Multi-scale topological optimization method based on meshless EFGM multi-material periodic heat transfer structure

Abstract

The invention discloses a meshless EFGM-based multi-scale topological optimization method for an anisotropic multi-material periodic steady-state heat transfer structure, and the method comprises the steps: (1) inputting the number of material types, a thermal conductivity factor, the number of design subdomains, a material direction angle and other parameters of an anisotropic structure, and employing EFGM nodes to discretize macro and micro design domains; (2) solving a microcosmic equivalent heat conduction matrix by using a homogenization method; (3) solving a macroscopic equivalent heat conduction matrix by using an SIMP multi-material interpolation method; (4) solving a macroscopic temperature field of the anisotropic structure based on meshless EFGM; (5) establishing a multi-scale topological optimization mathematical model of the multi-material periodic steady-state heat transfer structure based on a meshless EFGM and a homogenization method; (6) solving macro-micro volume sensitivity and thermal flexibility sensitivity, and applying macro periodic constraint; (7) solving the mathematical model, and updating macro and micro EFGM node relative density design variables by using an OC method; according to the method, multi-scale topological optimization of the anisotropic multi-material periodic steady-state heat transfer structure is carried out based on the meshless EFGM, and the structure is clear, smooth, practical and efficient.

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 heat transfer structures based on the meshless Galerkin Method (EFGM). Background Technology

[0002] Currently, the requirements for heat dissipation performance of materials are becoming increasingly stringent in fields such as aerospace, thermal equipment, and biomedicine. Designing and manufacturing materials with high thermal conductivity has attracted considerable research attention, and topology optimization, as an important method for structural design and performance optimization, has gained significant popularity. Topology optimization is a method that adjusts the material distribution within a given design domain according to load conditions to achieve optimal structural performance. It can minimize material usage, reduce structural weight and cost, and has broad application prospects. Currently, the main performance analysis and calculation methods for topology optimization include element-based analysis methods such as the finite element method (FEM), isogeometric analysis (IGA), and finite volume method (FVM). Due to the presence of meshes, these methods can introduce numerical instability problems such as checkerboard patterns, thus affecting the reliability and manufacturability of the topology optimization results. Meshless methods, on the other hand, avoid the cumbersome mesh generation process. By discretizing the computational domain using field nodes and constructing shape functions and approximate field functions, they can solve the problem of poor performance and response analysis caused by mesh distortion or deformation in element-based analysis methods. EFGM, as a relatively mature meshless method, is easy to construct high-order field functions, exhibits better computational stability and higher accuracy, and demonstrates good topological convergence without the need for sensitivity filtering techniques. It is widely used in structural performance analysis and topology optimization. However, most current meshless EFGM topology optimization designs for heat transfer structures are for isotropic single-material structures.

[0003] With the increasing demands for heat dissipation performance in engineering applications, single isotropic materials are insufficient to meet the design requirements of complex working conditions. Multi-material composite structures possess excellent thermal conductivity, but composite materials exhibit anisotropy, meaning they have different elastic moduli, thermal conductivity, and electrical conductivity in different material directions, giving them numerous advantages such as lightweight, high tensile strength, and excellent heat dissipation capabilities. By rationally designing and arranging these anisotropic materials, heat dissipation performance in specific directions can be optimized, fully leveraging the positive role of anisotropic multi-materials in practical engineering applications and effectively improving the overall performance of the structure. The complexity of multi-material structures increases the difficulty of material manufacturing. Introducing periodicity into multi-material design can not only enhance the heat dissipation performance of materials but also reduce manufacturing difficulty, making it a potentially valuable research and application area. Periodic structures are a class of uniquely configured structures with identical or similar repeating patterns at macroscopic or microscopic scales. They typically possess good thermal conductivity and stability due to the uniformity of heating resulting from the interaction between repeating units. This allows periodic structures to more uniformly distribute thermal stress when facing external thermal loads, enhancing the heat dissipation performance of materials and improving the overall structural stability, leading to their widespread application in many industrial fields. Currently, some scholars have conducted topology optimization studies on periodic multi-material heat transfer structures, but most of these studies only involve a single scale and do not consider the influence of macroscopic and microscopic structures on their heat dissipation performance from a multi-scale perspective.

[0004] Multi-scale topology optimization of periodic multi-material heat transfer structures not only allows for simultaneous structural optimization at both macro and micro levels, ensuring optimal material placement while deriving the best heat dissipation configuration, but also provides a comprehensive understanding of the impact of various parameters on material heat dissipation performance at different scales. This facilitates a holistic consideration of requirements at each scale during the design process, enhancing the structure's heat dissipation performance at both macro and micro levels. Compared to single-scale, single-material heat transfer structure topology optimization problems, this approach has greater theoretical research significance and engineering application value. Currently, there are few reports on multi-scale optimization design of multi-material periodic heat transfer structures, and the application of meshless EFGM to multi-scale topology optimization of multi-material periodic heat transfer structures is also unreported. This invention, based on meshless EFGM, performs multi-scale topology optimization design on anisotropic multi-material periodic steady-state heat transfer structures, aiming to maximize the heat dissipation performance of multi-material periodic structures at both macro and micro scales while deriving the optimal material placement and topology configuration. Summary of the Invention

[0005] The integrated macro-micro structure design method has demonstrated strong vitality in structural design and optimization, and multi-material periodic structures show promising application prospects in this field. The purpose of this invention is to establish a multi-scale topology optimization model for macro-micro structures with multi-material periodic heat transfer using a meshless EFGM. The optimization objective is to minimize the thermal compliance of the macrostructure. The relative densities of the meshless EFGM nodes in the macro and micro structures are used as design variables at the macro and micro scales, respectively, with the volume fractions of the multi-material periodic macrostructure and various microstructure types as constraints. Based on the theory, a computer program is written to perform topology optimization design for different numbers of multi-material types, design subdomains, thermal conductivity factors, and multi-material orientation angles, outputting the optimal material distribution at the macro level and the optimal topology configuration of the macro and micro structures.

[0006] The technical solution adopted by this invention to solve its technical problem is as follows: Discretize the steady-state heat transfer control equation of anisotropic structures based on meshless EFGM; to ensure periodicity at both scales, the macroscopic design domain is divided into multiple design subdomains at the macroscopic scale, forcing the relative density of EFGM nodes at the same location in all design subdomains to be exactly the same in the calculation, and combining this with a homogenization method to achieve the periodic distribution of the microstructure at the microscopic scale; based on the alternating active multiphase algorithm and meshless EFGM theory, construct a relative density field using the relative density at the macroscopic and microscopic EFGM nodes of multiple materials as the design variable through the SIMP multimaterial interpolation model; establish a macroscopic-microscopic integrated topology optimization model for steady-state heat transfer multimaterial periodic structures based on meshless EFGM and the energy homogenization method, and obtain different optimal topology configurations by adjusting different design parameters and material properties.

[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 initial design domain and input the macro- and micro-thermal parameters of the anisotropic multi-material system: the number of multi-material types q, and the thermal conductivity of various microstructures in the two directions of the material coordinate system. Thermal conductivity factor (TCF), multi-material orientation angle θ, and macro- and micro-volume fractions of various materials. (In this invention, the subscript ma represents the macroscopic level, mi represents the microscopic level, and the index of the material type i = 1, 2, 3, ..., q); (2) Discretize the macro- and micro-scale design domains using EFGM nodes, and arrange Gaussian points within and on the boundaries of the design domains; set the relative density vectors and matrices of EFGM nodes for macro-multi-material and initial multi-class microstructures; set the number of horizontal and vertical design subdomains; and set the iteration parameters: the minimum tolerance ch for the change in relative density 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 tmax 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 k for the r, s, t-th iterations: k = t max s max (r-1)+t max (s-1)+t, where r, s, and t are the first, second, and third iteration steps, respectively; (6) Based on the meshless EFGM theory and the idea of ​​alternating active multiphase material topology optimization, and by analogy with the SIMP single-material interpolation model, a SIMP multi-material macro-micro interpolation model is established: This invention constructs a relative density field using the relative density of multi-material EFGM nodes varying from 0 to 1 as the design variable. Under the r-th first-level iteration, the s-th second-level iteration, and the t-th third-level iteration, the hierarchical iteration is performed for a total of k = t. max s max (r-1)+t max (s-1)+t times, where t max This is the maximum limit for three-level iterations, which can be set to 1, 2, or 3 as needed. Let represent the number of two-phase subproblems to be solved in the alternating active phase algorithm. Then, the material properties and thermal conductivity λ of the Gaussian point under the k(r,s,t)th graded iteration for the i-th material are... i(0) The relationship between them is Let λ be the relative density of the Gaussian point after penaltying for the i-th material, p be the penalty intensity, and λ be the relative density of the i-th material. i(0) Let be the thermal conductivity of the i-th material in a completely solid state; and be the 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. Let φ be the relative density of the i-th EFGM node within the Gaussian point influence domain of the i-th material. I Let n be the MLS shape function corresponding to the EFGM node, and n be the number of EFGM nodes in the Gaussian point support domain. (7) Solving the macroscopic temperature field of anisotropic multimaterials based on meshless EFGM: (a) Discretizing and classifying the EFGM nodes of the microstructure unit cell model; (b) Applying heat transfer periodic boundary conditions according to the EFGM node classification, deriving the microscopic EFGM discrete heat transfer control equation to obtain the EFGM node temperature of the microstructure unit cell; (c) Solving for the EFGM equivalent thermal conductivity matrix of the corresponding microstructure. |Ω mi | represents the area of ​​a single cell of the microstructure. The macroscopic EFGM overall thermal stiffness matrix after introducing the SIMP multi-material interpolation model Transformation forms on single-material microstructure unit cells; (d) through relationships Find the thermal conductivity matrix of the i-th material in the k(r,s,t)th hierarchical iteration in the macroscopic structure. (e) Calculate the overall thermal stiffness matrix of the multi-material EFGM in the steady-state heat transfer periodic macrostructure; (f) Calculate the penalty term of the overall thermal stiffness matrix and the overall thermal load column vector of the EFGM using the penalty function method, and calculate the temperature value of the EFGM node in the macrostructure; (g) Output the EFGM node temperature, overall thermal load vector and its penalty term, and overall thermal stiffness matrix and its penalty term of the macrostructure. (8) Introducing the SIMP multi-material interpolation model, a macro-micro integrated topology optimization mathematical model for multi-material steady-state heat transfer periodic structures based on meshless EFGM and energy homogenization method is established. In the formula, Let be the relative density vector of the EFGM nodes of the i-th material in the macroscopic structure. For assembly The resulting multi-material EFGM node relative density matrix, Let M be the relative density of the EFGM nodes of the i-th material at the n-th EFGM node within the m-th design subdomain of a multi-material periodic macrostructure. s M represents the total number of design subdomains for a multi-material periodic macrostructure. s =M x ×M y M x M y N represents the number of design subdomains in the x and y directions, respectively. s N represents the total number of EFGM nodes within a single design subdomain of a multi-material periodic macrostructure. s =N x ×N y N x N y These represent the number of EFGM nodes in the x and y directions, respectively. Let be the relative density vector of EFGM nodes of the i-th type of microstructure. For assembly The resulting relative density matrices of EFGM nodes for various microstructures Let N be the relative density of the j-th EFGM node in the i-th type of microstructure. mi Let be the total number of EFGM nodes of the i-th type of microstructure; For the thermal compliance objective function of the EFGM of a multi-material periodic macrostructure, P ma and P α,maThe overall thermal load vector of EFGM and its penalty term. This is a global approximate temperature vector. Let T be the global EFGM node shape function matrix. ma This is a column vector of temperature parameters for the EFGM nodes; To introduce the overall thermal stiffness matrix of EFGM after the SIMP multi-material interpolation model, for The EFGM sub-thermal stiffness matrix, I,J=1,2,,TN are the global EFGM node indices, and TN is the solution domain Ω. ma The total number of EFGM nodes within the system. Here, γ is the thermal analysis geometry matrix, and γ is the coordinate transformation matrix. Let be the thermal conductivity matrix of the i-th material in the multi-material coordinate system 1-2. p is the relative density field at a Gaussian point. ma The macroscopic penalty intensity is given by h, the convective heat transfer coefficient is given by φ. I,ma ,φ J,ma Here, Γ3 represents the shape function at the corresponding EFGM node, and Γ3 represents the Robin boundary. Let be the thermal stiffness matrix of the i-th microstructure. Let be the column vector of temperature parameters for the EFGM nodes of the i-th microstructure. Let be the EFGM temperature load column vector for the i-th microstructure; Let i be the volume of the i-th material in a multi-material periodic macrostructure after the update. The 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. The volume fraction specified for the i-th type of microstructure. ρ represents the total volume of the i-th type of microstructure; min To prevent matrix singularity, the relative density of macro- and micro-EFGM nodes and Gaussian points is set to a minimum, and ρ is taken as... min =10 -3 The specific steps for establishing a macro-micro integrated topology optimization mathematical model for the periodic structure based on the steady-state heat transfer problem of EFGM are as follows: (a) Solve for the thermal compliance C of the macro-structure of EFGM based on the EFGM node temperature, overall thermal load vector and its penalty term, and overall thermal stiffness matrix and its penalty term of the macro-structure output in step (7). P (b) Output EFGM thermal flexibility C P ; (9) Solve for macroscopic volume sensitivity, microscopic volume sensitivity, macroscopic thermal flexibility sensitivity, and microscopic thermal flexibility sensitivity; apply macroscopic periodic constraints: in multi-material periodic macrostructures, periodic constraints are applied by the relative density of multi-material EFGM nodes with the same EFGM node number n in each design subdomain. This is achieved by forcing the settings to be exactly the same, ensuring this is done every time the relative density of the multi-material EFGM nodes is updated. The modified relative density of the multi-material EFGM nodes can be obtained by distributing them periodically and evenly. Solve for the thermal flexibility sensitivity after averaging; (10) Update the relative density of EFGM nodes of material a under the k-th hierarchical iteration according to the OC method, and solve the relative density of EFGM nodes of material b under this iteration based on the relationship that the sum of the relative densities of EFGM nodes of materials a and b remains unchanged before and after the update, and assemble the relative density matrix of EFGM nodes of multiple materials; update the relative density of EFGM nodes of microstructures a and b under the k-th hierarchical iteration, while keeping the relative density of EFGM nodes of other types of microstructures unchanged, and assemble the relative density matrix of EFGM nodes of multiple types of microstructures; calculate the infinite norm ch of the difference of macroscopic EFGM node relative density matrix. ma The infinite norm ch of the difference between the relative density matrix of the microscopic EFGM nodes mi ; (11) Determine if the iteration termination condition is met: (a) Compare t with t max The size relationship determines whether the third-level iteration should continue or end: if t < t max If t = t + 1, return to step (5) and continue the three-level iteration; if t < t max If this condition is not met, then the current three-level iteration ends; (b) Compare s with s max The size relationship determines whether the second-level iteration should continue or end: if s < s max And if b < q, then let s = s + 1, b = b + 1, and return to step (4); if s < s max However, if b < q is not true, then let s = s + 1, a = a + 1, b = a + 1, and return to step (4); if s < q max If this condition is not met, then the second-level iteration ends; (c) Compare r with r max , ch ma with ch min , ch mi with ch min The size relationship determines whether the first-level iteration should continue or end: if r < r max , ch ma >ch min And ch mi >ch minIf r = r + 1, then return to step (3); if r < r max Not valid, or ch ma >ch min Not valid, or ch mi >ch min If the condition is not met, the first-level iteration of this round ends, and step (12) is executed; (12) Output steady-state heat transfer problem: optimal topological configuration of multi-material periodic macrostructure and multi-type microstructure.

[0008] The beneficial effects of this invention are as follows: This invention uses the moving least squares method to construct high-order continuous shape functions, resulting in a smooth EFGM node relative density field, effectively eliminating checkerboard patterns and reducing the presence of intermediate relative densities. It yields a topology with continuous and relatively clear boundaries without relying on filters. This invention aims to maximize the heat dissipation performance of the macroscopic periodic structure while simultaneously optimizing the equivalent heat transfer performance of the periodic microstructure through homogenization methods. It also achieves parallel integrated topology optimization of macroscopic periodic structures and substrate multi-material multi-type microstructures from multiple scales (macroscopic and microscopic, material and structural) to further improve the heat dissipation performance of the structure. This invention can obtain structures with different heat transfer performances by adjusting the number of material types, the number of design subdomains, the thermal conductivity factor, and the multi-material orientation angles. It can perform steady-state heat transfer topology optimization design for anisotropic multi-material structures according to engineering requirements. The operation is simple and has significant theoretical research significance and application value. 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 multi-scale topology optimization design process of an anisotropic multi-material periodic heat transfer structure based on meshless EFGM. Figure 2 This is a schematic diagram of the global coordinate system and material coordinate system of the anisotropic structure of the present invention. Figure 3 This is a schematic diagram of the microstructure unit cell model discretized by EFGM nodes and its EFGM node classification according to an embodiment of the present invention. Figure 4 This is a schematic diagram of the boundary conditions and thermal loads of an embodiment of the present invention. Figure 5 This is a schematic diagram of the initial design domain of a microstructure unit cell according to an embodiment of the present invention. Figure 6 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1 / 4, material orientation angle θ is 0°, macroscopic multi-material volume fraction [0.3 0.3 0.4], volume fraction of microscopic multi-material The thermal conductivity of the microstructure in the 2nd direction of the material coordinate system is [0.6 0.55 0.5]. [10 6 10] -3 At that time, the macroscopic topological configuration of the multi-material periodic heat transfer structure based on the meshless EFGM Figure 7 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1 / 4, material orientation angle θ is 0°, macroscopic volume fraction 0.3, microscopic volume fraction The microstructure thermal conductivity is 0.6 in the 2-direction of the material coordinate system. When the value is 10, the topological configuration of a 3×3 array of multi-material periodic first-class microstructures based on meshless EFGM is... Figure 8 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1 / 4, material orientation angle θ is 0°, macroscopic volume fraction 0.3, microscopic volume fraction The thermal conductivity of the microstructure in the 2-axis direction of the material coordinate system is 0.55. When the value is 6, the 3×3 array topology of multi-material periodic second-type microstructures based on meshless EFGM is... Figure 9 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1 / 4, material orientation angle θ is 0°, macroscopic volume fraction 0.4, microscopic volume fraction Thermal conductivity of the microstructure in the 2-axis direction of the material coordinate system is 0.5. 10 -3 At that time, a 3×3 array topology of multi-material periodic third-type microstructures based on meshless EFGM. Figure 10 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1 / 2, material orientation angle θ is 0°, macroscopic multi-material volume fraction [0.3 0.3 0.4], volume fraction of microscopic multi-material The thermal conductivity of the microstructure in the 2nd direction of the material coordinate system is [0.6 0.55 0.5]. [10 6 10] -3 At that time, the macroscopic topological configuration of the multi-material periodic heat transfer structure based on the meshless EFGM Figure 11 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1 / 2, material orientation angle θ is 0°, macroscopic volume fraction 0.3, microscopic volume fraction The microstructure thermal conductivity is 0.6 in the 2-direction of the material coordinate system. When the value is 10, the topological configuration of a 3×3 array of multi-material periodic first-class microstructures based on meshless EFGM is... Figure 12 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1 / 2, material orientation angle θ is 0°, macroscopic volume fraction 0.3, microscopic volume fraction The thermal conductivity of the microstructure in the 2-axis direction of the material coordinate system is 0.55. When the value is 6, the 3×3 array topology of multi-material periodic second-type microstructures based on meshless EFGM is... Figure 13 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1 / 2, material orientation angle θ is 0°, macroscopic volume fraction 0.4, microscopic volume fraction Thermal conductivity of the microstructure in the 2-axis direction of the material coordinate system is 0.5. 10 -3 At that time, a 3×3 array topology of multi-material periodic third-type microstructures based on meshless EFGM. Figure 14 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1, material orientation angle θ is 0°, macroscopic multi-material volume fraction [0.3 0.3 0.4], volume fraction of microscopic multi-material The thermal conductivity of the microstructure in the 2nd direction of the material coordinate system is [0.6 0.55 0.5]. [10 6 10] -3At that time, the macroscopic topological configuration of the multi-material periodic heat transfer structure based on the meshless EFGM Figure 15 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1, material orientation angle θ is 0°, macroscopic volume fraction 0.3, microscopic volume fraction The microstructure thermal conductivity is 0.6 in the 2-direction of the material coordinate system. When the value is 10, the topological configuration of a 3×3 array of multi-material periodic first-class microstructures based on meshless EFGM is... Figure 16 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1, material orientation angle θ is 0°, macroscopic volume fraction 0.3, microscopic volume fraction The thermal conductivity of the microstructure in the 2-axis direction of the material coordinate system is 0.55. When the value is 6, the 3×3 array topology of multi-material periodic second-type microstructures based on meshless EFGM is... Figure 17 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1, material orientation angle θ is 0°, macroscopic volume fraction 0.4, microscopic volume fraction Thermal conductivity of the microstructure in the 2-axis direction of the material coordinate system is 0.5. 10 -3 At that time, a 3×3 array topology of multi-material periodic third-type microstructures based on meshless EFGM. Figure 18 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 2, material orientation angle θ is 0°, macroscopic multi-material volume fraction [0.3 0.3 0.4], volume fraction of microscopic multi-material The thermal conductivity of the microstructure in the 2nd direction of the material coordinate system is [0.6 0.55 0.5]. [10 6 10] -3 At that time, the macroscopic topological configuration of the multi-material periodic heat transfer structure based on the meshless EFGM Figure 19 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains Mx ×M y =1×4, thermal conductivity factor TCF is 2, material orientation angle θ is 0°, macroscopic volume fraction 0.3, microscopic volume fraction The microstructure thermal conductivity is 0.6 in the 2-direction of the material coordinate system. When the value is 10, the topological configuration of a 3×3 array of multi-material periodic first-class microstructures based on meshless EFGM is... Figure 20 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 2, material orientation angle θ is 0°, macroscopic volume fraction 0.3, microscopic volume fraction The thermal conductivity of the microstructure in the 2-axis direction of the material coordinate system is 0.55. When the value is 6, the 3×3 array topology of multi-material periodic second-type microstructures based on meshless EFGM is... Figure 21 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 2, material orientation angle θ is 0°, macroscopic volume fraction 0.4, microscopic volume fraction Thermal conductivity of the microstructure in the 2-axis direction of the material coordinate system is 0.5. 10 -3 At that time, a 3×3 array topology of multi-material periodic third-type microstructures based on meshless EFGM. Figure 22 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 4, material orientation angle θ is 0°, macroscopic multi-material volume fraction [0.3 0.3 0.4], volume fraction of microscopic multi-material The thermal conductivity of the microstructure in the 2nd direction of the material coordinate system is [0.6 0.55 0.5]. [10 6 10] -3 At that time, the macroscopic topological configuration of the multi-material periodic heat transfer structure based on the meshless EFGM Figure 23 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 4, material orientation angle θ is 0°, macroscopic volume fraction 0.3, microscopic volume fraction The microstructure thermal conductivity is 0.6 in the 2-direction of the material coordinate system. When the value is 10, the topological configuration of a 3×3 array of multi-material periodic first-class microstructures based on meshless EFGM is... Figure 24 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 4, material orientation angle θ is 0°, macroscopic volume fraction 0.3, microscopic volume fraction The thermal conductivity of the microstructure in the 2-axis direction of the material coordinate system is 0.55. When the value is 6, the 3×3 array topology of multi-material periodic second-type microstructures based on meshless EFGM is... Figure 25 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 4, material orientation angle θ is 0°, macroscopic volume fraction 0.4, microscopic volume fraction Thermal conductivity of the microstructure in the 2-axis direction of the material coordinate system is 0.5. 10 -3 At that time, a 3×3 array topology of multi-material periodic third-type microstructures based on meshless EFGM. Detailed Implementation

[0011] See Figure 1 and Figure 2 The multi-scale topology optimization method for anisotropic multi-material periodic steady-state heat transfer structures based on meshless EFGM mainly includes the following steps: First, determine the macroscopic and microscopic thermal parameters of the anisotropic multimaterial: the number of material types q, and the thermal conductivity of various microstructures in the two directions of the material coordinate system. Thermal conductivity factor (TCF), multi-material orientation angle θ, and macro- and micro-volume fractions of various materials. (In this invention, the subscript ma represents the macroscopic level, mi represents the microscopic level, and i = 1, 2, 3, ..., q are the indices of the material types). EFGM nodes are used to discretize the macroscopic and microscopic design domains, arranging Gaussian points within and on the boundaries of the design domains; the relative density vectors and matrices of the EFGM nodes for macroscopic multi-material and initial multi-class microstructures are set; the number of horizontal and vertical design subdomains is set. According to Fourier's law, the heat flux-temperature gradient relationship of the anisotropic multi-material heat transfer structure in the x and y directions is as follows: In the formula, Let x be the thermal conductivity matrix of the i-th material in the global coordinate system xy. Let be the thermal conductivity matrix of the i-th material in material coordinate system 1-2, and γ be the coordinate transformation matrix. middle, Let be the thermal conductivity of the i-th material in directions 1 and 2 of the material coordinate system 1-2, respectively. Define the thermal conductivity factor. Let TCF represent the anisotropic thermal conductivity. When TCF = 1 and θ = 0°, the multi-material structure exhibits isotropic thermal conductivity. By modifying TCF and θ, multi-material, multi-scale topology optimization structures with different thermal conductivity properties can be obtained.

[0012] Secondly, based on the meshless EFGM theory and the idea of ​​alternating active multiphase material topology optimization, and analogous to the SIMP single-material interpolation model, a SIMP multi-material macro-micro interpolation model is established: This invention constructs a relative density field using the relative density of multi-material EFGM nodes varying from 0 to 1 as the design variable. Under the r-th first-level iteration, the s-th second-level iteration, and the t-th third-level iteration, the hierarchical iteration is performed for a total of k = t. max s max (r-1)+t max (s-1)+t times, where t max This is the maximum limit for three-level iterations, which can be set to 1, 2, or 3 as needed. Let represent the number of two-phase subproblems to be solved in the alternating active phase algorithm. Then, the material properties and thermal conductivity λ of the Gaussian point under the k(r,s,t)th graded iteration for the i-th material are... i(0) The relationship between them is Let λ be the relative density of the Gaussian point after penaltying for the i-th material, p be the penalty intensity, and λ be the relative density of the i-th material. i(0) Let be the thermal conductivity of the i-th material in a completely solid state; and be the continuous Gaussian point relative density field. The MLS shape function, which can be approximated by the relative density of EFGM nodes within the Gaussian point support domain using the moving least squares method, is: Let φ be the relative density of the i-th EFGM node within the Gaussian point influence domain of the i-th material. I Let be the MLS shape function corresponding to the EFGM nodes, and n be the number of EFGM nodes within the Gaussian point support domain. This interpolation model is applicable at both macro and micro scales, and the parameters are calculated according to the macro and micro scales respectively.

[0013] By applying periodic boundary conditions to the microstructure, the temperature field of the steady-state heat transfer microstructure is solved: See [link to relevant documentation]. Figure 3 The microstructure unit cell is discretized into EFGM nodes, and all nodes are classified; a test temperature field T is applied to the nodes. i 0 For nodes A, B, C, and D of the EFGM, the first type of periodic boundary condition is: For the EFGM node sets I, II, III, and IV, the second type of periodic boundary conditions are: In the formula l x ,l y Given the length and width of the microstructure, and combining equations (6) and (7), the temperature field matrix T of the steady-state heat transfer microstructure is... mi It can be obtained from the following equilibrium equations. In the formula K Pij (i,j=1,2,3,4) is Figure 3 The thermal stiffness matrix of four types of EFGM nodes, and K Pij =K Pji (i,j=1,2,3,4), T2 and T3 are the temperatures of internal node V and outer boundary nodes I and IV, respectively, T1 is determined by periodic boundary conditions, W P pass Seeking, The side length of a single cell in the microstructure. T4 is a constant describing the displacement difference between a pair of parallel boundaries of a microstructure, where T4 = T3 + W. P .

[0014] Finally, temperature field analysis of anisotropic multi-material macrostructures was performed based on meshless EFGM. The optimization objective was to minimize the macroscopic thermal compliance of the multi-material periodic structure, and the relative density of EFGM nodes for the multi-material periodic macrostructure and various microstructures was used as the optimization parameters. and As design variables, the volume fractions of multi-material periodic macroscopic structures and multi-type microstructures are used. and As constraints, a macro-micro integrated topology optimization model for steady-state heat transfer of multi-material periodic structures is established based on meshless EFGM and the energy homogenization method. 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 resulting multi-material EFGM node relative density matrix, Let M be the nodal relative density of the i-th material at the n-th EFGM node within the m-th design subdomain of a multi-material periodic macrostructure. s M represents the total number of design subdomains for a multi-material periodic macrostructure. s =M x ×M y M x M y N represents the number of design subdomains in the x and y directions, respectively. s N represents the total number of EFGM nodes within a single design subdomain of a multi-material periodic macrostructure. s =N x ×N y N x N y These represent the number of EFGM nodes in the x and y directions, respectively. 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 Let be the total number of EFGM nodes of the i-th type of microstructure; For the thermal compliance objective function of the EFGM of a multi-material periodic macrostructure, P ma and P α,ma Let EFGM be the overall thermal load column vector and its penalty term. This is a global approximate temperature vector. Let T be the global EFGM node shape function matrix. ma This is a column vector of temperature parameters for the EFGM nodes; To introduce the overall thermal stiffness matrix of EFGM after the SIMP multi-material interpolation model, for The sub-thermal stiffness matrix, I,J=1,2,,TN, represents the global EFGM node indices, and TN represents the solution domain Ω. ma The total number of EFGM nodes within the system. Here, γ is the thermal analysis geometry matrix, and γ is the coordinate transformation matrix. Let be the thermal conductivity matrix of the i-th material in the multi-material coordinate system 1-2. p is the relative density field at a Gaussian point. ma The macroscopic penalty intensity is given by h, the convective heat transfer coefficient is given by φ. I,ma ,φ J,ma Here, Γ3 represents the shape function at the corresponding EFGM node, and Γ3 represents the Robin boundary. Let be the EFGM thermal stiffness matrix of the i-th microstructure. Let be the column vector of temperature parameters for the EFGM nodes of the i-th microstructure. The EFGM temperature load column vector of the i-th microstructure; Let be the Gaussian point relative density fields of the macro- and micro-structures of the i-th material, respectively. Let i be the volume of the i-th material in a multi-material periodic macrostructure after the update. The 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. The volume fraction specified for the i-th type of microstructure. ρ represents the total volume of the i-th type of microstructure; min To prevent matrix singularity, the relative density of macro- and micro-EFGM nodes and Gaussian points is set to a minimum, and ρ is taken as... min =10 -3 The macroscopic volumetric sensitivity, microscopic volumetric sensitivity, macroscopic thermal flexibility sensitivity, and microscopic thermal flexibility sensitivity are calculated and solved. A macroscopic periodicity condition is applied, and the relative density of macroscopic multi-material and multi-type microstructure EFGM nodes is updated using the OC method. Based on the iteration results and the magnitude of the infinite norm of the difference between the macroscopic and microscopic EFGM node relative density matrices, the optimal topological configurations of the macroscopic structure and multi-type microstructures are obtained.

[0015] See Figure 1 The specific steps of the multi-scale topology optimization method for multi-material periodic steady-state heat transfer structures based on meshless EFGM are as follows: (1) Based on the design requirements of the actual engineering structure, determine the initial design domain and input the macro- and micro-thermal parameters of the anisotropic multi-material system: the number of multi-material types q, and the thermal conductivity of the i-th microstructure in the 2-direction of the multi-material coordinate system. Material type index i = 1, 2, 3, ..., q, thermal conductivity factor TCF, multi-material orientation angle θ, macro- and micro-volume fraction of the i-th material. (2) The macro- and micro-scale design domains are discretized using EFGM nodes, and Gaussian points are arranged inside and on the boundaries of the design domains; the relative density vectors and matrices of the EFGM nodes for macro-scale multi-materials and initial multi-class microstructures are set. 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 iterations k for the r, s, t-th iterations: k = t max s max (r-1)+t max (s-1)+t, where r, s, and t are the first, second, and third iteration steps, respectively; (6) The shape function is constructed using the Moving Least Squares (MLS) method. Assuming the unknown field function is u(x), its MLS approximation function u in the computational domain Ω composed of N EFGM nodes is then... h (x) can be constructed as In the formula, p T (x)=[p1(x),p2(x),…,p m [x] is a basis function p of spatial coordinates. j The vector formed by (x), where m is the number of basis functions, a(x) = [a1(x), a2(x), ..., a... m (x)] T The unknown coefficient a j The vector formed by (x); a(x) can be obtained by minimizing the weighted L2 norm J(x). In the formula, the support domain Ω of any calculation point (Gaussian point) x x It covers n EFGM nodes x I (I = 1, 2, n), u I =u(x I ) is x I The EFGM node parameter value at that location, w(xx) I ) = w I (x) is x I For the compactly supported weight function at a given point, minimizing the normative function J(x) requires ensuring that Solving this equation yields A(x)a(x)=B(x)u (12) 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 A(x), B(x), and u into equation (12) yields a(x)=A -1 (x)B(x)u (13) Substituting equation (13) back into equation (10), we get In the formula, the support domain Ω x The MLS shape function Φ(x) of the n inner EFGM nodes can be expressed as: Φ(x)=[φ1(x),φ2(x),...,φ n [x]=p T (x)A -1 (x)B(x); For two-dimensional problems, this invention uses a linear basis p. T (x)=[1,x,y],m=3, using cubic spline weight function In the formula, for circular and rectangular support domains Ω x w I (x) can be represented as w(||xx) I || / d I ) and w(|xx I | / d Ix )w(|yy I | / d Iy ); where, ||xx I || / d I 、|xx I | / d Ix 、|yy I | / d Iy Ω x Regularized feature distance, ||xx I ||Calculate the distance from point x(x,y) to EFGM node x I (x I ,y I The distance d I Ω x radius, |xx I |、|yy I |for x and x I The distance d in the x and y directions Ix ,d Iy Ω x Half the length and width; The macroscopic equivalent thermal conductivity matrix (EFGM) of the i-th material is solved based on the meshless EFGM and energy homogenization method. The expression to be solved is as follows In the formula, Let be the equivalent thermal conductivity matrix of the EFGM for the i-th type of microstructure. Let n be the relative density of the i-th EFGM node of the i-th material within the Gaussian point influence domain on a periodic macrostructure. ma φ represents the number of EFGM nodes within the influence domain of the macroscopic Gaussian point. ma,I Let be the MLS-shaped function of the i-th EFGM node within the influence domain of the macroscopic Gaussian point. It can be obtained from the following formula. T mi For the temperature field of the EFGM node in the microstructure unit cell, |Ω mi | represents the area of ​​a single cell of the microstructure. For the geometric matrix of microstructure thermal analysis, The macroscopic EFGM overall thermal stiffness matrix after introducing the SIMP multi-material interpolation model The transformation form on a single-material microstructure unit cell, φ I,mi Let φ be the MLS shape function of the i-th EFGM node within the influence domain of the microscopic Gaussian point. I,mi,x ,φ I,mi,y Let x be the derivative of the shape function at the node of the microstructure with respect to x and y. (7) Calculate the macroscopic equivalent thermal conductivity matrix of the i-th material based on the meshless EFGM and energy homogenization method. The detailed steps are as follows: (7.1) Discretize and classify the nodes of the microstructure unit cell model using EFGM; (7.2) Apply heat transfer periodic boundary conditions according to the EFGM node classification, and derive the discrete heat transfer control equation of the microscopic EFGM through equation (8) to obtain the temperature T of the microstructure unit cell EFGM node. mi ; (7.3) Solve for the equivalent thermal conductivity matrix of the corresponding positional performance parameter EFGM using equation (18). (7.4) The EFGM thermal conductivity matrix of the i-th material in the k(r,s,t)th grade iteration is obtained by equation (16). (7.5) Solve for and output the macroscopic EFGM equivalent thermal conductivity matrix for all materials; Combining the SIMP multi-material interpolation model, the discrete heat transfer control equations of the EFGM for steady-state anisotropic multi-material macrostructures are rearranged as follows: In the formula, To introduce the overall thermal stiffness matrix of EFGM after the SIMP multi-material interpolation model, K Pα,ma T is the penalty term for the overall thermal stiffness matrix of EFGM. ma P is a column vector of parameter temperatures. ma and P α,ma Let K be the overall thermal load column vector of EFGM and its penalty term. P,ma K Pα,ma P ma P α,ma The submatrices are K PIJ,ma K PαIJ,ma P I,ma P αI,ma The integral expressions are shown in equations (22) to (25): In the formula, and K PαIJ,ma Let x be the thermal stiffness matrix of the EFGM node and its penalty term. I and x J Since they are not in the same influence domain, their integrals are 0; P I,ma and P αI,ma Let EFGM node thermal load vector and its penalty term be defined. For thermal analysis geometry matrix, φ I,ma,x ,φ I,ma,y The shape function at the EFGM node is the derivative with respect to x and y. α is called the penalty factor, typically taken as 10e5 to 10e7. h is the convective heat transfer coefficient. T ∞ The ambient temperature, Let q be the heat generation rate per unit area within the design domain, and q be the heat flux density on the boundary Γ2. The temperature is constant on boundary Γ1; (8) The detailed steps for solving the macroscopic temperature field based on the meshless EFGM are as follows: (8.1) Determine the overall EFGM thermal load vector of the design domain based on the magnitude of the thermal load applied to the design domain; (2.2) Based on the given thermal boundary conditions, various thermal boundary conditions are processed. The penalty function method is used to process the first type of heat transfer boundary, and the penalty term of the EFGM overall thermal stiffness matrix and the penalty term of the EFGM overall thermal load vector are obtained. (8.3) Substitute the equivalent thermal conductivity matrix of the multi-material macro EFGM output in step (7.5) into equation (22) to obtain the thermal stiffness matrix of each EFGM node. Then assemble the unit thermal stiffness matrix into the overall thermal stiffness matrix of EFGM. Solve the temperature parameter values ​​of the meshless EFGM node in each design domain according to the discrete heat transfer control equation of equation (21). (8.4) Solve for the temperature value of each unmesh node based on the temperature parameter values ​​of the unmesh EFGM node in the design domain; (8.5) Output the nodal temperature values ​​of the EFGM in the design domain and the overall thermal load vector of the EFGM; (9) Establish a macro-micro integrated topology optimization model for steady-state heat transfer multi-material periodic structures based on meshless EFGM and energy homogenization method, and solve the macro-structure thermal flexibility based on the meshless EFGM node temperature values ​​and the overall thermal load vector of EFGM output in step (8.5). (10) Solving for macroscopic volume sensitivity Microscopic volume sensitivity Macroscopic thermal flexibility sensitivity and microscopic thermal flexibility sensitivity Applying periodic constraints: Solving for the averaged relative density of nodes in a multi-material EFGM and thermal flexibility sensitivity The calculation formulas are shown below (26)-(35): In the above formula, In the above formula, (11) The OC method is used to update the relative density vector of macroscopic multi-material and multi-type microstructure EFGM nodes. The detailed steps are as follows: (11.1) Macrostructure: The relative density of the EFGM nodes of the a-th material in the k-th hierarchical iteration is updated according to the OC method. 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: Assembly of multi-material EFGM node relative density matrix Calculate the macroscopic infinite norm of the difference between the relative density matrices of macroscopic EFGM nodes. (11.2) Microstructure: Since the relative densities of the EFGM nodes of materials a and b in the macroscopic multi-material system are updated, the relative densities of the EFGM nodes of the microstructures a and b under k hierarchical iterations are updated according to the OC method as follows: The relative density of EFGM nodes remains unchanged for other types of microstructures, and the relative density matrix of EFGM nodes for assembling multiple types of microstructures is shown. Calculate the infinite norm of the difference between the relative density matrices of microscopic EFGM nodes. (12) Determine whether the iteration termination condition is met: (12.1) Compare t with t max The size relationship determines whether the third-level iteration should continue or end: if t < t max If t = t + 1, return to step (5) and continue the three-level iteration; if t < t max If this condition is not met, then the current three-level iteration ends. (12.2) Compare s with s max The size relationship determines whether the second-level iteration should continue or end: if s < s max And if b < q, then let s = s + 1, b = b + 1, and return to step (4); if s < s max However, if b < q is not true, then let s = s + 1, a = a + 1, b = a + 1, and return to step (4); if s < q max If this condition is not met, then the second-level iteration of this round ends; (12.3) Compare r with r max , ch ma with ch min , ch mi with ch min The size relationship determines whether the first-level iteration should continue or end: if r < r max , ch ma >ch min And ch mi >ch min If r = r + 1, then return to step (3); if r < r max Not valid, or ch ma >ch min Not valid, or ch mi >ch min If the condition is not met, the first-level iteration ends, and the optimal topology is output.

[0016] The following is an example of the application of the method of the present invention in engineering practice: See Figure 4In this embodiment, the structure is square, and a line region with a length of L / 6 in the middle of the lower boundary serves as a fixed temperature boundary. The remaining four sides form an adiabatic boundary, and a uniformly distributed thermal load is applied within the macroscopic design domain. The macroscopic design domain is set to a side length of L = 1.08 m, and is discretized using 108 × 108 EFGM nodes. For the microscopic design domain, see [link to microscopic design domain]. Figure 5 The micro-design domain has a side length of l = 0.01 m. It is discretized using 30 × 30 EFGM nodes, with an initial design aperture radius of l / 6. The number of design subdomains is set to M. x ×M y =1×4, number of material types q=3, multi-material orientation angle θ=0°, macroscopic penalty intensity p ma =3, micro-level penalty intensity p mi =5. To verify the effectiveness of the proposed method, the influence of the thermal conductivity factor on the topological configuration and thermal compliance objective function of the optimal periodic macrostructure and microstructure is discussed. The corresponding macro- and micro-volume fractions and the thermal conductivity of the microstructure in the 2-direction of the material coordinate system are as follows: Take values ​​of 0.3, 0.3, and 0.4. Take values ​​of 0.6, 0.55, and 0.5. (10W / (m·K)) is taken as 10, 6, 10 -3 The thermal conductivity factor (TCF) is taken as 1 / 4, 1 / 2, 1, 2, and 4.

[0017] The specific implementation steps of this invention for this example are as follows: (a) Determine the initial design domain and input the macro- and micro-thermal parameters of the anisotropic multi-material system: the number of multi-material types q, and the thermal conductivity of various microstructures in the two directions of the material coordinate system. Material type index i = 1, 2, 3, ..., q, thermal conductivity factor TCF, multi-material orientation angle θ, macro- and micro-volume fractions of each material. (b) Discretize the macro- and micro-scale design domains using EFGM nodes, and arrange Gaussian points inside and on the boundaries of the design domains; set the relative density vectors and matrices of EFGM nodes for macro-scale multi-materials and initial multi-class microstructures. Set the number of horizontal and vertical design subdomains: M x M y =1×1; Set iteration parameters: minimum tolerance ch for relative density change of multi-material EFGM nodes min =10 -5 Maximum number of steps in 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 in the r, s, t-th hierarchical iterations: k = t max s max (r-1)+t max (s-1)+t; (f) Discretize and classify the EFGM nodes of the microstructure unit cell model; (g) Apply heat transfer periodic boundary conditions according to the EFGM node classification, and derive the discrete heat transfer control equation of the microscopic EFGM using equation (8) to obtain the temperature T of the microstructure unit cell EFGM node. mi ; (h) Solve for the equivalent thermal conductivity matrix EFGM at the corresponding location of the microstructure using equation (18). (i) Obtain the EFGM thermal conductivity matrix of the i-th material in the k(r,s,t)th grade iteration in the macroscopic structure using Equation (16). (j) Solve for and output the macroscopic EFGM equivalent thermal conductivity matrix for all materials; (k) Determine the overall EFGM thermal load vector of the design domain based on the magnitude of the thermal load applied to the macroscopic design domain; (l) Based on the given thermal boundary conditions, various thermal boundary conditions are processed. Among them, the penalty function method is used to process the first type of heat transfer boundary, and the penalty term of the EFGM overall thermal stiffness matrix and the penalty term of the EFGM overall thermal load vector are obtained. (m) Substitute the multi-material macroscopic equivalent thermal conductivity matrix output in step (j) into equation (22) to obtain the thermal stiffness matrix of each EFGM node. Then assemble the thermal stiffness matrix of the EFGM unit into the overall thermal stiffness matrix of the EFGM. Solve the temperature parameter values ​​of the meshless EFGM node in each design domain according to the discrete heat transfer control equation of equation (21). (n) Solve for the temperature values ​​of each meshless EFGM node based on the temperature parameter values ​​and corresponding shape functions of the meshless EFGM node in the design domain; (o) Output the meshless EFGM node temperature values ​​and the overall EFGM thermal load vector in the design domain; (p) Establish a macro-micro integrated topology optimization model for steady-state heat transfer multi-material periodic structures based on meshless EFGM and energy homogenization method, and solve the macro-structure thermal flexibility of EFGM based on the nodal temperature values ​​of meshless EFGM and the overall thermal load vector of EFGM output in step (o). (q) Solve for macroscopic volume sensitivity, microscopic volume sensitivity, macroscopic thermal flexibility sensitivity, and microscopic thermal flexibility sensitivity; (r) Apply periodic constraints: Solve for the relative density and thermal compliance sensitivity of multi-material EFGM nodes after averaging. (s) The relative density vector of the macroscopic multi-material EFGM nodes is updated using the OC method, with a positive movement constraint of m0 = 0.2 and a damping coefficient of τ = 0.5. The infinite norm ch of the difference between the macroscopic EFGM node relative density matrices is calculated. ma ; (t) The relative density vectors of EFGM nodes of various microstructures are updated using the OC method (the values ​​of m0 and τ are the same as those of the macrostructure), and the infinite norm ch of the difference between the microstructure EFGM relative density matrices is calculated. mi ; (u) Determine whether the iteration termination condition is met. If not, repeat the iteration calculation until the condition is met and the result is output. (v) Output steady-state heat transfer problem: optimal topological configuration of multi-material periodic macrostructure and multi-type microstructure.

[0018] Figures 6-25 This embodiment presents the macroscopic topological configuration of a multi-material periodic heat transfer structure based on a meshless EFGM and its corresponding 3×3 array topology of various microstructure unit cells. Figures 6-9 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1 / 4, material orientation angle θ is 0°, macroscopic multi-material volume fraction [0.3 0.3 0.4], volume fraction of microscopic multi-material The thermal conductivity of the microstructure in the 2nd direction of the material coordinate system is [0.6 0.55 0.5]. [10 6 10] -3 At that time, the macroscopic topological configuration of the multi-material periodic heat transfer structure based on the meshless EFGM and its corresponding 3×3 array topology of three microstructure unit cells were presented. Figures 10-13 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1 / 2, material orientation angle θ is 0°, macroscopic multi-material volume fraction [0.3 0.3 0.4], volume fraction of microscopic multi-material The thermal conductivity of the microstructure in the 2nd direction of the material coordinate system is [0.6 0.55 0.5]. [10 6 10] -3At that time, the macroscopic topological configuration of the multi-material periodic heat transfer structure based on the meshless EFGM and its corresponding 3×3 array topology of three microstructure unit cells were presented. Figures 14-17 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 1, material orientation angle θ is 0°, macroscopic multi-material volume fraction [0.3 0.3 0.4], volume fraction of microscopic multi-material The thermal conductivity of the microstructure in the 2nd direction of the material coordinate system is [0.6 0.55 0.5]. [10 6 10] -3 At that time, the macroscopic topological configuration of the multi-material periodic heat transfer structure based on the meshless EFGM and its corresponding 3×3 array topology of three microstructure unit cells were presented. Figures 18-21 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 2, material orientation angle θ is 0°, macroscopic multi-material volume fraction [0.3 0.3 0.4], volume fraction of microscopic multi-material The thermal conductivity of the microstructure in the 2nd direction of the material coordinate system is [0.6 0.55 0.5]. [10 6 10] -3 At that time, the macroscopic topological configuration of the multi-material periodic heat transfer structure based on the meshless EFGM and its corresponding 3×3 array topology of three microstructure unit cells were presented. Figures 22-25 In this embodiment of the invention, the number of material types q is 3 and the number of design subdomains M x ×M y =1×4, thermal conductivity factor TCF is 4, material orientation angle θ is 0°, macroscopic multi-material volume fraction [0.3 0.3 0.4], volume fraction of microscopic multi-material The thermal conductivity of the microstructure in the 2nd direction of the material coordinate system is [0.6 0.55 0.5]. [10 6 10] -3 [At that time, the macroscopic topological configuration of the multi-material periodic heat transfer structure based on the meshless EFGM and its corresponding three microstructure unit cell 3×3 array topology diagrams.]

[0019] from Figures 6-25It is evident that the optimal macro- and micro-scale topologies obtained by the meshless EFGM-based anisotropic multi-material periodic heat transfer structure topology optimization method have clear and smooth boundaries, without numerical instabilities such as intermediate density and checkerboard patterns, facilitating subsequent performance analysis and manufacturing. The topological configurations of various microstructures and their positional distribution within the periodic macrostructure undergo adaptive dynamic evolution with the optimization of multi-material distribution at the macroscale. The increase in the thermal conductivity factor (TCF) causes the substrate materials of various microstructures to gradually arrange themselves from the longitudinal direction to the lateral direction to maximize the equivalent thermal conductivity of the microstructures. The periodic macrostructure will generate more refined lateral branches to enhance the structure's heat dissipation performance. The TCF value ranges from TCF = 2 to 4. In the steady-state heat transfer multi-material periodic macro- and micro-scale integrated topology optimization, the thermal conductivity... The decreasing number of materials corresponds to multiple types of microstructures with varying degrees of fineness, and the minimum thermal flexibility C decreases as the number of material types q increases. P,min As the number of microstructures increases, their topological configurations undergo adaptive and differentiated evolution, and q can be selected according to actual engineering requirements; when the number of transverse design subdomains M is fixed... x Increasing the number of vertical design subdomains M y At this time, the multiple materials in the macrostructure are periodically distributed within a flatter, allowable design space, resulting in a minimum thermal flexibility C. P,min The size increases, and the morphology of various microstructures undergoes adaptive changes; M x M y The selection can be made according to actual engineering requirements; the increase of the multi-material orientation angle θ causes the substrate material of various microstructures to rotate along the direction with stronger thermal conductivity, resulting in the generation of dendritic structures with different inclination angles within the periodic macrostructure, and the minimum thermal flexibility C P,min The variation trend is related to the location of thermal load and constraints, with θ ranging from 45° to 90°. Compared to traditional single-material, single-scale topology optimization, this method can fully utilize the performance gradients of multiple substrate materials at both macro and micro scales, achieving complementary advantages between strong and weak materials while seeking diverse microstructure combinations. This demonstrates that integrating meshless EFGM into the design of anisotropic multimaterial macro-micro integrated structures without using sensitivity filtering techniques has significant research and application value.

[0020] 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 based on meshless EFGM multi-material periodic heat transfer structures, characterized in that... Includes the following steps: (1) Based on the design requirements of the actual engineering structure, determine the initial design domain and input the macro- and micro-thermal parameters of the anisotropic multi-material system: the number of material types q, the index of the material type i = 1, 2, 3, ..., q, and the thermal conductivity of various microstructures in the two directions of the material coordinate system. Thermal conductivity factor (TCF), number of design subdomains (M) x ×M y Multi-material orientation angle θ, macro- and micro-volume fractions of each material (2) Discretize the macro- and micro-scale design domains using meshless EFGM nodes, and arrange Gaussian points inside and on the boundaries of the design domains; set the relative density vectors and matrices of the meshless EFGM nodes for macro-multi-material and initial multi-class microstructures; set the iteration parameter: the minimum tolerance ch for the change in the relative density of the multi-material meshless 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 k for the r, s, t-th iterations: k = t max s max (r-1)+t max (s-1)+t, where r, s, and t are the first, second, and third iteration steps, respectively; (6) Based on the meshless EFGM theory and the idea of ​​alternating active multiphase material topology optimization, and by analogy with the SIMP single-material interpolation model, a SIMP multi-material macro-micro interpolation model is established: This invention constructs a relative density field with the relative density of multi-material nodes varying from 0 to 1 as the design variable. Under the r-th first-level iteration, the s-th second-level iteration, and the t-th third-level iteration, the hierarchical iteration is performed k times in total. Let represent the number of two-phase subproblems to be solved in the alternating active phase algorithm. Then, the material properties and thermal conductivity λ of the Gaussian point under the k(r,s,t)th graded iteration for the i-th material are... i(0) The relationship between them is Let λ be the relative density of the Gaussian point after penaltying for the i-th material, p be the penalty intensity, and λ be the relative density of the i-th material. i(0) Let be the thermal conductivity of the i-th material in a completely solid state; and be the 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. Let φ be the relative density of the i-th EFGM 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 EFGM nodes in the Gaussian point support domain. (7) Solving the macroscopic temperature field of anisotropic multimaterials based on meshless EFGM: (a) Discretizing and classifying the nodes of the microstructure unit cell model using meshless EFGM; (b) Applying heat transfer periodic boundary conditions based on the meshless EFGM node classification, deriving the microstructure unit cell node temperature from the microstructure meshless EFGM discrete heat transfer control equation; (c) Solving for the EFGM equivalent thermal conductivity matrix of the corresponding microstructure. |Ω mi | represents the area of ​​a single cell of the microstructure. The macroscopic EFGM overall thermal stiffness matrix after introducing the SIMP multi-material interpolation model Transformation forms on single-material microstructure unit cells; (d) through relationships Find the EFGM thermal conductivity matrix of the i-th material in the k(r,s,t)th hierarchical iteration in the macroscopic structure. (e) Calculate the overall thermal stiffness matrix of the steady-state heat transfer multi-material periodic macrostructure EFGM; (f) Calculate the penalty term of the overall thermal stiffness matrix of EFGM and the penalty term of the overall thermal load column vector of EFGM using the penalty function method; (g) Calculate the temperature parameter values ​​of the discrete nodes of the macrostructure EFGM; (h) Obtain the temperature values ​​of the EFGM nodes by interpolation of the shape functions of the meshless EFGM nodes in each influence domain. (i) Output the temperature values ​​of the EFGM nodes in the macrostructure, the overall thermal load vector of the EFGM and its penalty term, and the overall thermal stiffness matrix of the EFGM and its penalty term; (8) Introducing the SIMP multi-material interpolation model, a macro-micro integrated topology optimization mathematical model for multi-material steady-state heat transfer periodic structures based on meshless EFGM and energy homogenization method 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 resulting multi-material EFGM node relative density matrix, Let M be the relative density of the EFGM node of the i-th material at the n-th node within the m-th design subdomain of a multi-material periodic macrostructure, where m is the number of the macroscopic design subdomain, n is the number of the EFGM node within the design subdomain, and M is the relative density of the EFGM node of the i-th material. s M represents the total number of design subdomains for a multi-material periodic macrostructure. s =M x ×M y M x M y N represents the number of design subdomains in the x and y directions, respectively. s N represents the total number of EFGM nodes within a single design subdomain of a multi-material periodic macrostructure. s =N x ×N y N x N y These represent the number of unmesh EFGM nodes in the x and y directions, respectively. Let be the relative density vector of EFGM nodes of the i-th type of microstructure. For assembly The resulting relative density matrices of EFGM nodes for various microstructures 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; For the thermal compliance objective function of the EFGM of a multi-material periodic macrostructure, P ma and P α,ma Let EFGM be the overall thermal load column vector and its penalty term. This is a global approximate temperature vector. Let T be the global node shape function matrix of EFGM. ma This is a column vector of temperature parameters for the EFGM nodes; To introduce the overall thermal stiffness matrix of EFGM after the SIMP multi-material interpolation model, Let be the overall thermal stiffness matrix of the EFGM for the i-th microstructure. Let be the column vector of temperature parameters for the EFGM nodes of the i-th microstructure. The column vector of temperature loads at the EFGM nodes of 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. The 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. The volume fraction specified for the i-th type of microstructure. ρ represents the total volume of the i-th type of microstructure; min To prevent matrix singularity, the relative density of macro- and micro-EFGM nodes and Gaussian points is set to a minimum, and ρ is taken as... min =10 -3 The specific steps for establishing a macro-micro integrated topology optimization mathematical model for a periodic structure with steady-state heat transfer problem based on a meshless EFGM are as follows: (a) Solve for the macro-structure EFGM thermal compliance C based on the EFGM node temperature, overall thermal load vector, and overall thermal stiffness matrix of the macro-structure and their corresponding penalty terms output in step (7). P (b) Output thermal flexibility C P ; (9) Solve for macroscopic volume sensitivity, microscopic volume sensitivity, macroscopic thermal flexibility sensitivity, and microscopic thermal flexibility sensitivity; apply macroscopic periodic constraints: in multi-material periodic macrostructures, periodic constraints are applied by the relative density of multi-material EFGM nodes with the same node number n in each design subdomain. This is achieved by forcing the settings to be exactly the same, ensuring this is done every time the relative density of the multi-material EFGM nodes is updated. The modified relative density of the multi-material EFGM nodes can be obtained by distributing them periodically and evenly. Solve for the thermal flexibility sensitivity after averaging; (10) Update the relative density of EFGM nodes of material a under the k-th hierarchical iteration according to the OC method, and solve for the relative density of nodes of material b under this iteration based on the relationship that the sum of the relative densities of EFGM nodes of materials a and b remains unchanged before and after the update, and assemble the relative density matrix of EFGM nodes of multiple materials; update the relative density of EFGM nodes of microstructures a and b under the k-th hierarchical iteration, while keeping the relative density of EFGM nodes of other types of microstructures unchanged, and assemble the relative density matrix of EFGM nodes of multiple types of microstructures; calculate the infinite norm ch of the difference of macroscopic EFGM node relative density matrix. ma The infinite norm ch of the difference between the relative density matrix of the microscopic EFGM nodes mi ; (11) Determine if the iteration termination condition is met: (a) Compare t with t max The size relationship determines whether the third-level iteration should continue or end: if t < t max If t = t + 1, return to step (5) and continue the three-level iteration; if t < t max If this condition is not met, then the current three-level iteration ends; (b) Compare s with s max The size relationship determines whether the second-level iteration should continue or end: if s < s max And if b < q, then let s = s + 1, b = b + 1, and return to step (4); if s < s max However, if b < q is not true, then let s = s + 1, a = a + 1, b = a + 1, and return to step (4); if s < q max If this condition is not met, then the second-level iteration ends; (c) Compare r with r max , ch ma with ch min , ch mi with ch min The size relationship determines whether the first-level iteration should continue or end: if r < r max , ch ma >ch min And ch mi >ch min If r = r + 1, then return to step (3); if r < r max Not valid, or ch ma >ch min Not valid, or ch mi >ch min If the condition is not met, the first-level iteration of this round ends, and step (12) is executed; (12) Output steady-state heat transfer multi-material periodic macrostructure and multi-type microstructure EFGM optimal topology configuration.

2. The multi-scale topology optimization method based on a meshless EFGM multi-material periodic heat transfer structure according to claim 1, characterized in that... In step (1), the number of material types q, thermal conductivity factor TCF, and number of subdomains M can be adjusted. x ×M y By optimizing the orientation angle θ of multiple materials, materials with different heat transfer properties are obtained. Meshless EFGM parallel integrated topology optimization is achieved from multiple scales, including macroscopic and microscopic, material and structure, for macroscopic periodic structures and substrate multi-material multi-type microstructures. While seeking diverse microstructure combination methods, the advantages of strong and weak materials are complemented to further improve the heat dissipation performance of the structure.

3. The multi-scale topology optimization method based on a meshless EFGM multi-material periodic heat transfer structure according to claim 1, characterized in that... In step (8), the relative density of meshless EFGM nodes in multi-material periodic macrostructures and multi-type microstructures is used. and The design variables are used, and the overall relative density field is constructed based on the SIMP multi-material interpolation model and the moving least squares method. Without using sensitivity filtering techniques, a clear and smooth macro- and micro-scale heat transfer structure topology can be obtained.

4. The multi-scale topology optimization method based on a meshless EFGM multi-material periodic heat transfer structure according to claim 1, characterized in that... In step (9), in the multi-material periodic macrostructure, the periodic constraint is achieved by adjusting the relative density of multi-material EFGM nodes with the same node number n in each design subdomain. This is achieved by forcing the settings to be exactly the same, ensuring that the relative density of the multi-material EFGM nodes is updated every time. It is distributed cyclically and equally.

5. The multi-scale topology optimization method based on a meshless EFGM multi-material periodic heat transfer structure according to claim 1, characterized in that... The increase of thermal conductivity factor (TCF) causes the substrate materials of various microstructures to gradually arrange themselves from the longitudinal direction to the transverse direction in order to maximize the equivalent thermal conductivity of the microstructure. The periodic macrostructure will generate more fine transverse branches to enhance the heat dissipation performance of the structure. The TCF value ranges from 2 to 4.

6. The multi-scale topology optimization method based on a meshless EFGM multi-material periodic heat transfer structure according to claim 1, characterized in that... In the steady-state heat transfer multi-material periodic macro-micro integrated topology optimization, thermal conductivity The decreasing number of materials corresponds to multiple types of microstructures with varying degrees of fineness, and the minimum thermal flexibility C decreases as the number of material types q increases. P,min As the number of microstructures increases, their topological configurations undergo adaptive and differentiated evolution; when the number of lateral design subdomains M is fixed... x Increasing the number of vertical design subdomains M y At this time, the multiple materials in the macrostructure are periodically distributed within a flatter, allowable design space, resulting in a minimum thermal flexibility C. P,min The size increases, and the morphology of various microstructures undergoes adaptive changes.

7. The multi-scale topology optimization method based on a meshless EFGM multi-material periodic heat transfer structure according to claim 1, characterized in that... The increase in the orientation angle θ of multiple materials causes various types of microstructure materials to deflect towards the direction with stronger thermal conductivity, resulting in the formation of tree-like branch structures with different inclination angles within the periodic macrostructure, and the minimum thermal flexibility C P,min The trend of change is related to the specific location of the thermal load and constraint, and the value of θ ranges from 45° to 90°.

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