Negative Poisson's ratio anisotropic multi-material microstructure topological optimization method based on EFGM

Abstract

The invention discloses an EFGM-based topological optimization method for a negative Poisson's ratio anisotropic multi-material microstructure, and the method comprises the steps: (1) inputting a Poisson's ratio factor, a material direction angle and other material attributes of the anisotropic multi-material microstructure, and carrying out the discrete design of a domain based on a meshless EFGM; (2) initializing the relative density of multi-material meshless EFGM nodes, and setting iteration parameters of each stage; (3) setting initial parameters of an active alternating phase algorithm; (4) calculating the total number of times of hierarchical iteration; (5) establishing a negative poisson ratio anisotropic multi-material microstructure topological optimization mathematical model based on the meshless EFGM, and calculating a meshless EFGM node displacement field and an equivalent tensor matrix; (6) carrying out sensitivity analysis on the EFGM topological optimization model; (7) updating meshless EFGM node relative density by using an OC method and a multi-material two-phase competition cycle; and (8) judging whether a convergence condition is met or not, and outputting an EFGM optimal negative Poisson's ratio metamaterial topological structure. The topological optimization method for the anisotropic multi-material microstructure based on the EFGM negative Poisson's ratio is stable in convergence and good in practicability.

Description

Technical Field

[0001] This invention belongs to the field of structural optimization design in computer-aided engineering, specifically involving a topology optimization method for negative Poisson's ratio anisotropic multi-material microstructures based on the element-free Galerkin method (EFGM). Background Technology

[0002] Negative Poisson's ratio metamaterials are artificial materials with a unique microstructure and a Poisson's ratio less than 0. When subjected to tension or compression in the longitudinal direction, they expand or contract in the transverse direction, exhibiting excellent mechanical properties that bring new possibilities for solving numerous engineering and scientific problems. Therefore, they have broad application prospects in aerospace, marine, mechanical automation, and biomedicine. The design of negative Poisson's ratio metamaterials primarily starts from the microstructure, achieving precise control and optimization of material properties by altering the material distribution within the microstructure to adapt to different application requirements. However, designing the microstructure of negative Poisson's ratio metamaterials using traditional design methods is quite difficult. Structural topology optimization, as a cutting-edge design method, provides an excellent option for the design of negative Poisson's ratio metamaterials.

[0003] Structural topology optimization utilizes performance analysis and optimization theory to obtain the optimal material distribution within the design domain. Due to its excellent designability for materials, this method has been applied in numerous engineering fields. Currently, widely used numerical analysis methods in structural performance analysis include the finite element method, finite volume method, finite difference method, and isogeometric analysis methods. These methods all relate surrounding nodes based on a mesh. However, due to the special nature of the mesh, topology optimization requires the introduction of sensitivity filtering, density filtering, and size constraints to eliminate checkerboard phenomena, mesh dependence, and numerical instability in the topology. Meshless methods, on the other hand, only require discretizing the computational domain using nodes during structural performance analysis, eliminating the need for mesh generation and thus avoiding the drawbacks associated with meshes. Furthermore, meshless methods are easier to construct higher-order field functions compared to other methods, thus providing higher accuracy in performance analysis. The relatively mature meshless Galerkin method has been applied to the optimization design of negative Poisson's ratio metamaterial structures. However, most current topology optimization methods for negative Poisson's ratio metamaterials are designed for single isotropic materials. As the demands for material performance continue to increase, the performance limitations of single isotropic materials have gradually become apparent.

[0004] Multi-material composite structures combine two or more materials. Compared to single-material structures, multi-material structures can adapt materials with corresponding properties according to the performance requirements of different locations within the structure. The performance advantages of various materials can complement each other, thus overcoming the shortcomings of single-material structures. Unlike most isotropic materials, a significant characteristic of multi-material composite structures is their anisotropy, meaning they exhibit different physical properties in different directions, such as elastic modulus, thermal conductivity, and tensile strength. Introducing anisotropic multimaterials into the topology optimization design of negative Poisson's ratio metamaterial microstructures provides a larger design space, combining the negative Poisson's ratio characteristics with the superior properties of other materials to better leverage the advantages of multimaterials. Furthermore, most topology optimizations of negative Poisson's ratio metamaterial microstructures rarely consider the periodicity of the structure. Introducing periodic structures into the design of negative Poisson's ratio microstructures can further improve the stability and load-bearing capacity of the structure, while reducing its weight and manufacturing costs. In summary, this invention proposes a topology optimization method for negative Poisson's ratio anisotropic multi-material microstructures based on meshless EFGM. Summary of the Invention

[0005] With the rapid development of industrial technology, more and more industries are constantly raising their requirements for structural design, no longer satisfied with traditional structural design, and urgently needing a superior structural design method to meet stringent performance requirements. Structural topology optimization is also well-suited for lightweight design under complex conditions, demonstrating outstanding performance in structural design. This invention uses EFGM (Extended Front-Oriented Generalized Mechanism) to perform performance calculations on the computational domain, and constructs a structural topology optimization model using the Solid Isotropic Material with Penalization (SIMP) method based on the variable density method; it uses the relative density of EFGM nodes as the design variable and volume fraction as the constraint condition; it employs an alternating active phase algorithm to conduct two-phase competition among multiple materials; it establishes a topology optimization model for a negative Poisson's ratio anisotropic multi-material periodic microstructure based on EFGM, with the minimum negative Poisson's ratio as the objective; and it writes a topology optimization program that can optimize the design for different Poisson's ratio factors, initial design domains, number of multiple material types, and orientation angles of multiple materials, and outputs the iterative process and the optimal topology.

[0006] The technical solution adopted by this invention to solve its technical problem is as follows: a topology optimization method for negative Poisson's ratio anisotropic multi-material periodic microstructures based on meshless EFGM is used to discretize the computational domain, and a topology optimization mathematical model is constructed by combining the SIMP method and the discrete static control equation of EFGM; multi-material two-phase competition is carried out through alternating active phase algorithm; and the material density distribution is explored by adjusting the Poisson's ratio factor, the initial design domain shape, the number of multi-material types, and the orientation angle of multi-materials.

[0007] The specific implementation steps of the technical solution described in this invention are as follows: (1) Based on the design requirements of the actual structure, determine the initial design domain d of the microstructure and the total number of material types q; input the mechanical parameters of each material: elastic modulus Poisson's ratio shear modulus Poisson's ratio factor PRF, relative volume fraction ν i and multi-material orientation angle θ; (2) Based on the initial design domain, the design domain is discretized using EFGM, and Gaussian points are arranged inside and on the boundary of the design domain; the initial multi-material EFGM node relative density vector is set. And assembled into a matrix Set the number of horizontal and vertical design subdomains: M x M y Set iteration parameters: minimum tolerance ch for the relative density difference of multi-material EFGM nodes. min Maximum number of steps in first-level iteration r max Maximum number of steps in the second-level iteration s max Maximum number of steps t in three-level iterations max The initial number of first-level iterations is r; (3) Set the initial number of second-level iterations: s = 1; Initial parameters for the two-phase material competitive optimization subproblem: a = 1, b = a + 1; (4) Set the initial number of three-level iterations: t = 1; (5) Calculate the total number of iterations for the first, second, and third levels: k = t max s max (r-1)+t max (s-1)+t; (6) Calculation of the elastic matrix of anisotropic multimaterials based on meshless EFGM: Orthogonal anisotropic materials have independent and unaffected single values ​​in terms of elastic modulus and Poisson's ratio in two perpendicular directions. In order to better characterize the properties of orthogonal anisotropic multimaterials, a multimaterial rectangular coordinate system 1-2 is established, and the multimaterial direction angle θ is defined as the angle between the x-axis and the 1-axis in the Cartesian global coordinate system xy. According to the generalized Hooke's law, the mechanical constitutive relation of orthogonal anisotropic multimaterials is constructed. in In the above formula, σ x , σ y , ε x , ε y Let τ be the stress and strain in the x and y directions of the structure. xy and γ xy These are the shear stress and shear strain of the material, respectively. Let Q be the elasticity matrix of the i-th material (i = 1, 2, ..., q) in the global coordinate system xy. i Let T be the elastic matrix of the i-th material in the multi-material coordinate system 1-2. s To be With Q i The associated coordinate transformation matrix; in Q i middle, in, Let μ be the elastic modulus of the i-th material in directions 1 and 2 in the multi-material coordinate system 1-2. 12 μ 21 These are the Poisson's ratios in directions 1 and 2 of the multi-material coordinate system 1-2, respectively. Let i be the shear modulus of the i-th material, expressed by the formula Approximate calculation; definition of Poisson's ratio factor Indicates the orthotropic strength of mechanical properties; (7) Combining the SIMP multi-material interpolation model and the EFGM discrete static control equations, a mathematical model for topology optimization of periodic microstructures of negative Poisson's ratio metamaterials based on meshless EFGM is established. In the formula, N s The total number of discrete EFGM nodes in a multi-material microstructure. Let be the relative density of the EFGM nodes of the ith material at the j-th node of the multi-material microstructure. The objective function for constructing the minimum negative Poisson's ratio is expressed as follows: Where β∈(0,1) is a constant value. It is the equivalent elastic tensor, obtained by the homogenization method. K is the global EFGM nodal stiffness matrix with penalty, and F is the EFGM nodal load matrix with penalty. V is the relative density at the Gaussian point of the i-th material. i V represents the total integral of the i-th material. 0 The initial total integral is given; by applying periodic boundary conditions to the microstructure, the discrete static control equation KU = F of the EFGM is solved to obtain the nodal displacements U of the EFGM; the equivalent elastic tensor matrix is ​​then solved. (8) Solve for the volume sensitivity and the objective function sensitivity; wherein, the volume sensitivity analysis is performed through... The minimum negative Poisson's ratio objective function sensitivity was obtained through... Therefore, β in the formula is obtained. k Define a fixed parameter for the k-th iteration; (9) The design variables are updated using the Optimality Criteria (OC) method. The relative density matrix of the a-th material in the k-th iteration of the multi-material EFGM node is: in The following formula can be used to calculate it. In the formula Let be the material relative density matrix of the ith material (type a) in the ith iteration, τ = 1 be the damping coefficient, and m0 = 0.2 be the positive displacement constraint. Defined as The relative density matrix of the b-th material in the k-th iteration of the multi-material EFGM node is: in, The infinite norm of the difference in the relative density matrix of the meshless EFGM nodes obtained by calculation iteration (10) Compare t with t max The size relationship determines whether the third-level iteration should continue or end: if t < t max If the condition is met, update the third-level iteration step number t = t + 1, and return to step (5) to continue the third-level iteration; if t < t max If this condition is not met, then the current three-level iteration ends. (11) Compare s with s max The size relationship determines whether the second-level iteration should continue or end: if s < s max If both s and b < q hold true, then let s = s + 1, b = b + 1, and return to step (4) to continue the second-order iteration; if s < s max If the condition is true but b < q is false, then let s = s + 1, a = a + 1, b = a + 1, and return to step (4) to continue the second-order iteration; if s < s max If this condition is not met, then the second-level iteration of this round ends; (12) Compare r with r max and ch min The size relationship determines whether the first-level iteration should continue or end: if r < r max It holds true and ch > ch min If true, update the first-level iteration step number r = r + 1, and return to step (3) to continue executing the first-level iteration; if r < r max Not true or ch > ch min If the condition is not met, the first-level iteration ends, and step (13) is executed. (13) Output the optimal topological configuration of the periodic microstructure of the negative Poisson's ratio metamaterial.

[0008] The beneficial effects of this invention are as follows: Based on the meshless Galerkin method and the SIMP method, this invention can effectively eliminate the problems of mesh dependence, checkerboard patterns, and numerical instability that easily occur in topologies obtained based on common topology optimization methods; by setting the minimum negative Poisson's ratio objective function, this invention can obtain the microstructure topology of negative Poisson's ratio metamaterials, which can meet the special requirements of engineering problems; this invention can control the mechanical properties of anisotropic structures by adjusting the orientation angles of multiple materials and the Poisson's ratio factor; based on the alternating active phase algorithm, this invention conducts two-phase competition among multiple materials, and controls the multi-material distribution of the final topology by setting the number of multiple material types; this invention can design periodic negative Poisson's ratio microstructures through microstructure periodic constraints; this invention can adjust the initial design domain according to actual engineering conditions and can adapt to diverse design domain requirements; this invention integrates multiple factors such as meshless methods, multiple materials, anisotropy, and negative Poisson's ratio metamaterial microstructures, enabling the optimal topology to meet numerous requirements while achieving the required mechanical performance. Attached Figure Description

[0009] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0010] Figure 1 This is a flowchart illustrating the topology optimization method for negative Poisson's ratio anisotropic multi-material microstructures of the present invention. Figure 2 The global coordinate system and material coordinate system of the anisotropic structure of this invention Figure 3 This is a schematic diagram showing the dimensions under which the periodic boundary conditions are applied according to the present invention. Figure 4 A schematic diagram of the classification of rectangular microstructure nodes. Figure 5 This is a schematic diagram of the multi-material design domain in an embodiment of the present invention. Figure 6 This is a schematic diagram of the shape of the initial design domain d=1 in an embodiment of the present invention. Figure 7 This is a schematic diagram of the shape of the initial design domain d=2 in an embodiment of the present invention. Figure 8 This is the optimal topological configuration of a negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 1, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 3. Figure 9 This is the optimal topological configuration of a negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 1, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4. Figure 10This is the optimal topological configuration of a negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 2, the Poisson's ratio factor PRF is 1, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4. Figure 11 This is the optimal topological configuration of a negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 1 / 4, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4. Figure 12 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 1 / 3, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4. Figure 13 This is the optimal topological configuration of a negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 1 / 2, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4. Figure 14 This is the optimal topological configuration of a negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 1, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4. Figure 15 This is the optimal topological configuration of a negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 2, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4. Figure 16 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 3, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4. Figure 17 This is the optimal topological configuration of a negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 4, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4. Figure 18 This is the optimal topological configuration of a negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4. Figure 19 This is the optimal topological configuration of a negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 15°, and the number of multi-material types q is 4. Figure 20 This is the optimal topological configuration of a negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 30°, and the number of multi-material types q is 4. Figure 21 This is the optimal topological configuration of a negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 45°, and the number of multi-material types q is 4. Figure 22 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 60°, and the number of multi-material types q is 4. Figure 23 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 75°, and the number of multi-material types q is 4. Figure 24 This is the optimal topological configuration of a negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 90°, and the number of multi-material types q is 4. Figure 25 In this embodiment of the invention, when the initial design domain shape d is 1, the number of multiple material types q is 4, and the multiple material orientation angle θ is 0, different Poisson's ratio factors are used. The minimum value of the objective function and the minimum negative Poisson's ratio under the given conditions Figure 26 These are the minimum objective function values ​​and the minimum negative Poisson's ratio values ​​under the following conditions in this embodiment of the invention: initial design domain shape d = 1, number of material types q = 4, and Poisson's ratio factor PRF = 1. Detailed Implementation

[0011] See Figure 1 and Figure 2 The topology optimization method for negative Poisson's ratio anisotropic multi-material microstructures based on EFGM mainly includes the following steps: First, based on the actual engineering requirements, determine the initial design domain d of the microstructure and the total number of material types q, and input the mechanical parameters of the orthogonal anisotropic multimaterial structure: elastic modulus. Poisson's ratio shear modulus Poisson's ratio factor PRF, relative volume fraction ν iAnd the multi-material orientation angle θ; according to the generalized Hooke's law, the stress-strain relationship can be constructed under the anisotropic multi-material coordinate system 1-2 and the global coordinate system xy as follows: In the formula, Let Q be the elasticity matrix of the i-th material (i = 1, 2, ..., q) in the global coordinate system xy. i Let T be the elastic matrix of the i-th material in the multi-material coordinate system 1-2. s To be With Q i Associated coordinate transformation matrix; Q i middle, in, μ 12 μ 21 Let be the elastic modulus and Poisson's ratio of the i-th material in directions 1 and 2 of the multi-material coordinate system 1-2, respectively. Let i be the shear modulus of the i-th material, expressed by the formula Approximate calculation; definition of Poisson's ratio factor If PRF represents the orthotropic strength, then when PRF = 1 and θ = 0°, the structure is an isotropic structure.

[0012] Based on the initial design domain, a meshless EFGM discretization domain is adopted, and Gaussian points are arranged inside the design domain; the initial multi-material EFGM node relative density vector is predefined based on the initial material relative volume fraction. And assembled into a total multi-material density matrix Set the number of horizontal and vertical design subdomains: M x M y Set iteration parameters: minimum tolerance ch for relative density change at multi-material EFGM nodes. min Maximum number of steps in first-level iteration r max Maximum number of steps in the second-level iteration Maximum number of steps in the third-level iteration t max .

[0013] Secondly, based on the SIMP method for interpolation of multiple materials, the relationship between the Gaussian point material properties of the i-th material in the k-th iteration and the initially defined elastic modulus and shear modulus is as follows: In the formula, Let p be the Gaussian point relative density after penaltying for the i-th material, and p be the penalty intensity, typically p≥3; a continuous Gaussian point relative density field. The relative density of EFGM nodes within the Gaussian point support domain can be approximated by the MLS shape function constructed using the moving least squares method. In the formula, Let φ be the relative density of the i-th node within the Gaussian point influence domain of the i-th material. I Let n be the MLS shape function of the corresponding node, and n be the number of nodes in the Gaussian point support domain.

[0014] Finally, combining the SIMP multi-material interpolation model and the EFGM discrete static control equations, a mathematical model for topology optimization of periodic microstructures of negative Poisson's ratio metamaterials based on meshless EFGM is established. The nodal displacements of the meshless EFGM are solved, and the relative density of the nodes is updated using the OC method. The relative density analysis of the multi-material structure is then performed using the alternating active phase algorithm. The topology optimization mathematical model is as follows: In the formula, N s The total number of discrete EFGM nodes in a multi-material microstructure. Let be the relative density of the ith material at the j-th node of the multi-material microstructure. The objective function for minimizing the negative Poisson's ratio is expressed as follows: Where β∈(0,1) is a constant value. It is the equivalent elastic tensor, obtained by the homogenization method. K is the global EFGM nodal stiffness matrix with penalty, and F is the EFGM load matrix with penalty. V is the relative density at the Gaussian point of the i-th material. i V is the total integral of the i-th material. 0 The initial total integral is given; by applying periodic boundary conditions to the microstructure, the nodal displacements U are obtained by solving the discrete static control equations of the EFGM; the equivalent elastic tensor matrix is ​​then solved. The material is subjected to a two-phase competition cycle based on the alternating active phase algorithm to obtain the relative density of EFGM nodes in a single first-level iteration, thereby completing one iteration cycle; finally, the final topology structure is output by judging whether the set iteration termination condition is met.

[0015] See Figure 1 The specific steps of the topology optimization method for negative Poisson's ratio anisotropic multi-material microstructures based on EFGM are as follows: (1) Based on the design requirements of the actual engineering structure, determine the initial design domain d and the total number of material types q for the anisotropic structure, and input the elastic modulus of the anisotropic structure. Poisson's ratio shear modulus Poisson's ratio factor PRF, relative volume fraction νi and multi-material orientation angle θ; (2) Discretize the design domain using EFGM nodes, and arrange Gaussian points inside and on the boundary of the design domain; set the initial multi-material EFGM node relative density vector. And assembled into a matrix Set the number of horizontal and vertical design subdomains: M x M y Set iteration parameters: minimum tolerance ch for relative density change at multi-material EFGM nodes. min Maximum number of steps in first-level iteration r max Maximum number of steps in the second-level iteration Maximum number of steps in the third-level iteration t max The initial number of first-level iterations is r = 1; (3) Set the initial number of second-level iterations: s = 1; Initial parameters for the two-phase material competitive optimization subproblem: a = 1, b = a + 1; (4) Set the initial number of three-level iterations: t = 1; (5) Calculate the total number of times the current hierarchical iteration is executed, k = t max s max (r-1)+t max (s-1)+t, begin executing the k(r,s,t)th hierarchical iteration; (6) Combining the SIMP multi-material interpolation model with the EFGM discrete static control equation, the mathematical model for topology optimization of negative Poisson's ratio metamaterial periodic microstructure is given by equation (7); applying mechanical periodic boundary conditions according to the EFGM node classification, solving the micro-EFGM discrete static control equation yields the microstructure unit cell node displacement: U; solving for the corresponding positional performance parameters Assembled into an equivalent elasticity matrix: Q H ; (6.1) Based on the meshless Galerkin method, the shape function is constructed using the Moving Least Squares (MLS) method. The MLS approximation function u in the computational domain Ω consisting of N nodes is obtained. h (x) can be constructed as In the formula, p T (x)=[p1(x),p2(x),...,p m [x] is a basis function p j The vector consisting of (x), where m is the number of terms in the basis functions, is a linear basis, and a(x) = [a1(x), a2(x), ..., a...]. m (x)] T A vector of unknown coefficients; The unknown coefficient vector a(x) is obtained by minimizing the functional J, where J is... In the formula, u I Let u(x) be the function to be solved I At calculation point x I The function value at the point where there are no grid nodes in the neighborhood, where n is the calculation point x. I The number of unmeshable nodes in the neighborhood, w(xx) I Let be the weight function, and choose a cubic spline weight function, the specific expression of which is: In the formula, r = ||xx I || / d mI The radius of the neighborhood of point x is calculated as d. mI =scale×s[k], where scale is a multiplier greater than 1, and s[k] is the distance between unmesh node i and its nearest k-th unmesh node; Therefore, a(x) can be derived from the following formula: a(x)=A -1 (x)B(x)u (11) In the formula, B(x)=[w1(x)p(x1),w2(x)p(x2),...,w n (x)p(x n )],u=[u1,u2,...,u n ] Τ ; Substituting equation (11) back into equation (8) yields the following result: In the formula, Φ(x) is the vector composed of MLS shape functions corresponding to the unmesh nodes within the influence domain of node x, Φ(x)=[φ1(x),φ2(x),...,φ n [x]=p T (x)A -1 (x)B(x); (6.2) The detailed steps for solving the displacement field of periodic microstructures based on meshless EFGM are as follows: Initial nodal test strain was applied Subsequently, the internal displacement field of the microstructure consists of a macroscopic displacement field and a periodic fluctuating displacement field. sum because Unknown and difficult to solve, consider the displacement on a pair of parallel boundaries of the microstructure described by equation (13). See Figure 3In the formula, k+ and k- represent the microstructure perpendicular to y, respectively. k Points on a pair of parallel boundaries in the direction (k=1,2), where the + and - signs represent the positive and negative directions of the coordinate axes respectively, can be eliminated by subtracting equation (13). Obtain explicit boundary conditions that can be directly applied to the microstructure. In the formula, The side length of a single cell in the microstructure. A constant describing the displacement difference between a pair of parallel boundaries of a microstructure; See Figure 4 To directly apply the explicit boundary conditions of equation (14) to the microstructure, the size is l x ×l y The microstructure unit cell is discrete into N s The number of EFGM nodes is determined, and all nodes are classified; where A, B, C, and D represent the EFGM nodes at the vertices of the microstructure unit cell, I, II, III, and IV represent the set of EFGM nodes on the boundary of the microstructure unit cell excluding the vertices, and V represents the set of EFGM nodes inside the microstructure unit cell. Test strain field ε at initial node 0 Decomposed into Figure 2 Constant strain field ε in coordinate system 0(xx) ε 0(yy) and shear strain field γ 0(xy) u and v represent the displacements of any node within the unit cell of the microstructure in the x and y directions, respectively. Therefore, according to... Figure 4 The EFGM node classification shown in equation (14) can be divided into two categories: periodic boundary conditions of microstructures. For nodes A, B, C, and D of the EFGM, the first type of periodic boundary condition is: For the node sets I, II, III, and IV of the EFGM, the second type of periodic boundary condition is: See Figure 4 The overall displacement field U of the microstructure unit cell is divided into four categories: the first category is the displacement U1 of node set A, B, C, and D; the second category is the displacement U2 of node set V; the third category is the displacement U3 of node sets I and IV; and the fourth category is the displacement U4 of node sets II and III. According to equation (14), U4 = U3 + W always holds, and W is obtained through... We obtain the following: At this point, the global displacement field equilibrium equation KU=F of the microstructure unit cell can be expanded to: In the formula, F1 represents the reaction forces on the given displacement boundary, i.e., nodes A, B, C, and D of EFGM; F2 represents the forces on node set V of EFGM; F3 represents the forces on node sets I and IV; and F4 represents the forces on sets II and III. By assuming periodic boundary conditions, F2 = 0, and F3 + F4 = 0. Due to the symmetry of the stiffness matrix, the stiffness matrices of the four types of nodes have a relationship K. ij =K ji (i,j=1,2,3,4); due to ε 0(xx) ε 0(yy) and γ 0(xy) It is known that, with periodic boundary conditions, the equilibrium equation shown in equation (17) can be further reduced to equation (18), from which the displacement field can be obtained: (6.3) Solving the corresponding location performance parameters based on meshless EFGM The detailed steps for assembling it into an equivalent elastic tensor matrix are as follows: Let x be the macroscopic structural scale in the global coordinate system and y be the microscopic structural scale in the local coordinate system. The ratio of the two scales is ε = x / y and 0 < ε << 1. Then, at any point x in the macroscopic scale, any structural field function that depends on coordinate changes is related to y = x / ε and exhibits periodicity. G ε (x)=G(x,y)=G(x,y+nY) (19) In the formula, Y is the period of the periodic field function, represents the size of the microstructure at the microscale, and n is any integer; For the equivalent elastic problem, the displacement function u ε (x,y) is asymptotically developed with a small parameter ε. Substituting equation (20) into the weak integral form of the governing equations of the statics problem, and retaining only the first-order variational terms containing ε, we can solve the equations to obtain the equivalent elastic tensor for the d-dimensional problem as follows: In the formula, i,j=1,2,...,f, in the two-dimensional problem, f=2, |Ω| is the area of ​​the microstructure unit cell, Q pqrs For the intrinsic elastic tensor of solid materials, The initial nodal test strain field was applied; For applied The characteristic strain field of the EFGM nodes to be solved on the post-microstructure The following linear elastic equilibrium equation is obtained by solving it. This invention is based on the theory of average stress and strain, and applies nodal test strain fields at the boundaries of microstructure cells. by a superimposed strain field If we substitute, then equation (21) becomes In the numerical analysis based on EFGM, the unit cell of the multi-material microstructure is discretized into N EFGM nodes. Then, the equivalent elastic tensor can be written as the interaction energy e based on the EFGM nodes in conjunction with equation (22). ijkl integral form In the formula, u A(ij) The displacement of the EFGM nodal parameters to be solved corresponds to the nodal test strain field. To introduce a submatrix of the global stiffness matrix after penalizing the elastic modulus and shear modulus using the SIMP material interpolation model, the following formula is used to calculate the submatrix of the overall stiffness matrix. In the formula, B I For static analysis, the geometric matrix is ​​given; for two-dimensional problems, the matrix form of the elastic tensor represented by equation (23) is: (7) Solve for the volume sensitivity and the objective function sensitivity, wherein the volume sensitivity analysis is performed through... The minimum negative Poisson's ratio objective function sensitivity was obtained through... It can be concluded that; (8) The OC method updates the multi-material design variables. For the EFGM node relative density of the a-th material in the k-th iteration, the following is... Based on the relationship that the sum of the relative densities of the EFGM nodes of materials a and b remains unchanged before and after the update, the relative density of the EFGM nodes of material b in this iteration is calculated as follows: The obtained multi-material relative density matrix is ​​then assembled into a multi-material EFGM node relative density matrix. (8.1) When i = a, substitute the initial relative density of the EFGM nodes in the multi-material microstructure into the OC formula and update... in for For the requirements of this invention, in the formula, m0 = 0.2 is the positive movement limit, τ = 1 is the damping coefficient, and ρ min =10 -3 This is the lower limit of relative density. pass Define; (8.2) When i = b, the relative densities of the EFGM nodes of materials a and b remain unchanged before and after the update. The relative densities of the EFGM nodes of material b in the k-th hierarchical iteration are obtained. in for (8.3) When i≠a and i≠b, this hierarchical iteration does not involve updating the relative density of the EFGM nodes of the i-th material. That is, the relative density of the EFGM nodes of the i-th material should be the same in the k-th and k-1-th iterations. Update the relative density of the EFGM nodes of the i-th material in the k-th hierarchical iteration. in for (8.4) Combining Sections 8.1, 8.2, and 8.3, calculate the infinite norm of the relative density difference between EFGM nodes. (9) Determine whether to continue or end the first, second, and third level iterations based on the iteration parameters; (9.1) Compare t with t max The size relationship determines whether the third-level iteration should continue or end: if t < t max If the condition is met, update the third-level iteration step number t = t + 1, and return to step (5) to continue the third-level iteration; if t < t max If this condition is not met, then the current three-level iteration ends. (9.2) Compare s with s max The size relationship determines whether the second-level iteration should continue or end: if s < s max If both s and b < q hold true, then let s = s + 1, b = b + 1, and return to step (4) to continue the second-order iteration; if s < s max If the condition is true but b < q is false, then let s = s + 1, a = a + 1, b = a + 1, and return to step (4) to continue the second-order iteration; if s < s max If this condition is not met, then the second-level iteration of this round ends; (9.3) Compare r with r max and ch min The size relationship determines whether the first-level iteration should continue or end: if r < r max It holds true and ch > ch min If true, update the first-level iteration step number r = r + 1, and return to step (3) to continue executing the first-level iteration; if r < r max Not true or ch > ch min If the condition is not met, the first-level iteration ends, and step (10) is executed. (10) Output the optimal topology of negative Poisson's ratio metamaterial based on EFGM.

[0016] The following is an example of the application of the method of the present invention in engineering practice: See Figure 5 This embodiment considers a l x ×l y The rectangular microstructure design domain, where multiple materials are mixed and stacked on each EFGM node, here let l x =l y =8.4cm, discretized into 85×85 EFGM nodes using EFGM; two different initial design domains d=[1,2] are selected, and their specific design domain shapes are shown in [reference]. Figure 6 and Figure 7 Let the total number of material types be q = [3, 4], and when q is 3, the relative volume fraction v of each material is... i = [0.3, 0.3, 0.4], the elastic modulus in the 2-direction for each material. When q is 4, the relative volume fraction v of each material i = [0.2, 0.2, 0.2, 0.4], the elastic modulus in direction 2 for each material. Set Poisson ratio uniformly The micro-penalty strength p = 5; the Poisson's ratio factor PRF is taken as 1 / 4, 1 / 3, 1 / 2, 1, 2, 3, 4 respectively; and the material orientation angle θ is taken as 0°, 15°, 30°, 45°, 60°, 75°, 90° respectively.

[0017] The specific implementation steps of this invention for this example are as follows: (a) Determine the initial design domain, the number of material types q, and input the elastic modulus of the anisotropic structure. Poisson's ratio shear modulus Poisson's ratio factor (PRF) and multi-material orientation angle (θ); (b) Set the iteration parameters for the instance: minimum tolerance of relative density difference between multi-material EFGM nodes ch min =10 -5 Maximum number of steps in the first-level iteration, r max =100, maximum number of steps in the second-level iteration Maximum number of steps in the third-level iteration t max =2, initial first-level iteration step number r = 1; (c) Set the initial number of second-level iterations: s = 1; Initial parameters for the two-phase material competitive optimization subproblem: a = 1, b = a + 1; (d) Set the initial number of three-level iterations: t = 1; (e) Calculate the total number of iterations k for levels one, two, and three: k = t max s max (r-1)+t max (s-1)+t (f) Using the generalized Hooke's law and the set mechanical parameters, the coordinate transformation matrix T is solved through the multi-material direction angle θ. s Through Poisson's ratio elastic modulus Calculating the elastic matrix Q of anisotropic multimaterials using the Poisson's ratio factor (PRF) i ; (g) Discretize the design domain using EFGM and set Gaussian points through the discrete EFGM nodes; (h) Preprocess the Gaussian point and EFGM node, solve for the EFGM node number and degree of freedom number in the influence domain of the Gaussian point, solve for the MSL shape function φ of the Gaussian point and EFGM node, and solve for the stiffness matrix without penalty. (i) See also Figure 4 The discrete EFGM nodes are divided into five categories, and W is calculated. (j) Set the initial relative density vector of the multi-material EFGM nodes. and total relative density matrix (k) Using the EFGM node numbers, degrees of freedom numbers, shape functions and stiffness matrix obtained in step (e), solve for the global stiffness matrix K under the penalty case; (l) To ensure the symmetry of the stiffness matrix, this method, after calculating the overall stiffness matrix, further processes it to make K = (K + K T ) / 2, to ensure perfect symmetry; (m) Based on the periodic boundary conditions and the EFGM equilibrium equation, and according to the classification of the total displacement of the microstructure unit cell, solve for the four types of EFGM nodal displacements U1, U2, U3 and U4; (n) Using homogenization theory to apply the penalty to the elastic tensor matrix Q H Solve the problem; (o) Calculate the negative Poisson's ratio objective function of EFGM. (p) The volume sensitivity is analyzed for each material by differentiating the volume fraction with respect to the relative density of the EFGM nodes. (q) By differentiating the objective function with respect to the relative density of the EFGM nodes, the sensitivity of the objective function under each material is analyzed to obtain the sensitivity of the objective function; (r) Update the design variables using the OC method to obtain the relative density of the EFGM nodes of the material under the current iteration; (s) Determine t < t max If the condition is true, update the third-level iteration step t = t + 1 and repeat step (e) - (s); if the condition is false, end the current round of third-level iteration. (t) Determine s and smax Size relationship, if s < s max If both s and b < q hold true, then let s = s + 1, b = b + 1, and repeat step (d) - (t); if s < s max If the condition is true but b < q is false, then let s = s + 1, a = a + 1, b = a + 1, and repeat step (d) - (t); if s < s max If the condition is not met, then the current second-level iteration ends; (u) Determine whether the first-level iteration continues or ends: If r < r max It holds true and ch > ch min If true, then let r = r + 1 and repeat step (c) - (u); if r < r max Not true or ch > ch min If the condition is not met, then the current first-level iteration ends; (t) Output the optimal topological configuration of the periodic microstructure of the negative Poisson's ratio metamaterial.

[0018] Figures 8-24 This embodiment represents the optimal topology obtained through topology optimization of meshless, negative Poisson's ratio anisotropic multi-material microstructures, where... Figure 8 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 1, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 3 in this embodiment of the invention. Figure 9 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 1, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 10 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 2, the Poisson's ratio factor PRF is 1, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 11 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 1 / 4, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 12 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 1 / 3, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 13 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 1 / 2, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 14This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 1, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 15 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 2, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 16 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 3, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 17 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 4, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 18 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 0°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 19 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 15°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 20 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 30°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 21 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 45°, and the number of multi-material types q is 4. Figure 22 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 60°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 23 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 75°, and the number of multi-material types q is 4 in this embodiment of the invention. Figure 24 This is the optimal topological configuration of the negative Poisson's ratio metamaterial microstructure based on EFGM when the initial design domain shape d is 1, the Poisson's ratio factor PRF is 5 / 2, the multi-material orientation angle θ is 90°, and the number of multi-material types q is 4 in this embodiment of the invention. Figures 25-26These are the minimum values ​​of the objective function and the minimum negative Poisson's ratio under different Poisson's ratio factors PRF and multi-material orientation angles θ when the initial design domain shape d is 1 and the number of multiple material types q is 4 in the embodiments of the present invention. Figure 25 In this embodiment of the invention, when the initial design domain shape d is 1, the number of multiple material types q is 4, and the multiple material orientation angle θ is 0, different Poisson's ratio factors are used. The minimum value of the objective function and the value of the minimum negative Poisson's ratio under the given conditions; Figure 26 This refers to the minimum objective function and the minimum negative Poisson's ratio when the initial design domain shape d is 1, the number of material types q is 4, and the Poisson's ratio factor PRF is 1, under different multi-material orientation angles θ = [0°, 15°, 30°, 45°, 60°, 75°, 90°]. Figures 8-26 It is evident that the optimal topology obtained through topology optimization of anisotropic multi-material microstructures with negative Poisson's ratio using a meshless method exhibits no numerical instabilities such as intermediate density or checkerboard patterns, facilitating subsequent performance analysis and manufacturing. The optimal topology varies under different parameters, indicating differences in its overall performance. The number of materials in the anisotropic multi-material structure can be controlled by adjusting the number of multi-material types, q. The Poisson's ratio factor, PRF, primarily affects the structural flexibility, thus significantly influencing the minimum negative Poisson's ratio objective function. Compared to the Poisson's ratio factor, the multi-material orientation angle θ has a smaller impact on the objective function. Therefore, the reasonable range for the Poisson's ratio factor, PRF, is 2–4, and the material orientation angle θ is selected based on actual requirements. This demonstrates that the optimal topology and its performance obtained through topology optimization of anisotropic multi-material microstructures with negative Poisson's ratio can be controlled by adjusting the above parameters, possessing significant theoretical research and engineering application value.

[0019] Although the present invention has been described in detail with reference to this embodiment, the above description does not limit the scope of protection of the present invention. Any modifications and improvements made based on the concept of the present invention shall be considered within the scope of protection of the present invention.

Claims

1. A topology optimization method for negative Poisson's ratio anisotropic multi-material microstructures based on EFGM, characterized in that... Includes the following steps: (1) Based on the design requirements of the actual structure, determine the initial design domain d of the microstructure and the total number of material types q; input the mechanical parameters of the anisotropic structure: elastic modulus Poisson's ratio shear modulus Poisson's ratio factor PRF, relative volume fraction ν i and multi-material orientation angle θ; (2) Based on the initial design domain, the design domain is discretized using EFGM, and Gaussian points are arranged inside and on the boundary of the design domain; the initial multi-material EFGM node relative density vector is set. And assembled into a matrix Set the number of horizontal and vertical design subdomains: M x M y Set iteration parameters: minimum tolerance ch for the relative density difference of multi-material EFGM nodes. min Maximum number of steps in first-level iteration r max Maximum number of steps in the second-level iteration Maximum number of steps in the third-level iteration t max The initial number of first-level iterations is r = 1; (3) Set the initial number of second-level iterations: s = 1; Initial parameters for the two-phase material competitive optimization subproblem: a = 1, b = a + 1; (4) Set the initial number of three-level iterations: t = 1; (5) Calculate the total number of iterations for the first, second, and third levels: k = t max s max (r-1)+t max (s-1)+t; (6) Calculation of the elastic matrix of anisotropic multimaterials based on meshless EFGM: Orthogonal anisotropic materials have independent and unaffected single values ​​in terms of elastic modulus and Poisson's ratio in two perpendicular directions. In order to better characterize the properties of orthogonal anisotropic multimaterials, a multimaterial rectangular coordinate system 1-2 is established, and the multimaterial direction angle θ is defined as the angle between the x-axis and the 1-axis in the Cartesian global coordinate system xy. According to the generalized Hooke's law, the mechanical constitutive relation of orthogonal anisotropic multimaterials is constructed. in In the above formula, σ x , σ y , ε x , ε y Let τ be the stress and strain in the x and y directions of the structure. xy and γ xy These are the shear stress and shear strain of the material, respectively. Let Q be the elasticity matrix of the i-th material (i = 1, 2, ..., q) in the global coordinate system xy. i Let T be the elastic matrix of the i-th material in the multi-material coordinate system 1-2. s To be With Q i The associated coordinate transformation matrix; in Q i middle, in, Let μ be the elastic modulus of the i-th material in directions 1 and 2 in the multi-material coordinate system 1-2. 12 μ 21 These are the Poisson's ratios in directions 1 and 2 of the multi-material coordinate system 1-2, respectively. Let i be the shear modulus of the i-th material, expressed by the formula Approximate calculation, defining the Poisson's ratio factor This indicates the orthotropic strength of mechanical properties. (7) Combining the SIMP multi-material interpolation model and the EFGM discrete static control equations, a mathematical model for topology optimization of periodic microstructures of negative Poisson's ratio metamaterials based on meshless EFGM is established. In the formula, N s The total number of discrete EFGM nodes in a multi-material microstructure. Let be the relative density of the EFGM nodes of the ith material at the j-th node of the multi-material microstructure. The objective function for constructing the minimum negative Poisson's ratio is expressed as follows: Where β∈(0,1) is a constant value. It is the equivalent elastic tensor, obtained by the homogenization method. K is the global EFGM nodal stiffness matrix with penalty, and F is the EFGM nodal load matrix with penalty. Let V be the relative density at the Gaussian point of the i-th material. i V represents the total integral of the i-th material. 0 The initial total integral is given; by applying periodic boundary conditions to the microstructure, the discrete static control equation KU = F of the EFGM is solved to obtain the nodal displacements U of the EFGM; the equivalent elastic tensor matrix Q is then solved. H ; (8) Solve for the volume sensitivity and the objective function sensitivity; wherein, the volume sensitivity analysis is performed through... The minimum negative Poisson's ratio objective function sensitivity was obtained through... Therefore, β in the formula is obtained. k Define a fixed parameter for the k-th iteration; (9) The design variables are updated using the optimization criterion method. The relative density matrix of the a-th material in the k-th iteration of the multi-material EFGM node is: in It can be obtained using the following formula: In the formula, m0 = 0.2 is the positive movement limit, and τ = 1 is the damping coefficient. Defined as The relative density matrix of the b-th material in the k-th iteration of the multi-material EFGM node is: in, The infinite norm of the difference in the relative density matrix of the meshless EFGM nodes obtained by calculation iteration (10) Compare t with t max The size relationship determines whether the third-level iteration should continue or end: if t < t max If the condition is met, update the third-level iteration step number t = t + 1, and return to step (5) to continue the third-level iteration; if t < t max If this condition is not met, then the current three-level iteration ends. (11) Compare s with s max The size relationship determines whether the second-level iteration should continue or end: if s < s max If both s and b < q hold true, then let s = s + 1, b = b + 1, and return to step (4) to continue the second-order iteration; if s < s max If the condition is true but b < q is false, then let s = s + 1, a = a + 1, b = a + 1, and return to step (4) to continue the second-order iteration; if s < s max If this condition is not met, then the second-level iteration of this round ends; (12) Compare r with r max and ch min The size relationship determines whether the first-level iteration should continue or end: if r < r max It holds true and ch > ch min If true, update the first-level iteration step number r = r + 1, and return to step (3) to continue executing the first-level iteration; if r < r max Not true or ch > ch min If the condition is not met, the first-level iteration ends, and step (13) is executed. (13) Output the optimal topological configuration of the periodic microstructure of the negative Poisson's ratio metamaterial.

2. The method for topology optimization of negative Poisson's ratio anisotropic multi-material microstructures based on EFGM according to claim 1, characterized in that... In step (9), the alternating active phase algorithm based on SIMP interpolation is used to perform two-phase competition calculations on multiple materials. For rectangular pore microstructures, the number of multiple material types is 4.

3. The method for topology optimization of negative Poisson's ratio anisotropic multi-material microstructures based on EFGM according to claim 1, characterized in that... The Poisson's ratio factor (PRF) mainly affects the structural flexibility value, thus having a significant impact on the minimum negative Poisson's ratio objective function. For rectangular porous microstructures, the PRF value ranges from 2 to 4.

4. The method for topology optimization of negative Poisson's ratio anisotropic multi-material microstructures based on EFGM according to claim 1, characterized in that... The orientation angle θ of multiple materials has little impact on the compliance value, and therefore little impact on the objective function of minimizing the negative Poisson's ratio. For rectangular microstructures with holes, the material orientation angle θ depends on the Poisson's ratio μ in both directions. 12 and μ 21 Choose according to your needs, using μ 12 When θ takes values ​​in the range of 60° to 90°; using μ 21 When θ is in the range of 0° to 30°, it takes values.

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