Anisotropic multi-material structure multi-scale topological optimization method based on meshless EFGM

Abstract

The invention discloses an anisotropic multi-material periodic structure multi-scale topological optimization method based on meshless EFGM, and the method comprises the steps: (1) inputting the number of types of anisotropic multi-materials and the attributes of basic materials, and determining the number of periodic design subdomains; (2) dispersing a macro-micro design domain by adopting an EFGM node to obtain information of each node; (3) establishing an EFGM statics equilibrium equation of the macrostructure and the microstructure, and solving an anisotropic equivalent elastic matrix; (4) establishing a mathematical model for multi-scale topological optimization of the anisotropic multi-material periodic structure based on a meshless EFGM theory; (5) updating macro and micro EFGM node density by adopting an OC method on the basis of an alternative active multi-phase material topological optimization thought; and (6) inputting a termination condition, judging whether multi-stage iteration is ended or not, and outputting an optimal configuration. According to the method, multi-scale topological optimization of the anisotropic multi-material periodic structure is carried out based on the meshless EFGM, the method is simple and practical, the phenomenon of numerical instability is eliminated, the boundary of the multi-scale topological structure is clear and smooth, and the bearing capacity of the structure can be effectively improved.

Description

Technical Field

[0001] This invention belongs to the field of structural optimization design in computer-aided engineering, specifically involving a multi-scale topology optimization method for anisotropic multi-material periodic structures based on the meshless Galerkin Method (EFGM). Background Technology

[0002] Structural topology optimization, based on numerical analysis and optimization theory, determines the optimal distribution of void structures within a design domain to achieve the optimal target performance functions (such as stiffness, strength, temperature, and weight) under specific boundary and constraint conditions. It is widely used in innovative engineering structural design. Currently, mesh-based numerical methods, such as the finite element method, finite volume method, and finite difference method, are commonly used in structural topology optimization for performance or response analysis. However, due to the presence of the mesh, the topology optimization results suffer from numerical instability phenomena such as mesh dependence, checkerboard patterns, and intermediate densities. Meshless methods eliminate the need for mesh generation. During preprocessing, the computational domain is discretized using a series of arbitrarily distributed meshless nodes. Shape functions are constructed using the information from these meshless nodes to approximate the field function, and the relationships between meshless nodes are established through their influence domains, which can intersect or overlap. Compared to mesh-based methods, meshless methods offer advantages such as the ability to construct higher-order shape functions, higher computational accuracy, and freedom from mesh constraints, thus avoiding phenomena like unclear boundaries, intermediate densities, and checkerboard patterns in the topology optimization results. Among them, the Element-free Galerkin Method (EFGM), as a relatively mature meshless method, has been applied in structural topology optimization. However, most structural topology optimization based on the meshless EFGM is currently aimed at single isotropic materials.

[0003] With the increasing application of composite materials and the growing demand for integrated structural design, topology optimization research on multi-material structures is gaining more and more attention. However, topology optimization methods for anisotropic multi-material structures based on meshless methods are rare. Compared to single-material structures, multi-material composite structures are composed of different anisotropic materials arranged in a certain pattern, achieving multiple functions such as high strength, high hardness, and high thermal conductivity. A significant characteristic is anisotropy, meaning that the same material has different elastic moduli and thermal conductivity in different directions. This makes the mechanical and thermal properties of anisotropic multi-material structures exhibit obvious directional independence and controllability, presenting enormous potential application value. Meanwhile, most methods for multi-material topology optimization research do not consider the periodicity of the structure. Periodic structures have high designability and good predictability, and multi-material periodic structures are now widely used in aerospace, automotive, electronics, and biomedical fields. It is also worth noting that many current topology optimization methods based on meshless EFGM are performed on a single scale, which to some extent limits the high performance of the structures that can be obtained.

[0004] Multiscale topology optimization simultaneously considers both macroscopic and microscopic scales to optimize the topology of structures, unlocking the performance potential of lightweight structures within a broader design space. Inspired by the multi-scale morphology of porous and dense structures in nature, such as animal skeletons and bamboo, multiscale structures have attracted increasing attention due to their superior lightweight and robust properties. Furthermore, the combination of multiscale topology optimization and advanced additive manufacturing technologies has created increasingly mature conditions for shaping and handling subtly yet robust macro-microscopic integrated topologies. With the rapid development of industrial technologies across various fields, the requirements for the structural performance and stability of industrial products are becoming increasingly stringent. Traditional multi-material periodic structures designed only at the macroscopic level are gradually failing to meet these evolving needs. Therefore, starting from multiple scales and employing appropriate and efficient methods to optimize material distribution and arrangement at the macroscopic level while focusing on the material configuration design of the microstructure can better improve the performance of anisotropic multi-material periodic structures. However, current topology optimization methods based on meshless EFGM are mostly designed at a single macroscopic or microscopic scale, and often only optimize the structure by deeply exploring the material distribution. There are no publicly reported studies on the topology optimization of anisotropic multi-material periodic structures from a multi-scale perspective using meshless EFGM. Therefore, based on the above research background, this invention proposes a multi-scale topology optimization method for anisotropic multi-material structures based on meshless EFGM. Summary of the Invention

[0005] With the advancement of modern technology, society has increasingly higher requirements for the structural performance of fields such as mechanical engineering, power engineering, and aerospace. Anisotropic structures, due to their excellent mechanical and thermal properties, have been widely used in mechanical engineering, power engineering, and aerospace, and their working environments often involve multiphysics fields. To address the issues of numerical instability such as sawtooth patterns, checkerboard structures, and intermediate densities in topologies obtained using common topology optimization methods, and the difficulty in fabricating optimal topologies, this invention proposes a multi-scale topology optimization method for anisotropic multi-material periodic structures based on meshless EFGM nodes. This method discretizes the design domain using meshless EFGM nodes, uses the relative density of EFGM nodes in the multi-material periodic macrostructure and various microstructures as design variables, uses the volume fractions of the multi-material periodic macrostructure and various microstructures as constraints, and aims to minimize the compliance of the integrated macro-micro topology optimization of the multi-material periodic structure. A mathematical model for multi-scale topology optimization of static anisotropic multi-material periodic structures based on meshless EFGM is established. A computer program for the algorithm is then developed to perform topology optimization design on anisotropic multi-material periodic structures with different numbers of material types and anisotropic material property parameters, outputting the optimal topology.

[0006] The technical solution adopted by this invention to solve its technical problem is as follows: a topology optimization method for anisotropic multi-material periodic structures based on meshless EFGM, utilizing meshless EFGM to discretize the static control equations of macro- and micro-anisotropic multi-material periodic structures, and introducing SIMP multi-material interpolation model and alternating active multiphase material topology optimization method, establishing a mathematical model for multi-scale topology optimization of static anisotropic multi-material periodic structures based on meshless EFGM and energy homogenization EBHM; controlling the mechanical properties of the anisotropic level of the structure by adjusting the Poisson's ratio factor, controlling the mechanical properties of the multi-material level of the structure by adjusting the number of material types, and controlling the mechanical properties of the periodic level of the structure by adjusting the number of design subdomains.

[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 engineering structure, determine the model geometry information and multi-material volume fraction of the macroscopic static structure. Determine the geometric design domain and multi-material volume fraction of the microstructure. Input the two-directional elastic modulus of the anisotropic multimaterial periodic structure in a multimaterial coordinate system. Number of material types q, Poisson's ratio of each material Shear modulus of each material Poisson's ratio factor Bt and multi-material orientation angle θ material properties, where the superscript i = 1, 2, ..., q is the index of the material type, the subscript ma represents the macroscopic level parameter, and the subscript mi ​​represents the microscopic level parameter. Set the design domain boundary conditions and load magnitude of the macroscopic and microscopic structures. Based on the meshless node information, background mesh and set boundary conditions of the corresponding design domain, obtain the EFGM Gaussian point information of the macroscopic and microscopic design domains respectively. (2) Based on the initial multi-material volume fraction of the macro-microstructure, the density of each meshless node in the macro-design domain is determined. and the density of each meshless node within the micro-design domain Perform initialization and set the number of subdomains M. x ×M y The plan is to utilize the alternating active multiphase material topology optimization method to achieve the target optimization by updating and iterating the design variables. The iteration parameter is set as follows: the minimum tolerance ch of the relative density difference between the multi-material EFGM nodes. min The maximum number of first-level iterations r that affects the minimum tolerance update max Maximum number of second-order iterations for competitive optimization of multiphase materials The maximum number of iterations t in the third level of the internal alternating multiphase cycle setting max Set the initial first-level iteration step number r = 1; (3) Set the initial second-level iteration step s = 1, and select the two phase materials in the two-phase material competitive optimization subproblem as a = 1, b = a + 1; (4) Set the initial number of third-level iterations t = 1; (5) Calculate the current total number of iterations k = t max s max (r-1)+t max (s-1)+t, begin executing the k(r,s,t)th iteration; (6) Based on the EFGM theory and the idea of ​​alternating active multiphase material topology optimization, a SIMP multi-material interpolation model is established. q kinds of materials with variable relative densities in the range of 0 to 1 are introduced. The relative density fields are constructed with the relative densities of the meshless EFGM nodes of macroscopic multi-materials and multi-type microstructures as design variables. The material properties of the EFGM Gaussian point under the k-th iteration of the i-th material in the macroscopic structure and under the k-th iteration of the i-th type of microstructure in the microscopic scale are solved. (7) At the microscale, based on the energy homogenization method, the equivalent elastic matrix is ​​solved, and the discrete static control equations of the meshless EFGM at the microscale are derived: (a) According to the mean stress and strain theory, the strain field of the EFGM nodes is tested at the boundary of the microstructure cell. The equivalent elastic tensor expression is obtained as follows: In the formula |Ω mi| represents the area of ​​a single cell of the microstructure. To superimpose the strain field, Q pqrs (a) Based on the meshless EFGM numerical analysis, the unit cell of the multi-material microstructure is discretized into N EFGM nodes, then combined with the interaction energy e based on the EFGM nodes... ijkl The equivalent elasticity matrix is ​​expressed as In the formula u A(ij) The displacement of the EFGM node parameter to be solved corresponds to the strain field of the EFGM node test. (c) Solve for the stiffness matrix of the unit cell of the multi-material microstructure based on the anisotropic mechanical properties. In the formula in, as well as Let be the elastic modulus and Poisson's ratio of the i-th material in the x and y directions of the multi-material coordinate system, respectively, and satisfy the following relationship: The shear modulus is taken as the factor, and the density of each meshless node within the micro-design domain is also considered. Stiffness matrix of distributed assembled multi-material microstructure unit cell According to the equivalent elastic tensor expression, the matrix form of the elastic tensor is obtained as follows: (d) The initial EFGM nodal test strain field determined in step (a) is decomposed into two constant strain fields and one shear strain field. Explicit boundary conditions are applied to the microstructure, and the discrete static governing equations K for the global displacement field EFGM of the microstructure unit cell can be derived. Fmi U mi =F mi In the formula, K Fmi U is the overall stiffness matrix of the microstructure. mi Let F be the nodal displacement matrix of the microstructure EFGM. mi The load matrix of the EFGM nodes in the microstructure; (8) At the macroscopic scale, derive the discrete static control equations of the meshless EFGM and calculate the macroscopic structural displacements and objective functions: (a) Determine the overall force load vector F of the meshless EFGM in the design domain based on the magnitude of the applied force loads within the design domain. ma (b) Based on the given force and displacement boundary conditions, various force and displacement boundary conditions are processed, among which the penalty function method is used to process the displacement boundary, and the penalty term K of the overall force stiffness matrix of EFGM is obtained. Fα,ma and the overall force load vector penalty term F of EFGM α,ma (c) Based on the equivalent elasticity matrix obtained in step (6) and the density of each multi-material EFGM node in the macroscopic design domain at this time. Calculate the EFGM nodal stiffness matrix And assemble it into a macroscopic anisotropic multi-material periodic structure overall stiffness matrix. Derive the meshless discrete statics governing equations for the anisotropic multi-material periodic structure statics problem, and combine them with the overall force load vector F of the EFGM determined in step (a). ma The penalty term K of the overall force stiffness matrix of EFGM obtained by the penalty function method in step (b) Fα,ma and the overall force load vector penalty term F of EFGM α,ma (d) Solve for the displacement parameter values ​​of the meshless nodes in the design domain; (d) Solve for the displacement values ​​of each meshless node based on the displacement parameter values ​​of the meshless nodes in the design domain and calculate the objective function, and output the meshless displacement parameter values, meshless displacement values, EFGM overall force load vector and objective function of the design domain. (9) Based on the meshless EFGM and multi-material SIMP interpolation model, a multi-scale topology optimization model for anisotropic multi-material periodic structures integrating macro- and micro-scales is established. In the formula, Let be the relative density vector of the EFGM nodes of the i-th material in a multi-material periodic macrostructure. For assembly The obtained EFGM node relative density matrix of the multi-material Let M be the relative density of the EFGM node of the i-th material at the n-th EFGM node within the m-th design subdomain of a multi-material periodic macrostructure. s N represents the total number of design subdomains for a multi-material periodic macrostructure. s The total number of EFGM nodes within a single design subdomain of a multi-material periodic macrostructure; Let be the relative density vector of EFGM nodes of the i-th type of microstructure. For assembly The relative density matrices of EFGM nodes in various microstructures were obtained. Let N be the relative density of the j-th EFGM node in the i-th type of microstructure. mi The total number of EFGM nodes for the i-th type of microstructure; Let be the relative density of the EFGM Gaussian point of the material in the i-th macrostructure. Let be the relative density of the Gaussian points of the EFGM for the i-th type of microstructure. Let i be the volume of the i-th material in a multi-material periodic macrostructure after the update. Let be the initial volume fraction of the i-th material in a multi-material periodic macrostructure. The total volume of a multi-material periodic macroscopic structure; Let i be the volume of the i-th type of microstructure after the update. Let i be the initial volume fraction of the i-th type of microstructure. C represents the total volume of the i-th type of microstructure; F Let ρ be the compliance objective function for a meshless EFGM multi-material periodic macrostructure. min To prevent matrix singularity, the relative density of macro- and micro-level EFGM nodes and Gaussian points is set to a minimum. The detailed steps for establishing a macro- and micro-level integrated anisotropic multi-material periodic structure multi-scale topology optimization model based on the meshless EFGM and multi-material SIMP interpolation model are as follows: (a) Establish constraint conditions based on the micro-structure discrete static control equations obtained in step (5) and the macro-structure discrete static control equations obtained in step (6), constraining the volume fraction of each material in the macro- and micro-structures under the multi-material structure; (b) Solve for the compliance value C of the anisotropic multi-material periodic structure based on the macro-structure meshless displacement value and the overall force load vector of the EFGM output in step (6). F (c) Output the compliance value C of the objective function. F ; (10) Solve for the macroscopic volume sensitivity and macroscopic compliance sensitivity, and solve for the microscopic volume sensitivity and microscopic compliance sensitivity; apply periodic constraints to the multi-material periodic macroscopic structure, and force the relative density of multi-material EFGM nodes with the same number in each design subdomain to be the same, to obtain the averaged macroscopic meshless EFGM node relative density. (11) Based on the idea of ​​alternating active multiphase topology optimization, the OC criterion is used to update the relative density vector of macroscopic multimaterial and multi-type microstructure nodes: The relative density of EFGM nodes of macroscopic and microstructure is updated according to the OC criterion. In the multimaterial binary phase topology optimization sub-model, it is determined whether b < q is true. If it is true, then b = b + 1, forming a new binary phase topology optimization sub-model to solve and update the design variables. If it is not true, then a = a + 1, b = a + 1, forming another new binary phase topology optimization sub-model to solve and update the design variables. This process continues until all materials are calculated alternately, and the updated relative density of macroscopic multimaterial and multi-type microstructure EFGM nodes is output. (12) Determine if the iteration termination condition is met: (a) Determine if the current third-level iteration number t satisfies t < t max If satisfied, let t = t + 1 and return to step (5); if not satisfied, proceed to the second-level iteration condition judgment; (b) Determine whether the second-level iteration condition s < s is satisfied. max If s < s max If b < q, then update s = s + 1, b = b + 1, and return to step (4); if s < q, then update s = s + 1, b = b + 1, and return to step (4); max However, if b < q is not satisfied, then update s = s + 1, a = a + 1, b = a + 1, and return to step (4); if s < q is not satisfied... maxThen proceed to the first-level iteration judgment; (c) Determine whether the first-level iteration satisfies the following three conditions: r≤r max ch ma <ch min ch mi <ch min If any one of the three conditions is met, return to step (3). If none of the three conditions are met, the overall iterative process ends. Output the optimal topological configuration of the anisotropic multi-material periodic macroscopic structure and multiple microstructures of the static problem in the form of an image.

[0008] The beneficial effects of this invention are as follows: Based on the meshless Galerkin method and the solid anisotropic multimaterial penalty model, this invention effectively eliminates numerical instability problems such as sawtooth, checkerboard, and intermediate density issues that easily occur in topologies obtained based on common topology optimization methods; based on the idea of ​​alternating active multiphase material topology optimization, this invention retains the comprehensive characteristics of multimaterials, optimizes material layout, and makes the performance of anisotropic structures more prominent; this invention simultaneously considers the integrated topology optimization design of structures at both macro and micro scales, deeply exploring the performance potential that single-scale structural design cannot achieve within a wider design space and broadening the performance boundaries achievable by multi-scale structural design, further meeting the integrated needs of modern industrial products for lightweight, multifunctional, and high-performance structures; this invention controls the mechanical properties of anisotropic multimaterial periodic structures by adjusting the number of material types, Poisson's ratio factor, material orientation angle, and number of design subdomains, and can perform topology optimization design of anisotropic multimaterial periodic structures according to engineering requirements, with simple operation; this invention can obtain topologies with optimal comprehensive performance according to different design requirements, and has important theoretical research significance and application value in the innovative design of complex engineering structures. 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 of the static multi-scale topology optimization method for anisotropic multi-material periodic structures according to the present invention. Figure 2 The global coordinate system xy and material coordinate system 1-2 of the anisotropic multi-material periodic structure of this invention are... Figure 3 This is a schematic diagram of the microstructure unit cell model and EFGM node classification of the present invention. Figure 4 This is a schematic diagram of displacement constraints and load application of the macroscopic structure according to an embodiment of the present invention. Figure 5 This is a schematic diagram of the initial design domain of a microstructure unit cell in an embodiment of the present invention. Figure 6In this embodiment of the invention, the number of material types q = 2, and the elastic moduli of each material are E1 = 10 GPa and E2 = 10 GPa, respectively. -3 GPa, principal Poisson's ratio ν 12 =0.12, Poisson's ratio factor Bt=1, number of design subdomains M x ×M y =2×2, the macroscopic multi-material periodic optimal topology obtained when the material orientation angle θ = 0°. Figure 7 In this embodiment of the invention, the number of material types q = 2, and the elastic moduli of each material are E1 = 10 GPa and E2 = 10 GPa, respectively. -3 GPa, principal Poisson's ratio ν 12 =0.12, Poisson's ratio factor Bt=1, number of design subdomains M x ×M y =2×2, the first type of periodic microstructure obtained when the material orientation angle θ = 0° Figure 8 In this embodiment of the invention, the number of material types q = 2, and the elastic moduli of each material are E1 = 10 GPa and E2 = 10 GPa, respectively. -3 GPa, principal Poisson's ratio ν 12 =0.12, Poisson's ratio factor Bt=1, number of design subdomains M x ×M y =2×2, the second type of periodic microstructure obtained when the material orientation angle θ = 0° Figure 9 In this embodiment of the invention, the number of material types q = 3, and the elastic moduli of each material are E1 = 10 GPa, E2 = 8 GPa, and E3 = 10 GPa, respectively. -3 GPa, principal Poisson's ratio ν 12 =0.12, Poisson's ratio factor Bt=1, number of design subdomains M x ×M y =2×2, the macroscopic multi-material periodic optimal topology obtained when the material orientation angle θ = 0°. Figure 10 In this embodiment of the invention, the number of material types q = 3, and the elastic moduli of each material are E1 = 10 GPa, E2 = 8 GPa, and E3 = 10 GPa, respectively. -3 GPa, principal Poisson's ratio ν 12 =0.12, Poisson's ratio factor Bt=1, number of design subdomains M x ×M y =2×2, the first type of periodic microstructure obtained when the material orientation angle θ = 0° Figure 11 In this embodiment of the invention, the number of material types q = 3, and the elastic moduli of each material are E1 = 10 GPa, E2 = 8 GPa, and E3 = 10 GPa, respectively. -3 GPa, principal Poisson's ratio ν12 =0.12, Poisson's ratio factor Bt=1, number of design subdomains M x ×M y =2×2, the second type of periodic microstructure obtained when the material orientation angle θ = 0° Figure 12 In this embodiment of the invention, the number of material types q = 3, and the elastic moduli of each material are E1 = 10 GPa, E2 = 8 GPa, and E3 = 10 GPa, respectively. -3 GPa, principal Poisson's ratio ν 12 =0.12, Poisson's ratio factor Bt=1, number of design subdomains M x ×M y =2×2, the third type of periodic microstructure obtained when the material orientation angle θ = 0° Figure 13 The number of subdomains M designed in this embodiment of the invention x ×M y =1×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 The optimal macroscopic multi-material periodic topology obtained when =0.12, Poisson's ratio factor Bt = 1, and material orientation angle θ = 0°. Figure 14 The number of subdomains M designed in this embodiment of the invention x ×M y =1×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 The first type of periodic microstructure obtained when the ratio is 0.12, the Poisson's ratio factor Bt = 1, and the material orientation angle θ = 0° Figure 15 The number of subdomains M designed in this embodiment of the invention x ×M y =1×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 The second type of periodic microstructure obtained when the ratio is 0.12, the Poisson's ratio factor Bt = 1, and the material orientation angle θ = 0° Figure 16 The number of subdomains M designed in this embodiment of the invention x ×M y =1×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12The third type of periodic microstructure obtained when the ratio is 0.12, the Poisson's ratio factor Bt = 1, and the material orientation angle θ = 0° Figure 17 The number of subdomains M designed in this embodiment of the invention x ×M y =3×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 The optimal macroscopic multi-material periodic topology obtained when =0.12, Poisson's ratio factor Bt = 1, and material orientation angle θ = 0°. Figure 18 The number of subdomains M designed in this embodiment of the invention x ×M y =3×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 The first type of periodic microstructure obtained when the ratio is 0.12, the Poisson's ratio factor Bt = 1, and the material orientation angle θ = 0° Figure 19 The number of subdomains M designed in this embodiment of the invention x ×M y =3×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 The second type of periodic microstructure obtained when the ratio is 0.12, the Poisson's ratio factor Bt = 1, and the material orientation angle θ = 0° Figure 20 The number of subdomains M designed in this embodiment of the invention x ×M y =3×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 The third type of periodic microstructure obtained when the ratio is 0.12, the Poisson's ratio factor Bt = 1, and the material orientation angle θ = 0° Figure 21 In this embodiment of the invention, the Poisson's ratio factor Bt = 0.5, and the number of subdomains M is designed. x ×M y =2×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 The optimal macroscopic anisotropic multimaterial periodic topology obtained when the material orientation angle θ = 0° is 0.12. Figure 22In this embodiment of the invention, the Poisson's ratio factor Bt = 0.5, and the number of subdomains M is designed. x ×M y =2×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 =0.12, and the material orientation angle θ = 0° yields the first type of anisotropic periodic microstructure. Figure 23 In this embodiment of the invention, the Poisson's ratio factor Bt = 0.5, and the number of subdomains M is designed. x ×M y =2×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 =0.12, and when the material orientation angle θ = 0°, the second type of anisotropic periodic microstructure is obtained. Figure 24 In this embodiment of the invention, the Poisson's ratio factor Bt = 0.5, and the number of subdomains M is designed. x ×M y =2×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 =0.12, and when the material orientation angle θ = 0°, the third type of anisotropic periodic microstructure is obtained. Figure 25 In this embodiment of the invention, the Poisson's ratio factor Bt = 2, and the number of subdomains M is designed. x ×M y =2×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 The optimal macroscopic anisotropic multimaterial periodic topology obtained when the material orientation angle θ = 0° is 0.12. Figure 26 In this embodiment of the invention, the Poisson's ratio factor Bt = 2, and the number of subdomains M is designed. x ×M y =2×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 =0.12, the first type of anisotropic periodic microstructure obtained when the material orientation angle θ = 0°. Figure 27 In this embodiment of the invention, the Poisson's ratio factor Bt = 2, and the number of subdomains M is designed. x ×M y=2×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 =0.12, the second type of anisotropic periodic microstructure obtained when the material orientation angle θ = 0°. Figure 28 In this embodiment of the invention, the Poisson's ratio factor Bt = 2, and the number of subdomains M is designed. x ×M y =2×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 =0.12, the third type of anisotropic periodic microstructure obtained when the material orientation angle θ = 0°. Detailed Implementation

[0011] See Figure 1 and Figure 2 The static topology optimization method for anisotropic multi-material periodic structures based on meshless EFGM mainly includes the following steps: First, based on the meshless EFGM theory, the shape function is constructed using the Moving Least Squares (MLS) method to approximately approximate the unknown field function. The MLS approximate expression for u(x) at x is defined as follows: In the formula, p Τ (x)=[p1(x),p2(x),…,p m (x)],p i Let a(x) be a basis function, m be the number of terms in the basis function, and a(x) be a vector of unknown coefficients consisting of a set of functions of x. A linear basis is chosen, and p... Τ (x) = [1, x, y], m = 3, using cubic spline interpolation. In the formula, r = ||xx i || / d mI The radius of the neighborhood of point x is calculated as d. mI =λ×s k λ is a multiplier greater than 1, s k Let i be the distance between the unmesh node i and the k-th unmesh EFGM node that is closest to it.

[0012] Therefore, a(x) is a(x)=A -1 (x)B(x)u (3) In the formula, B(x)=[ω1(x)p(x1)ω2(x)p(x2)···ω NP (x)p(x NP )], u=[u1 u2 u3…u n ] T .

[0013] Substituting formula (11) into formula (9) yields the following: In the formula, Φ(x) is the vector composed of MLS shape functions corresponding to the meshless EFGM nodes in the neighborhood of the calculation point x, and its specific matrix expression is Φ(x)=[φ1(x)φ2(x)···φ n [x]=p T (x)A -1 (x)B(x).

[0014] Determine the 2-direction elastic modulus E2 of an anisotropic multimaterial periodic structure in a multimaterial coordinate system. i , number of material types q, Poisson's ratio of each material Shear modulus of each material Poisson's ratio factor Bt and multi-material orientation angle θ are material properties; according to Hooke's law, in anisotropic multi-material structures, when... Figure 2 When the global coordinate system xy and the material coordinate system 1-2 have a material orientation angle θ, the stress-strain relationship of the anisotropic multimaterial periodic structure is as follows: In the formula, i = 1, 2, ..., q are the indices of the material types. For the elastic matrix of an orthogonal anisotropic multimaterial structure, T s for The coordinate transformation matrix, in, as well as Let be the elastic modulus and Poisson's ratio of the i-th material in the x and y directions of the multi-material coordinate system, respectively, and satisfy the following relationship: Let be the shear modulus; define Poisson's ratio as . When Bt = 1 and θ = 0, the structure is an isotropic multimaterial structure.

[0015] Secondly, based on EFGM theory and the topology optimization concept of alternating active multiphase materials, a SIMP multi-material interpolation model is established. The relative density fields are constructed using the relative densities of multi-material EFGM nodes varying from 0 to 1 at both the macroscopic and microscopic scales as design variables. The elastic moduli at the macroscopic and microscopic scales can be expressed as follows: In the formula, i is the material type index, and k represents the total number of iterations. Let E represent the elastic modulus of the i-th material in the k-th iteration at both the macroscopic and microscopic scales, respectively. i(0) Let be the initial elastic modulus of the i-th material. Let p represent the macroscopic and microscopic EFGM Gaussian point relative densities after the i-th material penalty, respectively. ma p mi These refer to the intensity of macro and micro penalties, respectively. Continuous EFGM Gaussian point relative density field The relative density of EFGM nodes within the EFGM Gaussian point support domain can be approximated by the MLS shape function constructed using the moving least squares method. In the formula The relative densities of the i-th node within the Gaussian point influence domain of the i-th material at the macroscopic and microscopic levels are φ. I,ma φ I,mi , respectively, are the MLS shape functions of the corresponding nodes at the macro and micro levels, and n is the number of nodes in the support domain of the macro and micro EFGM Gaussian points.

[0016] Finally, based on the meshless EFGM, macroscopic and microscopic displacement field analyses of anisotropic multi-material periodic structures were completed. Combining the alternating active multiphase material topology optimization method, with the relative density of EFGM nodes of the multi-material periodic macrostructure and various microstructures as design variables and the volume fraction of the multi-material periodic macrostructure and various microstructures as constraints, a multi-scale topology optimization model of the static anisotropic multi-material periodic structure based on EFGM was established as follows: In the formula, Let be the relative density vector of the EFGM nodes of the i-th material in a multi-material periodic macrostructure. For assembly The obtained EFGM node relative density matrix of the multi-material Let M be the relative density of the EFGM node of the i-th material at the n-th EFGM node within the m-th design subdomain of a multi-material periodic macrostructure. sN represents the total number of design subdomains for a multi-material periodic macrostructure. s The total number of EFGM nodes within a single design subdomain of a multi-material periodic macrostructure; Let be the relative density vector of EFGM nodes of the i-th type of microstructure. For assembly The relative density matrices of EFGM nodes in various microstructures were obtained. Let N be the relative density of the j-th EFGM node in the i-th type of microstructure. mi The total number of EFGM nodes for the i-th type of microstructure; Let be the relative density of the EFGM Gaussian point of the material in the i-th macrostructure. Let be the relative density of the Gaussian points of the EFGM for the i-th type of microstructure. Let i be the volume of the i-th material in a multi-material periodic macrostructure after the update. Let be the initial volume fraction of the i-th material in a multi-material periodic macrostructure. The total volume of a multi-material periodic macroscopic structure; Let i be the volume of the i-th type of microstructure after the update. Let i be the initial volume fraction of the i-th type of microstructure. C represents the total volume of the i-th type of microstructure; F Let ρ be the compliance objective function for a meshless EFGM multi-material periodic macrostructure. min To prevent matrix singularity, the relative density of macro- and micro-EFGM nodes and Gaussian points is set to a minimum; the OC method is used to update the design variables and solve for the optimal anisotropic multi-material periodic topology.

[0017] See Figure 1 The specific steps of the static topology optimization method for anisotropic multi-material periodic structures based on meshless EFGM are as follows: (1) Based on the design requirements of the actual engineering structure, determine the model geometric information and the number of design subdomains M of the macroscopic static structure. x ×M y and multiple material volume fractions Determine the geometric design domain of the microstructure and the volume fraction of various microstructure types. Input the two-directional elastic modulus of the anisotropic multimaterial periodic structure in a multimaterial coordinate system. Number of material types q, Poisson's ratio of each material Shear modulus of each material Poisson's ratio factor Bt and multi-material orientation angle θ material properties, where the superscript i = 1, 2, ..., q is the index of the material type, the subscript ma represents the macroscopic level parameter, and the subscript mi ​​represents the microscopic level parameter. Set the design domain boundary conditions and load magnitude for macroscopic and microscopic structures. Based on the meshless node information, background mesh and set boundary conditions of the corresponding design domain, obtain the EFGM Gaussian point information of the macroscopic and microscopic design domains respectively. (2) Based on the initial multi-material volume fraction of the macro-microstructure, the density of each meshless node in the macro-design domain is determined. and the density of each meshless node within the micro-design domain Perform initialization; set iteration parameters: minimum tolerance ch for relative density difference between multi-material EFGM nodes. min The maximum number of first-level iterations r that affects the minimum tolerance update max Maximum number of second-order iterations for competitive optimization of multiphase materials The maximum number of iterations t in the third level of the internal alternating multiphase cycle setting max Set the initial first-level iteration step number r = 1; (3) Set the initial second-level iteration step s = 1, and select the two phase materials in the two-phase material competitive optimization subproblem as a = 1, b = a + 1; (4) Set the initial number of third-level iterations t = 1; (5) Calculate the current total number of iterations k = t max s max (r-1)+t max (s-1)+t, begin executing the k(r,s,t)th iteration; (6) Based on the EFGM theory and the idea of ​​alternating active multiphase material topology optimization, a SIMP multi-material interpolation model is established. q kinds of materials with variable relative densities in the range of 0 to 1 are introduced. The relative density fields are constructed with the relative densities of the meshless EFGM nodes of macroscopic multi-materials and multi-type microstructures as design variables. The material properties of the EFGM Gaussian point under the k-th iteration of the i-th material in the macroscopic structure and under the k-th iteration of the i-th type of microstructure in the microscopic scale are solved. (7) Apply micro-periodic boundary constraints to the microstructure, solve for the equivalent elasticity matrix of various microstructures, and derive the discrete static control equations of the meshless EFGM at the microscale. The specific steps are as follows: (7.1) At the microscale, the discrete static governing equations of anisotropic multi-material periodic microstructures are derived based on meshless EFGM and energy homogenization, and the equivalent elastic matrix is ​​calculated. To ensure the displacement continuity of all unit cells at the boundaries of the periodic microstructure, see [reference needed]. Figure 3The diagram shows the classification of EFGM nodes in the microstructure unit cell. I represents the EFGM nodes at the vertices of the microstructure unit cell; I, II, III, and IV represent the sets 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. Given an initial test strain, explicit boundary conditions are obtained directly applied to the microstructure. In the formula, For the positive and negative displacements of the EFGM nodes on a pair of parallel boundaries of the microstructure, denoted as . A constant describing the displacement difference between a pair of parallel boundaries of a microstructure; (7.2) See also Figure 3 The total displacement field U of the microstructure unit cell mi The displacements are divided into four categories: the first category is the displacement U1 of EFGM node sets A, B, C, and D; the second category is the displacement U2 of EFGM node set V; the third category is the displacement U3 of EFGM node sets I and IV; and the fourth category is the displacement U4 of EFGM node sets II and III. Wherein, U3 + U4 = W F Always true, W F pass Seek; (7.3) Based on the meshless EFGM and SIMP multi-material interpolation models, the discrete statics governing equation K for anisotropic multi-material periodic structures at the microscale can be derived. Fmi U mi =F mi Its specific expansion is as follows In the formula, F1 represents the reaction forces at nodes A, B, C, and D of the given displacement boundary EFGM; F2 represents the force of node set V of EFGM; F3 represents the force of node sets I and IV; and F4 represents the force of sets II and III. 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. Fij =K Fji (i,j = 1,2,3,4); (7.4) Analogous to the elasticity matrix of anisotropic multi-material structures at the macroscopic scale, the elasticity matrix of the i-th material before and after the k(r,s,t)-th hierarchical iteration at the microscopic scale is calculated based on the SIMP multi-material interpolation model. The equivalent elasticity matrix of the i-th type of microstructure is obtained. In the formula, u A(ij) The displacement of the EFGM node parameter to be solved corresponds to the strain field of the EFGM node test. Stiffness matrix of a unit cell of a multi-material microstructure (8) On a macroscopic scale, the discrete static governing equations of the macroscopic anisotropic multi-material periodic structure EFGM are derived based on the meshless EFGM, and the macroscopic compliance is solved. The specific steps are as follows: (8.1) Determine the overall force-load vector F of the design domain EFGM based on the magnitude of the applied force load within the design domain. ma Based on the given force and displacement boundary conditions, various force and displacement boundary conditions are processed. Among them, the penalty function method is used to process the displacement boundary, and the penalty term K of the overall force stiffness matrix of EFGM is obtained. Fα,ma and the overall force load vector penalty term F of EFGM α,ma ; (8.2) Combining the equivalent elastic tensor obtained in step (7.4), the elasticity matrix on the periodic macrostructure is calculated as follows: The overall stiffness matrix of EFGM is assembled by combining the SIMP multi-material interpolation model and the relative density distribution function. The meshless EFGM discrete statics governing equations for the statics problem of orthotropic structures are derived as follows: The submatrices of each term in the formula are as follows: In the formula, For static analysis geometric matrix, For EFGM node shape functions, S is a diagonal matrix. When displacement constraints are applied in the x or y direction, S x or S y If it is 1, otherwise S x or S y =0; (8.3) Derivation of the discrete static control equations of the macroscopic structure EFGM Obtain the EFGM node displacement U ma ; (9) Based on the meshless EFGM, the macroscopic and microscopic displacement fields of the anisotropic multi-material periodic structure are analyzed. Combined with the alternating active multiphase material topology optimization method, the relative density of EFGM nodes of the multi-material periodic macrostructure and the multi-type microstructure are used as design variables, and the volume fraction of the multi-material periodic macrostructure and the multi-type microstructure are used as constraints. The multi-scale topology optimization model of the static anisotropic multi-material periodic structure based on EFGM is established as shown in formula (12). The macroscopic structural compliance C is solved according to the macroscopic structural EFGM node displacement and node force load solved in step (8).F ; (10) Solve for the macroscopic volume sensitivity and macroscopic compliance sensitivity, and solve for the microscopic volume sensitivity and microscopic compliance sensitivity; apply periodic constraints to the multi-material periodic macroscopic structure, set the relative density of multi-material EFGM nodes with the same number in each design subdomain to be the same, and obtain the averaged macroscopic EFGM node relative density; the steps are as follows: (10.1) Solving for the volume sensitivity of anisotropic multimaterial periodic macroscopic structures: In the formula φ I,ma Given the nodal shape functions of EFGM, solve for the compliance C of a statically anisotropic multimaterial periodic macrostructure. F For EFGM node density The sensitivity is: In the formula, EFGM is the overall force stiffness matrix. right The sensitivity is: In the formula B FJ,ma For the geometric matrix of the macroscopic structure static analysis, p ma To increase the severity of penalties for macroeconomic structures; (10.2) Solving for the volume sensitivity of anisotropic microstructures: Solving for the compliance C of anisotropic multimaterial periodic macrostructures F For microscopic EFGM node density Sensitivity: In the formula, EFGM is the overall force stiffness matrix. right The sensitivity is: In the formula, the equivalent elasticity matrix of the i-th type of microstructure right The sensitivity is: In the formula, the stiffness matrix of the i-th type of microstructure right The sensitivity is: (10.3) In anisotropic multimaterial periodic macrostructures, periodic constraints are achieved by adjusting the relative density of multimaterial EFGM nodes with the same node number n in each design subdomain. This is achieved by forcing the settings to be exactly the same; therefore, it must be ensured that the relative density of the multi-material EFGM nodes is updated every time. The modified relative density of the multi-material EFGM nodes, after being periodically and evenly distributed, is: (11) Based on the topology optimization idea of ​​alternating active multiphase materials, the relative density of macro and micro EFGM nodes is updated using the OC optimization criterion. The specific steps are as follows: (11.1) Update the relative density of macroscopic multi-material EFGM nodes. When i = a, update the relative density of EFGM nodes of the a-th material in the k-th hierarchical iteration as follows: in for: In the formula, m0 = 0.2 is the positive displacement limit, τ = 0.5 is the damping coefficient, and to avoid matrix singularity, the lower limit of relative density is taken as ρ. min =10 -3 According to the OC criterion, the definition of a statics problem is as follows: for λ ma The median is calculated based on the upper and lower limits of the bisection interpolation points after the periodic macrostructure update; when i = b, the relative density of the nodes of material b in this iteration is solved based on the relationship that the sum of the relative densities of materials a and b before and after the update remains unchanged. Assembled multi-material node relative density matrix When i≠a and i≠b, the relative density of macroscopic nodes of the i-th material remains consistent with the previous iteration. (11.2) Update the design variables under the microstructure. When i = a or i = b, update the relative density of EFGM nodes of the i-th type of microstructure under the k-th hierarchical iteration as follows: in for According to the OC criterion, the definition of a statics problem is... for λ mi The median is calculated based on the upper and lower limits of the bisection interpolation points after the periodic macrostructure update. When i≠a and i≠b, this hierarchical iteration does not involve updating the relative density of the EFGM nodes of the i-th type of microstructure, that is, the relative density of the EFGM nodes of the i-th type of microstructure after the k-th iteration should be the same as that after the (k-1)-th iteration. Therefore, the relative density of the EFGM nodes of the i-th type of microstructure under the k-th iteration is... in (11.3) Output the macroscopic multi-material EFGM node relative density matrix and the relative density matrix of EFGM nodes in various microstructures Calculate the infinite norm of the difference between the relative density matrices of macroscopic EFGM nodes. The infinite norm of the difference between the relative density matrix of microscopic EFGM nodes (12) Determine if the iteration termination condition is met: (a) Determine if the current third-level iteration number t satisfies t < t max If satisfied, let t = t + 1 and return to step (5); if not satisfied, proceed to the second-level iteration condition judgment; (b) Determine whether the second-level iteration condition s < s is satisfied. max If s < s max If b < q, then update s = s + 1, b = b + 1, and return to step (4); if s < q, then update s = s + 1, b = b + 1, and return to step (4); max However, if b < q is not satisfied, then update s = s + 1, a = a + 1, b = a + 1, and return to step (4); if s < q is not satisfied... max Then proceed to the first-level iteration judgment; (c) Determine whether the first-level iteration satisfies the following three conditions: r≤r max ch ma <ch min ch mi <ch min If any one of the three conditions is met, return to step (3). If none of the three conditions are met, the overall iterative process ends. Output the optimal topological configuration of the anisotropic multi-material periodic macroscopic structure and multiple microstructures of the static problem in the form of an image.

[0018] The following is an example of the application of the method of the present invention in engineering practice: See Figure 4 This embodiment is a cantilever beam structure with a macroscopic design domain side length of L = 0.72m. It is discretized using 72×72 meshless nodes, with its left boundary fixed and a downward force F = 10 applied to the midpoint of its right boundary. 5 N; see also Figure 5 The micro-design domain has a side length of l = 0.01 m. EFGM nodes are used to discretize the micro-design domain, and the initial design aperture radius is l / 6. Considering different engineering design requirements, this example performs static multi-scale topology optimization on anisotropic multi-material periodic cantilever beam structures under the following three conditions. In the first case, the number of material types q is 2 and 3, respectively. When the number of material types q = 2, the elastic moduli are E1 = 10 GPa and E2 = 10 GPa, respectively. -3 GPa, and the initial volume fractions of the macroscopic multimaterials are respectively Initial volume fraction of various microstructures When the number of material types q = 3, the elastic moduli are E1 = 10 GPa, E2 = 8 GPa, and E3 = 10 GPa, respectively. -3 GPa, and the initial volume fractions of the macroscopic multimaterials are respectively Initial volume fraction of various microstructures Set the principal Poisson ratio ν uniformly 12 =0.12, Poisson's ratio factor Bt is 1, and the number of design subdomains is M. x ×M y =2×2, with the material orientation angle θ set to 0°; In the second case, with 3 types of materials, the number of design subdomains is set to M respectively. x ×M y =1×2 and M x ×M y =3×2, and the initial volume fractions of macroscopic multi-materials are uniformly set as follows: Initial volume fraction of various microstructures Main Poisson's ratio ν 12 =0.12, Poisson's ratio factor Bt is 1, and material orientation angle θ is 0°; the third case has 3 types of materials and M design subdomains. x ×M y When the ratio is 2×2, we take the Poisson's ratio factor Bt = 0.5 and Bt = 2 respectively, and uniformly set the initial volume fraction of the macroscopic multi-materials as follows: Initial volume fraction of various microstructures Main Poisson's ratio ν 12 =0.12, and the material orientation angle θ is taken as 0°.

[0019] The specific implementation steps of this invention for this example are as follows: (a) Input the geometric information of the cantilever beam model and the number of macroscopic design subdomains M. x ×M y and multiple material volume fractions Input the initial design domain size and multi-material volume fraction of the microstructure. Input the two-directional elastic modulus of the anisotropic multimaterial periodic structure in a multimaterial coordinate system. Number of material types q, Poisson's ratio of each material Shear modulus of each material Poisson's ratio factor Bt and multi-material orientation angle θ material properties, where the superscript i = 1, 2, ..., q is the index of the material type, the subscript ma represents the macroscopic level parameter, and the subscript mi ​​represents the microscopic level parameter. Set the design domain boundary conditions and load magnitude for macroscopic and microscopic structures. Based on the meshless node information, background mesh and set boundary conditions of the corresponding design domain, obtain the EFGM Gaussian point information of the macroscopic and microscopic design domains respectively. (b) Based on the initial macro- and micro-scale design domains, the relative density of each meshless node within the macro- and micro-scale design domains is initialized respectively, and the maximum number of iterations for the first, second, and third stages is set to r. max =100, t max =2, minimum tolerance ch for relative density difference of multi-material EFGM nodes min =10 -5 Set the initial first-level iteration step number r = 1; (c) Set the initial second-level iteration step s = 1, and select the two phase materials in the two-phase material competitive optimization subproblem as a = 1 and b = a + 1; (d) Set the initial number of third-level iterations t = 1; (e) Calculate the current total number of iterations k = t max s max (r-1)+t max (s-1)+t, begin executing the k(r,s,t)th iteration; (f) Based on the meshless theory, calculate the shape function of the EFGM node in the influence domain of the Gaussian point of the EFGM, introduce the SIMP multi-material interpolation model, and solve the elastic matrix of each material under the current iteration at the microscale. (g) Apply periodic boundary conditions to the microstructure, solve the overall force stiffness matrix, force matrix and displacement matrix of the EFGM node of the microstructure, and solve the equivalent elastic matrix of various microstructures based on the energy homogenization theory. (h) Based on the SIMP multi-material interpolation model, and combined with the equivalent elasticity matrices of various microstructures determined in step (e), solve the elasticity matrix of each EFGM node on the macrostructure under the current iteration; (i) Determine the overall force matrix of the macroscopic cantilever beam structure EFGM based on the given force load magnitude of the macroscopic cantilever beam structure. Process the force and displacement boundary conditions based on the given force and displacement boundary conditions of the macroscopic cantilever beam structure. Use the penalty function method to process the displacement boundary to obtain the penalty term of the overall force stiffness matrix of the macroscopic cantilever beam structure EFGM and the penalty term of the overall force load vector of EFGM. (j) Based on the elasticity matrix of each EFGM node on the macrostructure under the current iteration obtained in step (f), and combined with the relative density distribution function, assemble the overall force stiffness matrix of the EFGM in the macrostructure design domain. Based on the meshless EFGM discrete statics control equation of the anisotropic multi-material structure statics problem, and combined with the overall force load vector, the penalty term of the overall force stiffness matrix, and the penalty term of the overall force load vector of the EFGM determined in step (g), solve for the displacement parameter values ​​of the EFGM Gaussian point in the macrostructure design domain. (k) Stepwise search for the EFGM nodes in the influence domain of each EFGM Gaussian point in the macro design domain and find the MLS shape function of the corresponding EFGM node. Combine the displacement parameters of the EFGM Gaussian points in the design domain to solve the displacement value of each EFGM node and calculate the macro structural flexibility value. (l) Solve the volume sensitivity of the i-th material of the macrostructure and the volume sensitivity of various microstructures in the multi-scale topology optimization model of the orthogonal anisotropic multi-material periodic structure of the meshless EFGM, and solve the flexibility sensitivity of the macro and micro structures. (m) Apply macroscopic periodic constraints to force the relative density of multi-material EFGM nodes with the same number in each design subdomain to be the same, and obtain the averaged macroscopic EFGM node relative density. (n) The relative density of macroscopic multi-material EFGM nodes is updated using the OC method. When i = a, the relative density of EFGM nodes of the a-th material in the k-th hierarchical iteration is updated according to the OC formula. When i = b, the relative density of the EFGM nodes of material b under this iteration is calculated 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. Assembly of multi-material EFGM node relative density matrix When i≠a and i≠b, the relative density of macroscopic EFGM nodes of the i-th material remains consistent with the previous iteration. (o) The design variables under the microstructure are updated using the OC method. When i = a or i = b, the relative density of EFGM nodes of the i-th type of microstructure under the k-th hierarchical iteration is updated according to the OC formula. When i≠a and i≠b, the relative density of EFGM nodes of the i-th type of microstructure remains consistent with the previous iteration. (p) Output the macroscopic multi-material EFGM node relative density matrix and the microscopic multi-type microstructure EFGM node relative density matrix, and calculate the infinite norm of the difference between the macroscopic EFGM node relative density matrix and the infinite norm of the difference between the microscopic EFGM node relative density matrix. (q) Determine if the iteration termination condition is met: Determine if the current third-level iteration number t satisfies t < t max If the condition is met, let t = t + 1 and return to step (e). If not, proceed to the second-level iteration condition check; check if the second-level iteration condition s < s is met. max If s < s max If b < q, then update s = s + 1, b = b + 1, and return to step (d); if s < q, then update s = s + 1, b = b + 1, and return to step (d). max However, if b < q, then update s = s + 1, a = a + 1, b = a + 1, and return to step (d); if s < q, then update s = s + 1, a = a + 1, b = a + 1, and return to step (d). maxThen proceed to the first-level iteration judgment; determine whether the first-level iteration satisfies the following three conditions: r≤r max ch ma <ch min ch mi <ch min If any one of the three conditions is met, return to step (c). If none of the three conditions are met, the overall iterative process ends. Output the optimal topological configuration of the anisotropic multi-material periodic macroscopic structure and multiple microstructures of the static problem in the form of an image.

[0020] Figures 6 to 28 This embodiment presents the optimal topology and its zero-level set-plane topology configuration obtained through thermo-mechanical coupling topology optimization based on the meshless method and PLSM. Figure 6 In the first case of this embodiment of the invention, the number of material types q = 2, and the elastic moduli of each material are E1 = 10 GPa and E2 = 10 GPa, respectively. -3 GPa, principal Poisson's ratio ν 12 =0.12, Poisson's ratio factor Bt=1, number of design subdomains M x ×M y =2×2, the macroscopic multi-material periodic optimal topology obtained when the material orientation angle θ = 0°, Figures 7-8 These are the first and second type periodic microstructures obtained in the first case of the embodiments of the present invention, respectively; Figure 9 In the first case of this embodiment, the number of material types q = 3, and the elastic moduli of each material are E1 = 10 GPa, E2 = 8 GPa, and E3 = 10 GPa, respectively. -3 GPa, principal Poisson's ratio ν 12 =0.12, Poisson's ratio factor Bt=1, number of design subdomains M x ×M y =2×2, the macroscopic multi-material periodic optimal topology obtained when the material orientation angle θ = 0°, Figures 10-12 These are the first, second, and third types of periodic microstructures obtained under these values; Figure 13 The second case in this embodiment of the invention designs the number of subdomains M. x ×M y =1×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 The optimal macroscopic multi-material periodic topology obtained when the ratio is 0.12, the Poisson's ratio factor Bt = 1, and the material orientation angle θ = 0° is... Figures 14-16 These are the first, second, and third types of periodic microstructures obtained under these values; Figure 17 This is the second case in the embodiments of the present invention, which designs the number of subdomains M.x ×M y =3×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 The optimal macroscopic multi-material periodic topology obtained when the ratio is 0.12, the Poisson's ratio factor Bt = 1, and the material orientation angle θ = 0° is... Figures 18-20 These are the first, second, and third types of periodic microstructures obtained under these values; Figure 21 In the third case of this invention, the Poisson's ratio factor Bt = 0.5, and the number of subdomains M is designed. x ×M y =2×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 =0.12, the optimal macroscopic anisotropic multimaterial periodic topology obtained when the material orientation angle θ = 0°. Figures 22-24 These are the first, second, and third types of anisotropic periodic microstructures obtained under these values; Figure 25 In the third case of this invention, the Poisson's ratio factor Bt = 2, and the number of subdomains M is designed. x ×M y =2×2, number of material types q=3, elastic moduli are E1=10GPa, E2=8GPa, E3=10 -3 GPa, principal Poisson's ratio ν 12 =0.12, the optimal macroscopic anisotropic multimaterial periodic topology obtained when the material orientation angle θ = 0°. Figures 26-28 These are the first, second, and third types of anisotropic periodic microstructures obtained under these values.

[0021] In the multi-scale topology optimization of static multi-material periodic structures, the decreasing elastic modulus of multiple materials corresponds to multiple types of microstructures with varying degrees of robustness. Furthermore, as the number of material types q increases, the topological configuration of these microstructures adaptively adjusts. When the longitudinal direction is fixed while the number of transverse subdomains M is increased... xIn this process, the multiple materials of the macrostructure are periodically distributed within a more slender and allowable design space, resulting in increased flexibility and differentiated evolution of various microstructure morphologies. The increase in the Poisson's ratio factor Bt causes the base materials of various microstructures to gradually evolve from longitudinal to lateral directions, maximizing the equivalent elastic performance of the microstructures. The periodic macrostructure will generate more refined lateral supports to enhance structural rigidity; the value of Bt ranges from 2 to 4. The increase in the multi-material orientation angle θ causes the base materials of various microstructures to rotate in the direction with stronger deformation resistance, leading to the generation of truss support structures with different inclination angles within the periodic macrostructure. The trend of flexibility variation is related to the position of the force load and constraints; the value of θ ranges from 0° to 45°.

[0022] 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 multi-scale topology optimization method for anisotropic multi-material structures based on meshless EFGM, characterized in that... Includes the following steps: (1) Based on the design requirements of the actual engineering structure, determine the model geometry information and multi-material volume fraction of the macroscopic static structure. Determine the geometric design domain of the microstructure and the volume fraction of various microstructure types. Input the two-directional elastic modulus of the anisotropic multimaterial periodic structure in a multimaterial coordinate system. Number of material types q, Poisson's ratio of each material Shear modulus of each material Poisson's ratio factor Bt and multi-material orientation angle θ material properties, where i = 1, 2, ... q are the indices of material types, setting the design domain boundary conditions and load magnitudes for macroscopic and microscopic structures, and obtaining the Gaussian point information of the macroscopic and microscopic design domains respectively based on the EFGM node information of the corresponding design domain, background mesh and set boundary conditions; (2) Based on the initial multi-material volume fraction of the macro-microstructure, the density of each meshless node in the macro-design domain is determined. and the density of each meshless node within the micro-design domain Perform initialization and set the number of subdomains M. x ×M y The plan is to utilize an alternating active multiphase material topology optimization algorithm to achieve target optimization by updating and iterating the design variables. The iteration parameters are set as follows: the minimum tolerance ch of the difference in the relative density matrix of the multi-material EFGM nodes. min The maximum number of first-level iterations r that affects the minimum tolerance update max Maximum number of second-order iterations for competitive optimization of multiphase materials The maximum number of iterations t in the third level of the internal alternating multiphase cycle setting max Set the initial first-level iteration step number r = 1; (3) Set the initial second-level iteration step s = 1, and select the two phase materials in the two-phase material competitive optimization subproblem as a = 1, b = a + 1; (4) Set the initial number of third-level iterations t = 1; (5) Calculate the current total number of iterations k = t max s max (r-1)+t max (s-1)+t, begin executing the k(r,s,t)th iteration; (6) Based on EFGM theory and the idea of ​​alternating active multiphase material topology optimization, a SIMP multi-material interpolation model is established. q kinds of materials with variable relative densities in the range of 0 to 1 are introduced. The relative density fields are constructed with the relative densities of the meshless EFGM nodes of macroscopic multi-materials and multi-type microstructures as design variables. The Gaussian point material properties under the k-th iteration of the i-th material in macroscopic structure and under the k-th iteration of the i-th type of microstructure in microscale are solved. (7) At the microscale, based on the meshless EFGM and energy homogenization theory, a SIMP multi-material interpolation model is introduced to apply static periodic boundary conditions to the microstructure and solve the global stiffness matrix K of various microstructures. Fmi and displacement matrix U mi The discrete static governing equations of anisotropic multi-material periodic microstructures are derived and the equivalent elastic matrix is ​​calculated. (8) At the macroscopic scale, based on the meshless EFGM theory and by introducing the SIMP multi-material interpolation model, the force load matrix F of the EFGM node of the multi-material macrostructure is determined. ma Overall stiffness matrix and their penalty item K Fα,ma K Fα,ma The discrete static governing equations of the macroscopically anisotropic multi-material periodic structure EFGM are derived and the nodal displacements U of EFGM are solved. ma And softness value C F ; (9) Based on the meshless EFGM, the macroscopic and microscopic displacement field analysis of anisotropic multi-material periodic structures is completed. Combined with the alternating active multiphase material topology optimization algorithm, the relative density of EFGM nodes of multi-material periodic macrostructure and multi-type microstructure is used as the design variable, and the volume fraction of multi-material periodic macrostructure and multi-type microstructure is used as the constraint condition. A multi-scale topology optimization model of anisotropic multi-material periodic structures based on EFGM is established. (10) Solve for the macroscopic volume sensitivity and macroscopic compliance sensitivity, and solve for the microscopic volume sensitivity and microscopic compliance sensitivity; apply periodic constraints to the multi-material periodic macroscopic structure, and force the relative density of multi-material EFGM nodes with the same number in each design subdomain to be the same, to obtain the averaged macroscopic meshless EFGM node relative density. (11) Update macro- and micro-scale design variables based on the alternating active multiphase topology optimization idea and OC optimization criterion: (11.1) Update the macro-scale multi-material EFGM node relative density. When i = a, update the EFGM node relative density of the a-th material in the k-th hierarchical iteration as follows: When i = b, the relative density of the EFGM nodes of material b under this iteration is calculated 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. Assembly of multi-material EFGM node relative density matrix When i≠a and i≠b, the relative density of macroscopic EFGM nodes of the i-th material remains consistent with the previous iteration. (11.2) Update the design variables under the microstructure. When i = a or i = b, update the relative density of EFGM nodes of the i-th type of microstructure under the k-th hierarchical iteration as follows: When i≠a and i≠b, the relative density of EFGM nodes of the i-th type of microstructure remains consistent with the previous iteration. (11.3) Output the macroscopic multi-material EFGM node relative density matrix and the relative density matrix of EFGM nodes in various microstructures Calculate the infinite norm of the difference between the relative density matrices of macroscopic EFGM nodes. The infinite norm of the difference between the relative density matrix of microscopic EFGM nodes (12) Determine if the iteration termination condition is met: (a) Determine if the current third-level iteration number t satisfies t < t max If satisfied, let t = t + 1 and return to step (5); if not satisfied, proceed to the second-level iteration condition judgment; (b) Determine whether the second-level iteration condition s < s is satisfied. max If s < s max If b < q, then update s = s + 1, b = b + 1, and return to step (4); if s < q, then update s = s + 1, b = b + 1, and return to step (4); max If b≥q, then update s=s+1, a=a+1, b=a+1, and return to step (4); if s<s max Then proceed to the first-level iteration judgment; (c) Determine whether the first-level iteration satisfies the following three conditions: r≤r max ch ma <ch min ch mi <ch min If any one of the three conditions is met, return to step (3). If none of the three conditions are met, the overall iterative process ends. Output the optimal topological configuration of the anisotropic multi-material periodic macroscopic structure and multiple microstructures of the static problem in the form of an image.

2. The multi-scale topology optimization method for anisotropic multi-material structures based on meshless EFGM as described in claim 1, characterized in that... The EFGM nodes are used to discretize both the periodic macroscopic structure design domain and the multi-type microstructure unit cell design domain. The design variables at both the macroscopic and microscopic scales are constructed using the relative density of the EFGM nodes. The corresponding relative density of the EFGM nodes is obtained by interpolation using the SIMP multi-material interpolation model.

3. The multi-scale topology optimization method for anisotropic multi-material structures based on meshless EFGM according to claim 1, characterized in that... In step (7), the anisotropic elastic matrices of each material are equivalently represented based on the meshless EFGM and energy homogenization theory, and the anisotropic equivalent elastic matrix is ​​solved. Explicit periodic boundary conditions were applied to the microstructure. This ensures the continuity of displacement at the microstructure boundary, where, For the positive and negative displacements of the EFGM nodes on a pair of parallel boundaries of the microstructure, denoted as . It is a constant that describes the displacement difference between a pair of parallel boundaries of a microstructure.

4. The multi-scale topology optimization method for anisotropic multi-material structures based on meshless EFGM as described in claim 1, characterized in that... In step (8), deriving the discrete static control equations of the macroscopic structure EFGM requires introducing the anisotropic equivalent elastic matrix obtained from the microscopic scale. Establishing submatrices of the macroscopic overall stiffness matrix In the formula Let p be the Gaussian point relative density after penalty for the i-th material. ma To enhance the overall level of punishment, B FJ,ma The geometric matrix for macroscopic static analysis enables the interaction between topology optimization of anisotropic multi-material periodic macroscopic structures and microstructure topology optimization, achieving parallel design integrating macro and micro dimensions.

5. The multi-scale topology optimization method for anisotropic multi-material structures based on meshless EFGM according to claim 1, characterized in that... Step (9) establishes a multi-scale topology optimization model for anisotropic multi-material periodic structures based on EFGM, using the relative density of EFGM nodes in multi-material periodic macrostructures and multi-type microstructures as design variables and the volume fraction of multi-material periodic macrostructures and multi-type microstructures as constraints. The specific expression is as follows: In the formula, Let be the relative density vector of the EFGM nodes of the i-th material in a multi-material periodic macrostructure. For assembly The obtained EFGM node relative density matrix of the multi-material Let M be the relative density of the EFGM node of the i-th material at the n-th EFGM node within the m-th design subdomain of a multi-material periodic macrostructure. s N represents the total number of design subdomains for a multi-material periodic macrostructure. s The total number of EFGM nodes within a single design subdomain of a multi-material periodic macrostructure; Let be the relative density vector of EFGM nodes of the i-th type of microstructure. For assembly The relative density matrices of EFGM nodes in various microstructures were obtained. Let N be the relative density of the j-th EFGM node in the i-th type of microstructure. mi The total number of EFGM nodes for the i-th type of microstructure; Let be the relative density of the EFGM Gaussian point of the material in the i-th macrostructure. Let be the relative density of the Gaussian points of the EFGM for the i-th type of microstructure. Let i be the volume of the i-th material in a multi-material periodic macrostructure after the update. Let be the initial volume fraction of the i-th material in a multi-material periodic macrostructure. The total volume of a multi-material periodic macroscopic structure; Let i be the volume of the i-th type of microstructure after the update. Let i be the initial volume fraction of the i-th type of microstructure. C represents the total volume of the i-th type of microstructure; F Let ρ be the compliance objective function for a meshless EFGM multi-material periodic macrostructure. min To prevent matrix singularity, the relative density of macro- and micro-EFGM nodes and Gaussian points is set to a minimum.

6. The multi-scale topology optimization method for anisotropic multi-material structures based on meshless EFGM according to claim 1, characterized in that... Step (10) In an anisotropic multi-material periodic macrostructure, the macrostructure is divided into M... x ×M y Total M s Given several design subdomains, and forcibly setting the relative density of macroscopic EFGM nodes with the same EFGM node number n in each design subdomain to be exactly the same to achieve the periodic constraint of the macroscopic structure, then the relative density of macroscopic EFGM nodes satisfying the periodicity is:

7. The multi-scale topology optimization method for anisotropic multi-material structures based on meshless EFGM according to claim 1, characterized in that... In the multi-scale topology optimization of static multi-material periodic structures, the decreasing elastic modulus of multiple materials corresponds to multiple types of microstructures with varying degrees of robustness. Furthermore, as the number of material types q increases, the topological configuration of these microstructures adaptively adjusts. When the longitudinal direction is fixed while the number of transverse subdomains M is increased... x At that time, the multiple materials of the macrostructure are periodically distributed within a more slender allowable design space, which increases the flexibility and causes the morphology of various microstructures to evolve in a differentiated manner.

8. The multi-scale topology optimization method for anisotropic multi-material structures based on meshless EFGM according to claim 1, characterized in that... The increase of the Poisson's ratio factor Bt causes the substrate materials of various microstructures to gradually evolve from the longitudinal direction to the transverse direction in order to maximize the equivalent elastic performance of the microstructures. The value of Bt ranges from 2 to 4. The increase of the multi-material orientation angle θ causes the substrate materials of various microstructures to rotate in the direction with stronger resistance to deformation, resulting in the generation of truss support structures with different inclination angles within the periodic macrostructure. The trend of flexibility change is related to the position of force load and constraint. The value of θ ranges from 0° to 45°.

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