Metamaterial with vanishing shear modulus and topological optimization design method thereof

Abstract

The application discloses a metamaterial with vanishing shear modulus and a topological optimization design method thereof, and belongs to the technical field of aerospace structures and advanced material design. The method comprises the following steps: establishing a design domain of a periodic cell and discretizing the design domain; constructing a topological optimization mathematical model with a relaxation function as an optimization target, converting a problem of maximizing a volume modulus to shear modulus ratio which is difficult to directly process into a weighted optimization problem which can be stably solved; combining an energy-based homogenization method to predict equivalent performance, and iteratively solving by using an improved moving asymptote method and a new floating projection function; and outputting a geometric model with clear boundaries through error verification and parameterized reconstruction. The application can design novel metamaterials with high B / G ratio, such as two-dimensional quasi-single mode / dual mode and three-dimensional quasi-triple mode, ensures seamless connection from optimization design to additive manufacturing, and provides an innovative material design solution for high-performance vibration isolation and acoustic regulation components in the field of aerospace.

Description

Technical Field

[0001] This invention belongs to the field of aerospace structure and advanced materials design technology, specifically relating to a metamaterial with vanishing shear modulus and its topology optimization design method. Background Technology

[0002] Metamaterials are a class of materials whose unique physical properties are achieved through the artificial design of their microstructure rather than by altering their composition. Among them, metamaterials with vanishing (extremely low) shear modulus (such as two-dimensional quasi-bimodal and three-dimensional quasi-pentamodal materials) can achieve near-fluid mechanical behavior in solid materials, showing great application potential in the aerospace field, such as for lightweight broadband vibration isolation, acoustic wave manipulation, and multifunctional load-bearing structures.

[0003] A key metric for measuring the "fluid-like" behavior of such materials is their bulk modulus. B With shear modulus G ratio α = B / G The larger this ratio, the stronger the material's resistance to volumetric deformation and the weaker its resistance to shear deformation, meaning it is closer to the mechanical response of an ideal fluid. Currently, the design of such metamaterials mainly relies on experience-based forward design methods, such as constructing classic biconical rod structures. These methods heavily depend on the designer's prior knowledge, have limited design freedom, and struggle to efficiently discover novel, high-performance topological configurations. Furthermore, while biconical connections are ideally treated as point connections, point connections are difficult to manufacture, and increasing the area at the connection point reduces performance. α value.

[0004] Topology optimization, as a reverse design method, automatically finds the optimal material distribution through mathematical modeling and algorithms, providing a powerful tool for metamaterial design. However, its application to designing low-shear modulus metamaterials still faces significant challenges. Existing methods struggle to directly and efficiently address the core objective of maximizing the ratio of bulk modulus to shear modulus (B / G). This is because optimizing this ratio leads to numerous numerical problems, such as instability, non-convergence, and incorrect optimization directions. Therefore, some research has turned to Poisson's ratio to improve this ratio. However, the improvement is limited, and isotropic constraints are required, limiting the algorithms to two-dimensional quasi-bimodal and three-dimensional quasi-pentamodal materials, thus preventing the discovery of novel metamaterials proposed in this invention. More importantly, avoiding point connections and generating clear and fabricable metamaterials to improve engineering applicability remains a pressing technical challenge.

[0005] Therefore, it is necessary to provide a metamaterial with vanishing shear modulus and its topology optimization design method to solve the above problems. Summary of the Invention

[0006] This invention provides a metamaterial with vanishing shear modulus and its topology optimization design method, aiming to solve the problems of numerical instability and convergence difficulties caused by directly optimizing the bulk modulus-shear modulus ratio B / G in the prior art, as well as the design limitations brought about by indirect methods such as Poisson's ratio optimization to avoid these problems.

[0007] The purpose of this invention is to provide an inverse design method for metamaterial microstructures with highly separated bulk modulus and shear modulus responses. This method aims to directly obtain practical metamaterial unit cells with extremely high B / G ratios (i.e., shear modulus tends to disappear) and clear topological configurations, fundamentally overcoming the difficulty in fabricating ideal "point-connected" models.

[0008] To achieve the above objectives, the present invention provides the following technical solution:

[0009] A topology optimization design method for metamaterials with vanishing shear modulus includes the following steps:

[0010] Step S1: Establish a periodic design domain for the target metamaterial unit cell, discretize the design domain, and set the initial material density distribution.

[0011] Step S2: Introduce a set of control parameters, which include at least weight coefficients for constructing the relaxed objective function. γ Initial aperture size a and quality constraint values M ;

[0012] Step S3, construct the relaxation function A topology optimization model with the objective of, where, For design variables, Equivalent bulk modulus Equivalent shear modulus;

[0013] Step S4: Calculate the current design variables based on the energy homogenization method. Corresponding equivalent bulk modulus With equivalent shear modulus ;

[0014] Step S5: Calculate the relaxation objective function. For the design variables The sensitivity is determined, and the design variables are updated based on the sensitivity information. ;

[0015] Step S6, update the design variables Density filtering is performed, and a 0 / 1 constraint is applied using a floating projection function, which is:

[0016] ;

[0017] in, For design variables The variables obtained after density filtering The density after projection. For projection threshold, These are projection control parameters;

[0018] Step S7: Determine whether the optimization process meets the convergence condition. If not, return to step S4 for iteration.

[0019] Step S8: Post-process and verify the converged material density distribution, and output the final metamaterial unit cell geometric model.

[0020] Optionally, the topology optimization model constructed in step S3, with structural quality as a constraint, is expressed as follows:

[0021] ;

[0022] in, M f Indicates the current design variable x e Structural quality, K, U A 、F These represent the global stiffness vector, displacement vector, and force vector, respectively, with superscripts. kl Indicates operating parameters; Indicates the unit number; This represents the minimum value of the design variable; N This indicates the number of discrete units in the discretization process.

[0023] Optionally, in step S4, for two-dimensional metamaterials, the equivalent bulk modulus and equivalent shear modulus Equivalent stiffness tensor obtained by homogenization method The calculation yielded:

[0024] ;

[0025] For three-dimensional metamaterials, the equivalent bulk modulus and equivalent shear modulus Equivalent stiffness tensor obtained by homogenization method C pqrs The calculation yielded:

[0026] .

[0027] Optionally, in step S5, the design variables are updated using the moving asymptote method. x e Furthermore, the upper limit for updating the design variables in the moving asymptote method is set to 1.001, and the lower limit is set to -0.001.

[0028] Optionally, step S8 specifically includes:

[0029] S8.1, Based on the converged density distribution, extract the explicit structural boundary of the metamaterial unit cell;

[0030] S8.2, Project the explicit structural boundary back to the analysis mesh and calculate the relaxation objective function. f ( x e The relative change τ ,in τ = |( f ( x e ) - f ( x e )) / f ( x e ) |, f ( x e The value of the objective function before post-processing is denoted as . f ( x e ) The objective function value is obtained by projecting the explicit structural boundary back onto the analysis mesh and then recalculating it.

[0031] S8.3, Determine the relative change. τ Whether it is less than a preset tolerance, where the preset tolerance is 1%;

[0032] If so, then based on the explicit structural boundary, reconstruct and output the final geometric model of the metamaterial unit cell;

[0033] If not, increase the strictness of the constraints and projection processing, and return to step S4.

[0034] Optionally, in step S8.3, reconstructing and outputting the final geometric model of the metamaterial unit cell specifically includes:

[0035] Based on the explicit structural boundary, the skeleton centerline of the structure is extracted by means of structural section separation and coordinate acquisition;

[0036] Along the centerline of the skeleton, a cross-section of a preset shape is swept or reconstructed to generate a solid three-dimensional geometric model.

[0037] The present invention also provides a metamaterial with vanishing shear modulus, the unit cell structure of which is obtained by the aforementioned topology optimization design method.

[0038] Optionally, the unit cell structure is a two-dimensional quasi-monomodal structure, a two-dimensional quasi-dual-mode structure, or a three-dimensional quasi-trimode structure.

[0039] Compared with the prior art, the advantages of this invention are as follows:

[0040] (1) This paper provides an effective way to directly and stably optimize the ratio of bulk modulus to shear modulus (B / G). This is achieved by constructing a relaxation objective function. f ( x e ) = - B ( x e )+ γG ( x e This transforms the ratio maximization problem, which is difficult to handle directly, into a weighted optimization problem that can be solved robustly, fundamentally avoiding numerical instability and convergence difficulties caused by the nonlinearity of the objective function.

[0041] (2) The method, through targeted improvements to the core optimization algorithm, demonstrates unique adaptability to the design challenges of low shear modulus and can efficiently generate novel topological configurations. Specifically: First, a novel floating projection function is adopted. Compared with the traditional floating projection, this form has better numerical stability and convergence characteristics when driving the intermediate density to 0 / 1 discretization. This allows it to more effectively balance exploration and convergence during the iteration process when dealing with the strongly nonlinear problem of B / G ratio optimization, directly promoting the clarity and stability of the final structural topology. At the same time, the improved floating projection limits the minimum structural size, preventing the occurrence of structural point connection problems. Second, the variable update boundary of the moving asymptote method is adaptively adjusted, extending the upper and lower limits to 1.001 and -0.001, respectively, to explore material distributions with extremely high B / G ratios. These targeted algorithm improvements, combined with the linear material interpolation model, jointly ensure that the method of this invention can stably and efficiently automatically discover novel topological configurations with extreme performances such as "quasi-single-mode", "quasi-dual-mode", and "quasi-tri-mode" that are difficult to obtain by traditional methods.

[0042] (3) Significantly improved manufacturability and process adaptability of the design. This is achieved through a built-in post-processing error verification and adaptive adjustment mechanism (such as...). τWith a tolerance of <1%, the accuracy of the conversion from numerical optimization results to geometric models is ensured. Building upon this, the unique skeleton extraction and parametric model reconstruction steps provide a high degree of flexibility in the design results: allowing for the flexible selection of different cross-sectional shapes (such as circles and hexagons) and the minimum manufacturable dimensions for reconstruction based on additive manufacturing processes and shape requirements, thereby generating a 3D solid model that can be directly used for machining. This not only fundamentally solves the problem of the inability to manufacture traditional "point-connected" ideal configurations, but also enables the same optimized topology to quickly adapt to various manufacturing schemes and performance requirements.

[0043] (4) The novel metamaterial structures designed have shown great application potential in aerospace and other fields. The "fluid-like" metamaterials with extremely high B / G ratio (i.e. extremely low shear modulus) provide a new core material solution for designing next-generation lightweight broadband vibration isolation components, high-performance acoustic control devices and multifunctional load-bearing structures, and are expected to solve the problem of vibration and noise control and structural function integration under extreme service environments. Attached Figure Description

[0044] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort, wherein:

[0045] Picture 1 A flowchart of a metamaterial with vanishing shear modulus and its topology optimization design method provided by the present invention;

[0046] Picture 2 This is a diagram showing the design domain and optimized design results in Embodiment 1 of the present invention;

[0047] Picture 3 This is a diagram showing the design domain and optimized design results in Embodiment 2 of the present invention;

[0048] Picture 4 This is a diagram showing the design domain and optimized design results in Embodiment 3 of the present invention. Detailed Implementation

[0049] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of the embodiments of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this invention, and not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0050] This invention provides a topology optimization design method for metamaterials with vanishing shear modulus, comprising the following steps:

[0051] Step S1: Establish a periodic design domain for the target metamaterial unit cell, discretize the design domain, and set the initial material density distribution.

[0052] Step S2: Introduce a set of control parameters, which include at least weight coefficients for constructing the relaxed objective function. γ Initial aperture size a and quality constraint values M ;

[0053] Step S3, construct the relaxation function A topology optimization model with the objective of, where, For design variables, Equivalent bulk modulus Equivalent shear modulus;

[0054] Step S4: Calculate the current design variables based on the energy homogenization method. Corresponding equivalent bulk modulus With equivalent shear modulus ;

[0055] Step S5: Calculate the relaxation objective function. For the design variables The sensitivity is determined, and the design variables are updated based on the sensitivity information. ;

[0056] Step S6, update the design variables Density filtering is performed, and a 0 / 1 constraint is applied using a floating projection function, which is:

[0057] ;

[0058] in, For design variables The variables obtained after density filtering The density after projection. For projection threshold, These are projection control parameters;

[0059] Step S7: Determine whether the optimization process meets the convergence condition. If not, return to step S4 for iteration.

[0060] Step S8: Post-process and verify the converged material density distribution, and output the final metamaterial unit cell geometric model.

[0061] In step S3, the constructed topology optimization model, with structural quality as the constraint, is expressed as follows:

[0062] ;

[0063] in, M f Indicates the current design variable x e Structural quality, K, U A 、F These represent the global stiffness vector, displacement vector, and force vector, respectively, with superscripts. kl Indicates operating parameters; Indicates the unit number; This represents the minimum value of the design variable; N This indicates the number of discrete units in the discretization process.

[0064] In step S4, for two-dimensional metamaterials, the equivalent bulk modulus and equivalent shear modulus Equivalent stiffness tensor obtained by homogenization method The calculation yielded:

[0065] ;

[0066] in, It is the equivalent stiffness tensor The amount inside, of which .

[0067] For three-dimensional metamaterials, the equivalent bulk modulus and equivalent shear modulus Equivalent stiffness tensor obtained by homogenization method C pqrs The calculation yielded:

[0068] .

[0069] In step S5, the design variables are updated using the moving asymptote method. x e Furthermore, the upper limit for updating the design variables in the moving asymptote method is set to 1.001, and the lower limit is set to -0.001.

[0070] Step S8 specifically includes:

[0071] S8.1, Based on the converged density distribution, extract the explicit structural boundary of the metamaterial unit cell;

[0072] S8.2, Project the explicit structural boundary back to the analysis mesh and calculate the relaxation objective function. f( x e The relative change τ ,in τ = |( f ( x e ) - f ( x e )) / f ( x e ) |, f ( x e The value of the objective function before post-processing is denoted as . f ( x e ) The objective function value is obtained by projecting the explicit structural boundary back onto the analysis mesh and then recalculating it.

[0073] S8.3, Determine the relative change. τ If the value is less than a preset tolerance of 1%, then based on the explicit structural boundary, the final geometric model of the metamaterial unit cell is reconstructed and output; otherwise, the strictness of the constraint and projection processing is increased, and the process returns to step S4.

[0074] Furthermore, in step S8.3, reconstructing and outputting the final geometric model of the metamaterial unit cell specifically includes:

[0075] Based on the explicit structural boundary, the skeleton centerline of the structure is extracted by means of structural section separation and coordinate acquisition;

[0076] Along the centerline of the skeleton, a cross-section of a preset shape is swept or reconstructed to generate a solid three-dimensional geometric model.

[0077] The present invention also provides a metamaterial with vanishing shear modulus, the unit cell structure of which is obtained by the aforementioned topology optimization design method.

[0078] The unit cell structure is a two-dimensional quasi-monomodal structure, a two-dimensional quasi-dual-mode structure, or a three-dimensional quasi-trimode structure.

[0079] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be described in detail below with reference to specific embodiments. However, it should be understood that these embodiments are only for illustrating this invention and should not be construed as limiting this invention.

[0080] Example 1

[0081] This embodiment provides a topology optimization design method for metamaterials with vanishing shear modulus, specifically applied to the design of two-dimensional metamaterial unit cells.

[0082] Step S1: Establish a periodic design domain for the target metamaterial unit cell, discretize the design domain, and set an initial material density distribution. In this embodiment, the design domain has a side length of [missing information]. L A square region with a value of 1, such as Picture 2 As shown. The design domain is discretized using the finite element method, dividing it into 200 × 200 four-node rectangular elements. An initial material density distribution is set, and a radius of [missing information] is preset at the center of the design domain. a The circular hole has a relative density of 0 for the elements within the hole area, and the initial relative density of the elements in the rest of the design domain is the mass constraint value in subsequent steps.

[0083] Step S2 introduces a set of control parameters. Specifically, regarding material parameters, the matrix is ​​an isotropic linear elastic material with a Young's modulus of... E = 1, Poisson's ratio ν = 0.3; Objective function weight coefficient γ = 0.5, used to construct the relaxation objective function. Initial aperture size a = 1 / 3 L The constraint is the upper limit of the quality fraction. M = 0.1; Regarding projection parameters, the initial control parameters for floating projection are... β 0 = 4, parameter β The increment is 1, which expands the optimization space by using the extension concept and makes it less likely to get trapped in local optima.

[0084] Step S3, construct the relaxation function For the topology optimization model with the objective as the target, mass constraints are selected as the constraint conditions. The established optimization model can be expressed as:

[0085] ;

[0086] In the formula, x e Indicates design variables, N This indicates the number of discrete units in step S1 discretization. f ( x e ) represents the relaxation function established. B ( x e )and G ( x e ) represent the metamaterial unit cell under the current design variables.x e The equivalent bulk modulus and equivalent shear modulus M f Indicates the current design variable x e Structural quality, K, U A 、F These represent the global stiffness vector, displacement vector, and force vector, respectively, with superscripts. kl Indicates operating parameters, Indicates the unit number. This represents the minimum value of the design variable. N This represents the number of discrete units in the discretization. By introducing a relaxation objective function, this embodiment transforms the problem of "maximizing the ratio of bulk modulus to shear modulus B / G," which is originally difficult to handle directly and prone to numerical instability, into a weighted summation problem that can be robustly solved by standard optimization algorithms, thereby fundamentally avoiding the oscillation and convergence difficulties in the optimization process.

[0087] Step S4: Calculate the current design variables based on the energy homogenization method. x e The equivalent bulk modulus B of the corresponding metamaterial unit cell x e ) and equivalent shear modulus G( x e By applying periodic displacement constraints on the corresponding boundaries of the discretized design domain, the simulated elements can represent a portion of an infinite periodic array. By solving a series of micromechanical equilibrium equations under a unit test strain field, a periodic fluctuating displacement field is obtained, from which the equivalent elastic tensor is calculated. For two-dimensional metamaterials, bulk modulus and shear modulus The calculation is as follows:

[0088] .

[0089] The energy-based homogenization method used in this step avoids the complex multi-scale derivative solution process in traditional asymptotic homogenization methods by directly applying the unit test strain field and utilizing the energy equivalence principle, significantly improving computational efficiency and providing rapid performance feedback for subsequent sensitivity analysis and iterative optimization.

[0090] Step S5: Calculate the relaxation objective function. f ( x e For the design variables The sensitivity is determined, and the design variables are updated based on the sensitivity information. This embodiment employs the automatic difference method to calculate sensitivity, which can obtain gradient information of complex objective functions with high accuracy and efficiency. It also updates design variables based on the moving asymptote method. In this design, the upper limit for updating the design variables in each iteration is set to 1.001, and the lower limit is set to -0.001. This key improvement is an important adaptation to the highly nonconvex and nonlinear characteristics of low shear modulus optimization problems. It provides the necessary feasible space for drastic adjustments to the design variables in the early stages of optimization, ensuring that iterative updates can be carried out effectively, thereby enabling the optimization process to proceed stably and eventually converge.

[0091] Step S6, update the design variables Density filtering is performed, and a floating projection function is used to apply 0 / 1 constraints. First, a distance-weighted linear filtering method is used to smooth the design variables, resulting in density-filtered variables. This effectively suppresses unstable chessboard patterns. Subsequently, a novel floating projection function is employed to obtain clear topological boundaries:

[0092] ;

[0093] in, For design variables The variables obtained after density filtering The density after projection. For projection threshold, These are the projection control parameters. Compared to traditional floating projection, this novel projection function provides a comprehensive consideration of the design variables before projection in form. By comprehensively considering both information, the optimization process can more balancedly take into account both discretization-driven and topological evolution stability, thereby maintaining good convergence robustness while pursuing clear 0 / 1 boundaries.

[0094] Step S7: Determine if the optimization process meets the convergence condition. After each variable update and projection process in step S6, calculate the maximum change Δ of the design variables between the current iteration and the previous iteration. x max When Δ x max If the value is less than 0.01, the optimization process is deemed to meet the convergence condition, and the process proceeds to step S8 for post-processing and output; otherwise, the process returns to step S4 and restarts the homogenization analysis and subsequent update cycle based on the new design variables.

[0095] Step S8 involves post-processing and verifying the converged material density distribution. Specifically, this includes:

[0096] S8.1, Based on the converged density distribution, the explicit, smooth structural boundary of the metamaterial unit cell is extracted using the level set method;

[0097] S8.2, map the extracted explicit structural boundaries back to the original finite element mesh, and calculate the relaxation objective function. f ( x e The relative change τ ,in τ = |( f ( x e ) - f ( x e )) / f ( x e ) |, f ( x e The value of the objective function before post-processing is denoted as . f ( x e ) To evaluate the numerical errors introduced by post-processing, the objective function value is recalculated after projecting the explicit structural boundary back onto the analysis mesh.

[0098] S8.3, Determine the relative change. τ Whether it is less than a preset tolerance, where the preset tolerance is 1%;

[0099] If so, the final geometric model of the metamaterial unit cell is reconstructed and output based on the explicit structural boundary. Specifically, the skeleton centerline describing the structural topology is extracted by means of structural section separation and coordinate acquisition. Along the skeleton centerline, the model is swept or reconstructed according to the preset cross-sectional shape (such as a circle) and size to generate a three-dimensional geometric model that can be directly used for additive manufacturing.

[0100] If not, then increase the stringency of constraints and projection processing, for example, by adjusting the floating projection parameters. β Add a fixed increment and return to step S4 to start a new round of iterative optimization until the error tolerance is met.

[0101] Picture 2 The optimized structure is demonstrated, including the microstructure unit cell, periodic array, effective elastic tensor, and performance calculations. The final structure has a bulk modulus of 2.7e-2 and a shear modulus of 1.9e-4, with a ratio of 142, far exceeding the single-digit values ​​of solid materials, achieving a high degree of separation between compressive and shear properties. In terms of eigenvalues, they are 5.3e-2, 9e-4, and 1e-4, respectively, exhibiting one large eigenvalue and two smaller ones, classifying it as a novel two-dimensional quasi-bimodal metamaterial suitable for acoustic and mechanical stealth metamaterial applications.

[0102] Example 2

[0103] This embodiment changes the initial design domain based on Embodiment 1, and the initial aperture size is set to... a =1 / 4 L The optimization steps are consistent with those in Example 1. The initial design domain and the optimized results are as follows: Picture 3 As shown.

[0104] The final structure has a bulk modulus of 2.7e-2 and a shear modulus of 5.3e-5, with a ratio of 509, far exceeding the single-digit values ​​for solid materials and higher than that of Example 1. In terms of eigenvalues, they are 5.3e-2, 3.8e-3, and 5e-5, respectively, exhibiting one large eigenvalue, one secondary eigenvalue, and one small eigenvalue, classifying it as a novel two-dimensional quasi-single-mode metamaterial suitable for acoustic manipulation and the design of multifunctional materials. This embodiment demonstrates the good adaptability of the method of this invention to different initial designs and its ability to generate novel topological configurations with superior performance.

[0105] Example 3

[0106] This embodiment extends the method of Embodiment 2 to three-dimensional topology optimization. The initial design domain is as follows: Picture 4 As shown, a radius of is preset at the center of the design domain. a =1 / 3 L The spherical holes are meshed at 80 × 80 × 80. The objective function weight coefficient is set to γ ​​= 0.67, and the design and optimization steps for the remaining parameters are consistent with the above embodiment.

[0107] The optimized result is as follows Picture 4 As shown, the final structure has a bulk modulus of 3.6e-4 and a shear modulus of 1.5e-6, with a ratio of 240, far exceeding the single-digit values ​​of solid materials. In terms of eigenvalues, they are 1.07e-3, 8.52e-5, 8.52e-5, 1.56e-6, 1.56e-6, and 1.56e-6, respectively, exhibiting one large eigenvalue, two secondary eigenvalues, and three smaller eigenvalues. This constitutes a novel three-dimensional quasi-trimodal metamaterial, applicable to acoustic manipulation and the design of multifunctional load-bearing materials. This embodiment successfully extends the method of this invention from two dimensions to three dimensions, demonstrating its universality and effectiveness across different dimensions.

[0108] In summary, this invention provides a metamaterial with vanishing shear modulus and its topology optimization design method. By constructing a relaxed objective function, employing an improved moving asymptote method and a novel floating projection function, and combining an energy-based homogenization method with a post-processing verification mechanism, it can directly and stably design novel metamaterial topologies with extremely high B / G ratios, such as two-dimensional quasi-single-mode / dual-mode and three-dimensional quasi-trimode. This method not only avoids the artificial restrictions on isotropy in principle, allowing for the free exploration of entirely new topological configurations, but also ensures that the final design possesses both high fidelity and high engineering feasibility through built-in error control and post-processing mechanisms. It achieves a seamless transition from optimization design to additive manufacturing, providing an innovative material design solution for high-performance vibration isolation and acoustic control components in the aerospace field.

[0109] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A topology optimization design method for metamaterials with vanishing shear modulus, characterized in that, Includes the following steps: Step S1: Establish a periodic design domain for the target metamaterial unit cell, discretize the design domain, and set the initial material density distribution. Step S2: Introduce a set of control parameters, which include at least weight coefficients for constructing the relaxed objective function. γ Initial aperture size a and quality constraint values M ; Step S3, construct the relaxation function A topology optimization model with the objective of, where, For design variables, Equivalent bulk modulus Equivalent shear modulus; Step S4: Calculate the current design variables based on the energy homogenization method. Corresponding equivalent bulk modulus With equivalent shear modulus ; Step S5: Calculate the relaxation objective function. For the design variables The sensitivity is determined, and the design variables are updated based on the sensitivity information. ; Step S6, update the design variables Density filtering is performed, and a 0 / 1 constraint is applied using a floating projection function, which is: ； in, For design variables The variables obtained after density filtering The density after projection. For projection threshold, These are projection control parameters; Step S7: Determine whether the optimization process meets the convergence condition. If not, return to step S4 for iteration. Step S8: Post-process and verify the converged material density distribution, and output the final metamaterial unit cell geometric model.

2. The topology optimization design method according to claim 1, characterized in that, The topology optimization model constructed in step S3, with structural quality as the constraint, is expressed as follows: ； in, M f Indicates the current design variable x e Structural quality, K, U A 、F These represent the global stiffness vector, displacement vector, and force vector, respectively, with superscripts. kl Indicates operating parameters; Indicates the unit number; This represents the minimum value of the design variable; N This indicates the number of discrete units in the discretization process.

3. The topology optimization design method according to claim 1, characterized in that, In step S4, for two-dimensional metamaterials, the equivalent bulk modulus and equivalent shear modulus Equivalent stiffness tensor obtained by homogenization method The calculation yielded: ； For three-dimensional metamaterials, the equivalent bulk modulus and equivalent shear modulus Equivalent stiffness tensor obtained by homogenization method C pqrs The calculation yielded: 。 4. The topology optimization design method according to claim 1, characterized in that, In step S5, the design variables are updated using the moving asymptote method. x e Furthermore, the upper limit for updating the design variables in the moving asymptote method is set to 1.001, and the lower limit is set to -0.

001.

5. The topology optimization design method according to claim 1, characterized in that, Step S8 specifically includes: S8.1, Based on the converged density distribution, extract the explicit structural boundary of the metamaterial unit cell; S8.2, Project the explicit structural boundary back to the analysis mesh and calculate the relaxation objective function. f ( x e The relative change τ ,in τ = |( f ( x e ) - f ( x e )) / f ( x e ) |, f ( x e The value of the objective function before post-processing is denoted as . f ( x e ) The objective function value is obtained by projecting the explicit structural boundary back onto the analysis mesh and then recalculating it. S8.3, Determine the relative change. τ Whether it is less than a preset tolerance, where the preset tolerance is 1%; If so, then based on the explicit structural boundary, reconstruct and output the final geometric model of the metamaterial unit cell; If not, increase the strictness of the constraints and projection processing, and return to step S4.

6. The topology optimization design method according to claim 5, characterized in that, In step S8.3, reconstructing and outputting the final geometric model of the metamaterial unit cell specifically includes: Based on the explicit structural boundary, the skeleton centerline of the structure is extracted by means of structural section separation and coordinate acquisition; Along the centerline of the skeleton, a cross-section of a preset shape is swept or reconstructed to generate a solid three-dimensional geometric model.

7. A metamaterial with vanishing shear modulus, characterized in that, Its unit cell structure is obtained by the topology optimization design method according to any one of claims 1 to 6.

8. The metamaterial according to claim 7, characterized in that, The unit cell structure is a two-dimensional quasi-monomodal structure, a two-dimensional quasi-dual-mode structure, or a three-dimensional quasi-trimodal structure.

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