the design successively to obtain multiple negative Poisson's ratio microstructures with different equivalent densities.

[0028] Furthermore, the equivalent density value ranges from (0, 1].

[0029] Furthermore, for the level set function, uniformly sample one hundred points in the density interval [0.01, 1] as the equivalent density values of the sample lattice points, and use the shape interpolation technique to obtain the level set function of the sample lattice points. The formula corresponding to the shape interpolation is:

[0030]

[0031] Among them is the prototype level set function, is the interpolation coefficient, φ mn(x) is the level set function of the base microstructure after shape interpolation; the formula for the equivalent density value is:

[0032]

[0033] where H represents the Heaviside function, D represents the design domain where a base microstructure is located, and x represents the coordinate point within the design domain.

[0034] Furthermore, the homogenization method is adopted to calculate the equivalent elastic tensor of the negative Poisson's ratio microstructure. The calculation formula of the homogenization method is:

[0035]

[0036] where is the equivalent elastic tensor of the microstructure, Ω n and |Ω n | represent the design domain of the microstructure and the volume of the design domain respectively, represents the locally varying microscopic strain field, represents the given macroscopic test strain field, i, j, k, l = 1, 2, …, d, d is the spatial dimension, D pqrs represents the elastic tensor of the base material, H represents the Heaviside function, is the displacement field corresponding to , is calculated by the following formula:

[0037]

[0038] where represents the virtual displacement field, represents the displacement field that is statically admissible under periodic boundary conditions.

[0039] Furthermore, calculate the sensitivity information of the objective function with respect to the design variable ρ mn . The sensitivity calculation formula is:

[0040]

[0041] Furthermore, calculate the sensitivity information of the constraint function with respect to the design variables s mn and ρ mn . The sensitivity calculation formula is:

[0042]

[0043] Generally speaking, compared with the prior art by the above technical solutions conceived by the present invention, the cross-scale topology optimization method of the auxetic metamaterial with a specified deformation mode provided by the present invention mainly has the following beneficial effects:

[0044] 1. By combining the homogenization method with the topology optimization model, the present invention takes the combination of minimizing the elastic tensor components of the target as the objective function, combines the volume fraction dynamic regulation stiffness constraint, and automatically generates a series of microstructures with negative Poisson's ratio characteristics and different equivalent densities, improving the design efficiency of complex auxetic metamaterials. At the same time, it avoids the unreasonable structure problems prone to occur in topology optimization and improves the accuracy of the topology optimization results.

[0045] 2. Due to the adoption of the progressive strategy, the present invention gradually increases the volume ratio constraint (increasing by 0.05 each time) through morphological dilation operation. On the premise of ensuring the negative Poisson's ratio characteristics, it generates gradient microstructures with continuously changing equivalent densities, breaks through the limitation of the fixed configuration of traditional microstructures, realizes the smooth transition of the topological morphology between microstructures and the gradient adaptation of mechanical properties, and provides a highly connected material library for cross-scale mechanical regulation.

[0046] 3. By constructing a cross-scale material interpolation model based on the Kriging model, the present invention maps the equivalent elastic tensor of the microstructure into the displacement field difference objective function of the macroscopic auxetic metamaterial. Through the collaborative optimization of the gradient microstructure and the macroscopic deformation mode, it solves the problem of the single force transmission path of traditional homogeneous materials, enables the metamaterial to present the expected boundary expansion deformation under unit tensile load, and realizes the active regulation of macroscopic deformation behavior.

[0047] 4. Due to the introduction of density filtering technology (sensitivity calculation) and the moving asymptote method for solution, combined with the level set function to smooth the topological boundary, the present invention effectively suppresses the checkerboard effect and island phenomenon. While ensuring the optimization convergence, it generates a geometric model that can be directly used for additive manufacturing, with both computational robustness and engineering practicality.

[0048] 5. Through the joint optimization of the elastic tensor component weighting strategy and the volume fraction constraint, under the limitation of the material volume constraint, the present invention synchronously realizes the optimal balance between the negative Poisson's ratio characteristics and the stiffness performance of the microstructure. Compared with the traditional single-objective optimization method, this mechanism avoids the problem of mechanical property imbalance and provides a high-degree-of-freedom solution for the design of multifunctional metamaterials. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 is a flowchart of a cross-scale topology optimization method for an auxetic metamaterial with a specified deformation mode provided by the present invention;

[0050] Figure 2 is a schematic diagram of the design problem of an auxetic metamaterial with a specified deformation mode constructed by the present invention;

[0051] Figure 3 is the iterative optimization process of the first prototype negative Poisson's ratio microstructure (volume ratio constraint 0.2) constructed by the present invention;

[0052] Figure 4 In (a) and (b) respectively are Figure 3 The schematic diagram of the smooth post - processing and its equivalent properties of the first prototype auxetic microstructure in

[0053] Figure 5 The schematic diagram of the successive design of gradient auxetic microstructure by the progressive optimization strategy constructed in the present invention;

[0054] Figure 6 is Figure 5 The prototype auxetic microstructures with different volume ratios obtained by the progressive optimization strategy in

[0055] Figure 7 The iterative optimization process of the cross - scale topology optimization model of the auxetic metamaterial with a specified deformation mode constructed in the present invention;

[0056] Figure 8 In (a) and (b) respectively are Figure 7 The schematic diagram of the auxetic metamaterial with a specified deformation mode and its displacement field in Detailed implementation manners

[0057] In order to make the purpose, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0058] The present invention provides a cross - scale topology optimization method for an auxetic metamaterial with a specified deformation mode. Without relying on the designer's experience, the topology optimization method can efficiently give a sub - optimal solution, quickly determine the appropriate types of lattice materials inside the design domain and the reasonable distribution method of the equivalent density of the basic microstructure, and significantly improve the deformation mode matching accuracy and regulation ability under complex loads on the basis of ensuring the connectivity of the deformation path inside the metamaterial.

[0059] Please refer to Figure 1 , the topology optimization method mainly includes the following steps:

[0060] Step 1, set the objective function to minimize the combination of elastic tensor components of the initial microstructure of the auxetic metamaterial to be optimized, and introduce a constraint function to construct a topology optimization model of the initial microstructure.

[0061] The mathematical expression of the topology optimization model is:

[0062]

[0063] K mic U A(kl) = F (kl) , k, l = 1, ..., d

[0064]

[0065] where γ is a fixed weighting factor, set to 0.8, which can promote the structural topology in the optimization iteration process to have certain stiffness performance, avoid generating unreasonable structural topologies to a certain extent, be conducive to realizing the negative Poisson's ratio characteristic of the microstructure, and iter represents the current iteration step. v e represents the area / volume of the elements within the design domain. V mic represents the maximum volume fraction of the solid material within the microstructure design domain. K mic is composed of assembled to obtain the global stiffness matrix. U A(kl) and F (kl) respectively represent the global displacement vector and its corresponding load vector under the unit test strain (kl) condition during the application of periodic boundary conditions.

[0066] Among them, the volume fraction is set to a small value x to obtain the first negative Poisson's ratio initial microstructure.

[0067] Step 2: Gradually change the volume fraction of the topology optimization model to obtain a series of negative Poisson's ratio microstructures with different equivalent densities.

[0068] Adopt the sequential progressive optimization strategy, perform morphological dilation operation on the element density distribution of the initial microstructure obtained in Step 1 based on the imdilate function of Matlab software. Based on the dilated element density distribution, increase the volume fraction constraint of the microstructure. The increased value of the volume fraction constraint for the new prototype negative Poisson's ratio microstructure is y, and repeat the design successively to obtain multiple negative Poisson's ratio microstructures with different equivalent densities.

[0069] Step 3: After using the level set function to describe the negative Poisson's ratio microstructure, use the homogenization method to calculate the equivalent elastic tensor of the negative Poisson's ratio microstructure, and construct a cross-scale material interpolation model of the auxetic metamaterial to be optimized based on the equivalent elastic tensor.

[0070] The level set function describes the negative Poisson's ratio microstructure obtained from the topology optimization model.

[0071] The mathematical expression of the cross-scale material interpolation model is:

[0072]

[0073] Among them, denote the elastic tensor of the base microstructure, \(m = 1, 2, \ldots, M\), where \(M\) is the total number of prototype level set functions, and \(n = 1, 2, \ldots, N\), where \(N\) is the total number of finite - element cells in the design domain. denote the elastic tensor of the base microstructure, \(i, j, k, l = 1, 2, \ldots, d\), where \(d\) is the spatial dimension. is the continuous design variable \(\rho\) mn after density filtering, \(\widetilde{\rho}\) mn is the continuous design variable representing the equivalent density of the base microstructure, and \(\delta\) is a positive number much less than 1 used to avoid matrix singularity. is the discrete design variable representing the configuration of the base microstructure and satisfies \(p\) represents the penalty parameter.

[0074] Step 4: Construct a tensile - dilatational metamaterial multi - scale topology optimization model based on the target deformation mode and the multi - scale material interpolation model.

[0075] The mathematical expression of the tensile - dilatational metamaterial multi - scale topology optimization model is:

[0076] Find: \(s\) mn , \(\rho\) mn

[0077]

[0078] Subject to: \(K(s\) mn , \(\rho\) mn ) \(U = F\)

[0079] \(K_0U_0 = F\)

[0080] \(0\leq s\) mn \leq1,\)

[0081]

[0082] where \(J\) represents the objective function, which is used to measure the relative difference between the current iterative - step structural displacement field \(U\) and the desired tensile - dilatational metamaterial displacement field \(U_0\) under the action of the external load.

[0083] Step 5: Calculate the sensitivities of the objective function and constraint function of the tensile - dilatational metamaterial multi - scale topology optimization model with respect to the design variables, and update the design variables using the gradient - based moving asymptotes method until the output meets the convergence condition. The tensile - dilatational metamaterial multi - scale topology optimization model outputs the optimal negative - Poisson's - ratio gradient microstructure and the tensile - dilatational metamaterial with the target deformation mode based on the current design variables.

[0084] Among them, when calculating the sensitivities of the objective function and constraint function with respect to the design variables and updating the design variables using the gradient - based moving asymptotes method, update the design variables \(s\) mn and \(\rho\)mn Until the optimized result after the output meets the convergence condition.

[0085] In this embodiment, for the lattice structure to be optimized, its design domain, load, and boundary conditions are as Figure 2 shown. The left end of this design domain is subject to a fixed constraint, the lower end is subject to a sliding constraint, and a uniformly distributed external load is applied to the right end. The equivalent density in this embodiment refers to the percentage of the area or volume of the solid structure part in the design domain to the area or volume of the entire design domain, and its value range is between (0, 1]. Without loss of generality, all physical quantities used in this embodiment are assumed to be dimensionless.

[0086] In this implementation manner, x is 0.2; the number of selected prototype level set functions is 1.

[0087] The following uses specific embodiments to further elaborate on the present invention in detail.

[0088] The stretching-dominated metamaterial cross-scale topology optimization method with a specified deformation mode provided by the embodiment of the present invention includes the following steps:

[0089] Step 1, Discretize the design domain of the two-dimensional negative Poisson's ratio microstructure using 100x100 plane quadrilateral elements. Use a resin material with a Young's modulus of 2750 MPa and a Poisson's ratio of 0.38 as the solid material. Based on the inverse homogenization method, set the topology optimization objective function as the combination of the minimum elastic tensor components of the microstructure, and introduce a constraint function to construct the negative Poisson's ratio microstructure topology optimization model as follows:

[0090]

[0091] K mic U A(kl) = F (kl) , k, l = 1,..., d

[0092]

[0093] Among them, γ is a fixed weighting factor, set to 0.8, which can promote the structure topology in the optimization iteration process to have a certain stiffness performance, avoid generating unreasonable structure topologies to a certain extent, and is conducive to realizing the negative Poisson's ratio characteristics of the microstructure. iter represents the current iteration step. v e represents the area / volume of the elements within the design domain. V mic represents the maximum volume fraction of the solid material within the microstructure design domain. K mic is the overall stiffness matrix assembled from . U A(kl) and F (kl)They respectively represent the global displacement vector and its corresponding load vector under the unit test strain (kl) during the application of periodic boundary conditions.

[0094] Set the volume fraction constraint to 0.2 to obtain the first negative Poisson's ratio initial microstructure. Figure 3 Shows the iterative optimization process of the first prototype negative Poisson's ratio microstructure. The smooth iterative convergence curve verifies the effectiveness of the proposed topology optimization model for designing negative Poisson's ratio microstructures. The objective function will show a significant increase in the first few iterative steps. This is because in the optimization model, the Coefficient -γ iter will increase sharply with the increase in the number of iterative steps. Along with the progress of iteration, γ iter gradually approaches 0, the influence of the second term on the objective function decreases, and the objective function begins to gradually decrease and converge. Figure 4 Shows the first prototype negative Poisson's ratio microstructure designed by topology optimization and its equivalent properties.

[0095] Step 2: Adopt the sequential progressive optimization strategy. Based on the imdilate function of Matlab software, perform morphological dilation operation on the element density distribution of the initial microstructure obtained in Step 1. Based on the dilated element density distribution, increase the volume fraction constraint of the microstructure. The increased value of the volume fraction constraint for the new prototype negative Poisson's ratio microstructure is 0.05. Repeat the design successively to obtain multiple prototype negative Poisson's ratio microstructures with different equivalent densities, such as Figure 5 shown.

[0096] Figure 6 Shows other initial microstructures obtained by applying the progressive optimization strategy. It can be seen that the proposed optimization model can obtain microstructures with negative Poisson's ratio characteristics under different volume fraction constraint conditions, and good connectivity can be maintained between the gradient microstructures.

[0097] Step 3: Construct a level set function to post - process the topology optimization results based on the SIMP interpolation model to obtain a microstructure with a smooth boundary. Use the homogenization method to calculate the equivalent elastic tensor of the base microstructure, and use the elastic tensor data to construct a cross - scale material interpolation model for auxetic metamaterials, which specifically includes the following sub - steps:

[0098] (3.1) Take a prototype level set function obtained in Step 1, uniformly sample one hundred points in the density interval [0.01, 1] as the equivalent density values of the sample lattice points, and use shape interpolation technology to obtain the level set function of the sample lattice. The formula corresponding to shape interpolation is:

[0099]

[0100] where is the prototype level set function, is the interpolation coefficient, φ mn (x) is the level set function of the basic microstructure after shape interpolation. The value of the interpolation coefficient of the basic microstructure with a specific equivalent density value is determined by the bisection method. The formula for determining the equivalent density value of the basic microstructure from the level set function of the basic microstructure is:

[0101]

[0102] where H represents the Heaviside function, D represents the design domain where a basic microstructure is located, and x represents the coordinate point within the design domain.

[0103] (3.2) Calculate the equivalent elastic tensor of the basic microstructure (negative Poisson's ratio microstructure) using the homogenization method. The calculation formula of the homogenization method is:

[0104]

[0105] where is the equivalent elastic tensor of the basic microstructure, Ω n and |Ω n | respectively represent the design domain and the volume of the design domain of the basic microstructure, represents the locally varying microscopic strain field, represents the given macroscopic test strain field, i, j, k, l = 1, 2, …, d, d is the spatial dimension, D pqrs represents the elastic tensor of the base material, H represents the Heaviside function, is the displacement field corresponding to , which can be calculated by the following formula:

[0106]

[0107] where represents the virtual displacement field, represents the displacement field that is statically admissible under periodic boundary conditions.

[0108] (3.3) Construct a cross-scale material interpolation model for auxetic metamaterials based on the Kriging model multi-phase material interpolation model using the equivalent elastic tensor of the microstructure obtained from the above steps:

[0109]

[0110] where, represents the elastic tensor of the basic microstructure, m = 1, 2, …, M, M is the total number of prototype level set functions, n = 1, 2, …, N, N is the total number of finite element cells within the design domain, Denote the elastic tensor of the base microstructure, i, j, k, l = 1, 2, …, d, where d is the spatial dimension. is the continuous design variable ρ mn after density filtering, ρ mn is the continuous design variable representing the equivalent density of the base microstructure, and δ is a positive number much less than 1 used to avoid matrix singularity. is the discrete design variable representing the configuration of the base microstructure, and satisfies p represents the penalty parameter.

[0111] Step 4: Taking the ability of the structure to exhibit a specified deformation mode under loading as the optimization objective, construct a cross-scale topology optimization model of auxetic metamaterials based on multi-class gradient lattice structures as follows:

[0112] Find: s mn , ρ mn

[0113]

[0114] Subject to: K(s mn , ρ mn )U = F

[0115] K0U0 = F

[0116] 0 ≤ s mn ≤ 1,

[0117]

[0118] where J represents the objective function, which is used to measure the relative difference between the current iteration step structure displacement field U and the expected auxetic metamaterial displacement field U0 under the action of the external load.

[0119] Step 5: Calculate the sensitivities of the objective function and the constraint function with respect to the design variables, and use the gradient-based moving asymptote method to update the design variables. Solve the negative Poisson's ratio gradient microstructure and the auxetic metamaterial topology optimization model with a specified deformation mode respectively. The sensitivity calculation formula is:

[0120]

[0121] where H ne , H ei and N e are the indicator matrices in density filtering.

[0122] Calculate the sensitivity information of the objective function with respect to the design variable ρ mn . The sensitivity calculation formula is as follows:

[0123]

[0124]

[0125] Calculate the sensitivity information of the constraint function with respect to the design variables s mn and ρ mn The sensitivity calculation formula is as follows:

[0126]

[0127] Sensitivity filtering is used to avoid unstable phenomena such as checkerboard problems and isolated islands during the optimization process. At the same time, it makes the distribution regions of the basic microstructures with different configurations inside the current lattice structure clearly distinguishable. According to the optimization results, it is judged whether the objective function meets the set threshold, that is, the convergence condition. If the convergence condition is met, the current optimization results are output; otherwise, step five is continued to update the design variables s mn and ρ mn .

[0128] Step six, use the gradient microstructure obtained from the previous steps as the candidate lattice material for Figure 2 the cross-scale topology optimization of the auxetic metamaterial shown in dis The left end of this design domain is subject to a fixed constraint, the lower end is subject to a sliding constraint, and a uniformly distributed external load with a magnitude of F = 4000 N is applied to the right end. The design domain is discretized using 60x40 planar quadrilateral elements, and the size of each element is 5 mm x 5 mm. The candidate basic microstructure 1 with a density of 0.2 is used as the basic microstructure, and the optimization objective is set such that under the action of the external load, the y-axis displacement of the unit node on the upper edge of the design domain satisfies the function y Figure 7 as shown in Figure 8 Figure

[0129] The results show that the optimization model proposed in this method can obtain microstructures with negative Poisson's ratio characteristics under different volume ratio constraints, and the gradient microstructures can maintain good connectivity. Without relying on the designer's experience, it can efficiently give a suboptimal solution and quickly determine the appropriate types of lattice materials inside the design domain and the reasonable distribution method of the equivalent density of the basic microstructures. Under the action of tensile load, different from the contraction deformation mode of traditional materials, the auxetic metamaterial designed by this method can change the force transmission path and control the deformation mode through the reasonable cooperation of the basic microstructures, enabling its upper boundary to expand in the expected shape, which proves the effectiveness and feasibility of this method.

[0130] In summary, the cross-scale topology optimization method of the auxetic metamaterial with a specified deformation mode provided by the present invention realizes the cross-scale collaborative optimization of the macroscopic deformation behavior and the microscopic lattice distribution compared with the prior art. By constructing a gradient microstructure candidate library covering the equivalent elastic tensor space (based on the inverse homogenization method and the progressive optimization strategy), it can quickly match the anisotropy-adjustable microstructure configuration according to the target displacement field, and combine the Kriging cross-scale interpolation model and the multi-class microstructure collaborative optimization algorithm to synchronously regulate the macroscopic force transmission path and the microscopic equivalent elastic tensor distribution. Compared with the cross-scale design method of traditional auxetic metamaterials, this method not only reduces the computational complexity of multi-scale iterative optimization and avoids the convergence difficulties caused by the checkerboard effect and island phenomenon in traditional topology optimization by introducing the elastic tensor component weighting strategy and the volume ratio constraint dynamic adjustment mechanism, but also can accurately control the boundary expansion deformation mode of the metamaterial under tensile load, significantly improving the multi-functional adaptability and deformation controllability of the auxetic metamaterial.

[0131] It is easy for those skilled in the art to understand that the above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principles of the present invention shall be included in the protection scope of the present invention.

Claims

1. A cross-scale topology optimization method for auxetic metamaterials with a specified deformation mode, characterized in that, The method includes the following steps: (1) Set the objective function to minimize the combination of elastic tensor components of the initial microstructure of the auxetic metamaterial to be optimized, and introduce a constraint function to construct a topology optimization model for the initial microstructure; (2) Gradually change the volume fraction of the topology optimization model to obtain a series of negative Poisson's ratio microstructures with different equivalent densities; (3) After using the level set function to describe the negative Poisson's ratio microstructure, use the homogenization method to calculate the equivalent elastic tensor of the negative Poisson's ratio microstructure, and construct a cross-scale material interpolation model for the auxetic metamaterial to be optimized based on the equivalent elastic tensor; (4) Construct a cross-scale topology optimization model for the auxetic metamaterial based on the target deformation mode and the cross-scale material interpolation model; (5) Calculate the sensitivities of the objective function and the constraint function of the cross-scale topology optimization model of the auxetic metamaterial with respect to the design variables, and update the design variables using the gradient-based moving asymptotes method based on the obtained sensitivity results until the convergence condition is satisfied. The cross-scale topology optimization model of the auxetic metamaterial outputs the best negative Poisson's ratio gradient microstructure and the auxetic metamaterial with the target deformation mode based on the current design variables.

2. The cross-scale topology optimization method of the auxetic metamaterial with a specified deformation mode according to claim 1, characterized in that: The mathematical expression of the cross-scale material interpolation model is: Among them, represents the elastic tensor of the base microstructure, m = 1, 2, …, M, where M is the total number of prototype level set functions, n = 1, 2, …, N, where N is the total number of finite element cells in the design domain, represents the elastic tensor of the base microstructure, i, j, k, l = 1, 2, …, d, where d is the spatial dimension, is the continuous design variable ρ mn after density filtering, ρ mn is the continuous design variable representing the equivalent density of the base microstructure, and δ is a positive number used to avoid matrix singularity, is the discrete design variable representing the configuration of the base microstructure, and satisfies p represents the penalty parameter.

3. The cross-scale topology optimization method of the auxetic metamaterial with a specified deformation mode according to claim 2, characterized in that: The mathematical expression of the cross-scale topology optimization model of the auxetic metamaterial is: Find:s mn ,ρ mn Subject to: K(s mn , ρ mn ) U = F K0U0 = F 0≤s mn ≤1, where J represents the objective function, which is used to measure the relative difference between the current iteration step structure displacement field U and the expected auxetic metamaterial displacement field U0 under the action of the external load.

4. The cross-scale topology optimization method of the auxetic metamaterial with a specified deformation mode according to claim 1, characterized in that: The mathematical expression of the topology optimization model is: K mic U A(kl) = F (kl) , k, l = 1, ..., d where γ is a fixed weighting factor set to 0.8, and iter represents the current iteration step; v e represents the area / volume of the elements within the design domain; V mic represents the maximum volume fraction of the solid material within the microstructure design domain; K mic is the overall stiffness matrix assembled from ; U A(kl) and F (kl) represent the global displacement vector and its corresponding load vector under the unit test strain (kl) condition, respectively, during the application of the periodic boundary conditions.

5. The cross-scale topology optimization method of the auxetic metamaterial with a specified deformation mode according to claim 1, characterized in that: Perform a morphological dilation operation on the cell density distribution of the obtained initial microstructure. Based on the dilated cell density distribution, increase the volume fraction constraint of the microstructure. The increased value of the volume fraction constraint of the new prototype negative Poisson's ratio microstructure is y. Repeat the design successively to obtain multiple negative Poisson's ratio microstructures with different equivalent densities.

6. The cross-scale topology optimization method of the auxetic metamaterial with a specified deformation mode according to any one of claims 1-5, characterized in that: The equivalent density value range is between (0, 1].

7. The cross-scale topology optimization method of the auxetic metamaterial with a specified deformation mode according to claim 1, characterized in that: For the level set function, uniformly sample one hundred points in the density interval [0.01, 1] as the sample lattice equivalent density values, and use the shape interpolation technique to obtain the level set function of the sample lattice. The formula corresponding to the shape interpolation is: wherein is the prototype level set function, is the interpolation coefficient, and φ mn (x) is the level set function of the basic microstructure after shape interpolation; the formula for the equivalent density value is: where H represents the Heaviside function, D represents the design domain where a basic microstructure is located, and x represents the coordinate points within the design domain.

8. The cross-scale topology optimization method of the auxetic metamaterial with a specified deformation mode as described in claim 1, characterized in that: Use the homogenization method to calculate the equivalent elastic tensor of the negative Poisson's ratio microstructure. The calculation formula of the homogenization method is: where is the equivalent elastic tensor of the microstructure, Ω n and |Ω n | represent the design domain of the microstructure and the volume of the design domain, respectively, represents the locally varying microscopic strain field, represents the given macroscopic test strain field, i, j, k, l = 1, 2, …, d, where d is the spatial dimension, D pqrs represents the elastic tensor of the base material, H represents the Heaviside function, is related to the corresponding displacement field, is calculated by the following formula: wherein represents the virtual displacement field, represents the displacement field that is statically admissible under the periodic boundary conditions.

9. The cross-scale topology optimization method of the auxetic metamaterial with a specified deformation mode according to claim 3, characterized in that: Calculate the sensitivity information of the objective function with respect to the design variable ρ mn The sensitivity calculation formula is as follows:

10. The cross-scale topology optimization method of the auxetic metamaterial with a specified deformation mode according to claim 9, characterized in that: Calculate the sensitivity information of the constraint function with respect to the design variables s mn and ρ mn The sensitivity calculation formula is as follows:

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