A double-scale connectable topology optimization method based on material field series expansion

Abstract

The application discloses a double-scale connectable topology optimization method based on material field series expansion, and belongs to the field of double-scale design of aerospace structures. In the aspect of double-scale structure representation, two material field functions are introduced to represent the distribution of microstructures on the macro level and the topology configuration of different microstructures, respectively. Different double-scale topology configurations can be obtained by adjusting the correlation length of different material fields, and the microstructures can continuously evolve in the optimization process. Compared with the traditional double-scale optimization method of pre-defined microstructure layout form, the optimization result is not dependent on the initial guess, and the double-scale topology optimization design with connectivity can be obtained. The application does not have any limitation on the objective function and the constraint function, can process complex double-scale structure design problems, and the optimization efficiency is significantly improved due to the dimension reduction of the design variable of the material field series expansion. The application is expected to become a double-scale structure design method with strong innovation and applicability in the field of aerospace, and can be applied to complex problems such as buckling and energy absorption.

Description

Technical Field

[0001] This invention belongs to the field of dual-scale design of aerospace structures and provides a dual-scale connectable topology optimization method based on material field series expansion. Background Technology

[0002] Multiscale structures with porous microstructures possess excellent properties in many aspects, such as high specific stiffness / strength, high thermal conductivity, and high buckling strength. The dual-scale parallel design problem has attracted increasing attention because it allows for the simultaneous design of the macroscopic distribution of lattice materials and their corresponding microscopic microstructures. Through dual-scale design, in addition to macroscopic structural design, material properties can also be optimized according to design objectives, potentially further expanding the design space. Obtaining lighter topological configurations through dual-scale optimization design is one of the important methods for maximizing the load-bearing capacity of small aerospace components.

[0003] Despite a long period of development, multi-scale parallel topology optimization design still suffers from two main drawbacks:

[0004] (1) The computation time for homogenization and microscale sensitivity analysis is very long, and most multiscale topology optimization work is based on the classical variable density method (SIMP). There are many design variables in the optimization process, which makes the optimization computation time unbearable and difficult to apply directly in practical engineering.

[0005] (2) Due to the scale separation assumption in the homogenization method, the connectivity between different microstructures cannot be guaranteed, which brings great difficulties to the manufacturing and performance prediction of multi-scale structures. There will be a large error between the optimized full-scale reconstructed structure and the dual-scale predicted structure performance, which may even make it difficult to use.

[0006] Therefore, it is necessary to establish a new dual-scale topology optimization framework that simultaneously considers the connectivity of microstructures and can significantly reduce the scale of design variables. This framework effectively incorporates the connectivity requirements between different microstructures into topology optimization, thereby further improving the load-bearing capacity of aerospace equipment. This has profound significance in the aerospace field and is beneficial for subsequent applications in practical engineering. Summary of the Invention

[0007] This invention primarily addresses the dual-scale design problem of aerospace structures, proposing an optimization design method that considers structural characterization theories at both macro and micro scales and the connectability of microstructures. This invention enables the design of connectable microstructures at both scales within a topology optimization framework. Through material field series expansion theory and a single-field method for characterizing multiple microstructures, a single material field function is introduced to describe the microstructure configurations at multiple microscales. The entire structure is represented by the superposition of macro and micro material fields, achieving connectable microstructure design in dual-scale structures.

[0008] Current practical dual-scale design methods typically employ predetermined microstructure forms, which heavily rely on prior knowledge and a given initial configuration, thus reducing the design space for topology optimization. Therefore, dual-scale parallel optimization design has gained considerable attention from researchers. However, existing general dual-scale parallel optimization design methods not only suffer from low optimization efficiency but also face significant obstacles in practical engineering applications due to the scale separation assumption in homogenization theory, making it difficult to connect different microstructures. This invention proposes a dual-scale representation strategy based on material field series expansion. It introduces a material field at both the macroscopic and microscopic levels to represent the structural distribution, and by collecting all microstructures into a single material field for expansion, the connectivity requirements of the microstructures are implicitly incorporated into the optimization model, satisfying the connectivity between microstructures without complex constraints. Compared to dual-scale topology optimization methods that rely on constraints to guarantee microstructure connectivity, this method offers a more natural algorithm implementation and eliminates the need for constraints in the optimization model, thereby improving the efficiency of topology optimization. Furthermore, since material field series expansion can significantly reduce the dimensionality of design variables (typically by two orders of magnitude), it can significantly improve the optimization speed of dual-scale parallel topology optimization.

[0009] To achieve the above objectives, the specific technical solution of the present invention is as follows:

[0010] A two-scale connectable topology optimization method based on material field series expansion includes the following steps:

[0011] In terms of dual-scale structural characterization, two material field functions are introduced to characterize the macroscopic distribution of microstructures and the topological configurations of different microstructures, respectively. By adjusting the correlation lengths of different material fields, dual-scale topological configurations with different structures can be obtained. The microstructures can continuously evolve during the optimization process. Compared with the traditional dual-scale optimization method with predefined microstructure layouts, the optimization results of this method do not depend on the initial guess and can obtain a dual-scale topological optimization design with connectivity.

[0012] The first step is to predefine the number of macro-partitions and micro-structures.

[0013] In practical engineering applications, different components should have different main load-bearing forms. For example, in an aircraft wing, the wing spars mainly bear bending moment and shear force, the longitudinal walls mainly bear the wing torque, and the stringers mainly bear part of the axial force caused by the wing bending moment. Therefore, in the design of a dual-scale structure, different areas of the structure should be divided according to actual needs, and each area should have different microstructure forms to bear different types of loads.

[0014] This invention also employs the above scheme, predefining the partitioning and number of microstructures within the structure. For ease of description, the subscript mac indicates macroscale, and the subscript mic indicates microscale.

[0015] First, determine the design domain Ω for topology optimization based on the design object (such as wall panel, tie rod, load-bearing beam, etc.). des Then, set the number of pre-defined macroscopic partitions, and define the macroscopic design domain Ω. mac Pre-defined artificial subdomains Each partition has a different microstructure, denoted as:

[0016]

[0017] Here, i and j represent the i-th and j-th subdomains, respectively. According to the above formula, there should be no overlap between different subdomains, and the union of different subdomains should be the entire macroscopic design domain Ω of the topology optimization. mac .

[0018] Due to the scale separation assumption in homogenization theory, when the design variables of microstructures are independent, the different microstructures obtained through homogenization will have poor connectivity at the boundaries, making them unusable in practical engineering. A general approach to dual-scale parallel topology optimization is to introduce as many material fields as there are microstructures, using different material fields to describe the topological form of different microstructures. However, since the design variables of different material fields are not correlated, the connectivity between microstructures cannot be guaranteed, causing the dual-scale topology optimization method to fail.

[0019] To effectively utilize the spatial correlation function in the material field method, this invention assembles microstructures from different partitions into a microscopic material field based on their spatial connectivity, i.e.,

[0020]

[0021] in the formula Ω represents the material field region occupied by the i-th microstructure. mic This represents the unfolding region of a microscopic material field assembled from different microstructures.

[0022] Therefore, through Ω mic Define microscopic material fields on a regional scale At the micro level, a microscopic material field is used. Different regions represent different microstructures; macroscopically, through Ω... mac Define macroscopic material fields in a region Using macroscopic material field It indicates whether the microstructure exists at the corresponding location.

[0023] The second step involves expanding the material field series at both the macroscopic and microscopic scales.

[0024] Based on the foregoing, the macroscopic structure and the multiple microstructures at the microscopic level are each composed of two material fields. Characterization.

[0025] 2.1) Macroscopic material field series expansion

[0026] 1. Calculation of the material field correlation matrix

[0027] First, the material field series is expanded on a macroscopic scale, introducing a spatially correlated material field function. To describe the macroscopic design domain Ω mac The topology in, that is:

[0028]

[0029] Among them, Ω lattice This indicates a macroscopic region where microstructures exist, while Ω void This indicates an empty region where no microstructure exists.

[0030] For macroscopic material fields Macro design domain Ω mac Any two points x in the interior i and x j The correlation can be introduced using a correlation function that monotonically decreases with spatial distance, i.e.

[0031]

[0032] Where ||·|| is the 2-norm, l c Used to describe the material field in the macroscopic design domain Ω mac The degree of fluctuation.

[0033] In numerical practice, the macroscopic design domain Ω mac Many observation points x are evenly distributed i (i = 1, 2, ..., N) p According to the definition of the relevant function (4), the material field The correlation between any two points can be obtained by formula (4), and it is called the correlation matrix C, i.e.:

[0034]

[0035] 2. Material field series expansion

[0036] Use eigenvalue expansion to obtain the eigenvectors ψ of the correlation matrix C. k and eigenvalues ​​λ k Then the eigenvector ψ kIt can be used as a basis function to characterize the entire material field. By solving the eigenvalue problem of the correlation matrix C, the eigenvalue λ can be obtained. k (k = 1, 2, ..., N) p ) and the corresponding normalized eigenvector ψ k (k = 1, 2, ..., N) p Arranged in descending order, the material field can be re-represented as a series sum:

[0037]

[0038] in, Represents the segmentation of the correlation matrix; η k N represents the design variable of the macroscopic material field; p λ represents the number of observation points. k and ψ k These represent the k-th eigenvalue and eigenvector, respectively.

[0039] 3. Material field level reduction

[0040] Since very small eigenvalues ​​contribute little to the topological description of the material field, to improve the computational efficiency of the material field series expansion method, an artificial truncation error ε > 0 can be introduced to reduce the design variables in the material field series expansion method to the coefficients of the first M eigenfunctions, i.e.,

[0041]

[0042] Where ε is the truncation error. To avoid losing too much information and to achieve a significant reduction in design variables, ε is generally taken as 10. -4 .

[0043] When discarding the less influential eigenvalues ​​and their corresponding eigenvectors, the macroscopic material field (Eq.(6)) can be expressed in a reduced series expansion as follows:

[0044]

[0045] in, Λ represents the macroscopic material field design variable in the method proposed in this invention; mac =diag(λ1,λ2,…,λ) M ) represents an M×M diagonal matrix composed of eigenvalues; Φ mac ={ψ1,ψ2,…,ψ M} represents N consisting of eigenvectors p ×M matrix. By discarding several modal terms with less influence in the material field series expansion method, the design variable η mac The dimension M is much smaller than the observation point N. p Dimensions.

[0046] 2.2) Field series expansion of micromaterials

[0047] According to the description of the invention, various microstructures are placed in a micro-region Ω based on their spatial connectivity. mic Materials field Series expansion.

[0048] So in the microscopic Ω mic The above also introduces a material field function with a certain spatial correlation. To describe the micro-design domain Ω mic The topology in, that is:

[0049]

[0050] Among them, Ω solid This indicates the existence of microscopic regions within the material, while Ω void This indicates an empty region where no material exists.

[0051] Similarly, many observation points x are evenly distributed throughout the entire microscopic region. i (i = 1, 2, ..., N) p Then, the correlation between different observation points is calculated, and the micro-correlation matrix C is calculated. mic Then, the material field series expansion method is used to expand and reduce the micro-region, and the reduced expression of the micro-material field is obtained as follows:

[0052]

[0053] in, Λ represents the microscopic material field design variable in the method proposed in this invention. mic =diag(λ1,λ2,…,λ) M ) represents an M×M diagonal matrix composed of the eigenvalues ​​of the micro-correlation matrix; Φ mic ={ψ1,ψ2,…,ψ M} represents N, which consists of the eigenvectors of the micro-correlation matrix. p ×M matrix.

[0054] After obtaining the material field with microstructure, the microstructure of different macroscopic regions can be obtained from different regions of the micromaterial field. During the optimization process, a design variable η is used. mic To synchronize updates. As mentioned earlier, microscopic material fields are assembled from various microstructures based on their spatial connectivity. Therefore, due to the spatial correlation definition of the microscopic material field series expansion method, the connectivity problem between different microstructures can be naturally solved.

[0055] The third step involves calculating different elastic tensors in different regions of the microscopic material field using homogenization theory at the microscopic level, and then assembling the overall stiffness matrix using the microstructural elastic tensors from different regions at the macroscopic level for macroscopic finite element analysis.

[0056] First, the projection function that maps the material field to the structural topology, as shown in equations (1) and (9), is discontinuous, which introduces numerical problems into the optimization process. Therefore, a generalized Sigmoid projection function is introduced to suppress this numerical problem, denoted as:

[0057]

[0058] Here, β is the projection parameter that affects the steepness of the projection function. If β becomes larger, the Sigmoid projection function will have a clearer topological form. In order to improve numerical stability, the value of β is gradually increased during the optimization process. Macroeconomic Or microscopic material field

[0059] 3.1) Calculation of homogenization at the microscale

[0060] For the two-scale parallel topology optimization problem, the commonly used solid isotropic (SIMP) interpolation method (generally p=3) is adopted as the material interpolation scheme to push the material distribution to 0 / 1. Therefore, the constitutive matrix of each microscopic element can be expressed as follows:

[0061]

[0062] Where c is the lower density limit to avoid singularities in the finite element method, and is generally taken as c = 10. -9 D represents the substrate equivalent elastic tensor at the microscopic level.

[0063] Subsequently, the equivalent elastic tensor matrix of the i-th microstructure is calculated using homogenization theory. It can be written as:

[0064]

[0065] in, This represents the position of the i-th unit cell in the microscopic material field. This represents the volume of the i-th microstructure. This represents the given macroscopic prestrain.

[0066] 3.2) Macroscale Finite Element Analysis

[0067] Based on the SIMP interpolation method used, the macroscopic material stiffness matrix can be denoted as:

[0068]

[0069] Where B represents the strain constant matrix, and c is the lower density bound to avoid finite element singularities, also taken as c = 10. -9 , Let be the equivalent elastic tensor matrix of the i-th microstructure obtained by homogenization theory.

[0070] After obtaining the stiffness matrix of each element, the overall stiffness matrix is ​​assembled, and a macroscopic finite element analysis is performed to obtain the macroscopic displacement response.

[0071] The fourth step is to perform topology optimization, solving for the sensitivity of the two material field design variables separately using gradient-based methods, as detailed below:

[0072] (1) Based on the distribution forms of macroscopic and microscopic material fields, a finite element analysis model is constructed, and finite element solutions and sensitivity calculations are performed. The sensitivity of the design variables is derived according to the specific forms of the objective function and constraint function. The derivation of sensitivity adopts the chain rule. For example, compliance, which is most commonly used in topology optimization, is the objective function, and the constraints are the maximum volume fraction limit of the macroscopic and each microstructure. Its optimization model is as follows:

[0073] (a) Objective function: Minimize the overall flexibility of the two-scale structure;

[0074] (b) Constraint: The total volume of the macrostructure and each microstructure must not exceed a given proportion;

[0075] (c) Design variables: Design variables for the two material fields.

[0076] (2) Since the sensitivity can be derived from this problem, gradient-based optimization algorithms can be used to solve it, such as the moving asymptote method and the quasi-Newton method. In addition, when gradient methods are not applicable, non-gradient methods can also be used to solve the problem due to the large dimensionality reduction of material field series expansion methods, such as the KG-MFSE algorithm, the DNN-MFSE algorithm, genetic algorithms, and artificial intelligence algorithms.

[0077] (3) Check if the convergence criterion is met. If it is met, proceed to step 5; otherwise, proceed to step 3.

[0078] Step 5: Extract the final optimized topology of the dual-scale structure.

[0079] Based on the optimized material field design variables, the dual material field layout is projected to obtain the final optimized dual-scale topology. The optimized dual-scale structure STL patch file is extracted, and the model is automatically reconstructed in CAD software. This yields an optimized and connectable dual-scale structure layout that does not rely on initial guesses.

[0080] The beneficial effects of this invention are as follows:

[0081] (1) This invention overcomes the problems of excessive design variables and poor microstructure connectivity in traditional two-scale topology optimization methods, and proposes a two-scale parallel optimization design theory based on the series expansion of dual material fields. In the case of a cantilever beam under load at the lower end, the traditional two-scale method not only has a long optimization time, but also poor microstructure connectivity, resulting in a large performance prediction deviation of 18.13% for the reconstructed structure. In contrast, the proposed method has a short optimization time and, with the same amount of material, ensures good connectivity between microstructures, with a performance prediction deviation of only 0.76%.

[0082] (2) Meanwhile, the dual-scale topological characterization method proposed in this invention is a new form of topological characterization. Therefore, it does not impose any restrictions on the objective function and constraint function, and can handle complex dual-scale structural design problems. Furthermore, due to the dimensionality reduction of the design variables from the material field series expansion, the optimization efficiency is significantly improved.

[0083] (3) This invention does not rely on the specific PDE solution method used, such as finite element, discrete element, and boundary element methods, which can be combined with this method. The proposed dual-scale topology optimization model can fully explore the design space while ensuring connectivity, and its optimization performance is comparable to that of dual-scale prediction without guaranteed connectivity.

[0084] Therefore, this invention is expected to become a highly innovative and applicable dual-scale structural design method in the aerospace field, and can be extended to complex problems such as buckling and energy absorption. Attached Figure Description

[0085] Figure 1 The flowchart illustrates the implementation of a dual-scale connectable topology optimization method based on material field series expansion provided by this invention.

[0086] Figure 2 This is a schematic diagram of the dual-scale parameterization framework provided by the present invention.

[0087] Figure 3 This is a schematic diagram of the microstructure topology characterization provided by the present invention; Figure 3 (a) represents the microscopic material field; Figure 3 (b) is the projected microscopic material field; Figure 3 (c) shows the distribution and layout of microstructures in the macroscopic environment; Figure 3 (d) represents the connectivity between microstructures.

[0088] Figure 4 A schematic diagram of the load-bearing cantilever beam at the lower point and the macroscopic partitioning provided for an example of the present invention; Figure 4 (a) is the cantilever beam under load at the lower point, where L1 = 10mm and L2 = 20mm; Figure 4 (b) represents the macroscopic partition of the cantilever beam under load at the lower point.

[0089] Figure 5 A schematic diagram of the optimal material distribution provided for an example of the present invention. Figure 5 (a) shows the topology optimization results at the macro and micro levels. The left figure shows the macro structure distribution, and the right figure shows the micro structure distribution. Figure 5 (b) is a schematic diagram of the model reconstruction of the optimization results.

[0090] Figure 6 This is a comparison diagram of the strain energy of the optimized structure and the simple optimized structure that ensures connectivity in the example of this invention. Figure 6 (a) is the optimized strain energy distribution diagram of the present invention; Figure 6 (b) Strain energy distribution diagram of the optimized structure to simply ensure connectivity. Detailed Implementation

[0091] To make the problems addressed, the methods proposed, and the effects achieved by this invention clearer, the invention will be further described in detail below with reference to the technical solutions and accompanying drawings. It should also be noted that, for ease of description, only the parts relevant to this invention are shown in the accompanying drawings, not all of them. The specific implementation steps are as follows:

[0092] Example: Dual-scale parallel optimization design of cantilever beam structure under single-point load

[0093] The first step is to design the object in this example as follows: Figure 4 As shown, this is a cantilever beam structure subjected to a single-point load. First, the macroscopic structure is pre-defined as a 3×3 region, where each region is occupied by a microstructure. Then, microstructures from different locations are collected into the same microscopic material field, and material field series expansions are performed simultaneously at both the macroscopic and microscopic scales. A schematic diagram of its dual-scale parameterization is shown below. Figure 2 As shown.

[0094] The second step is to perform material field unfolding to realize the material field representation of macroscopic and microscopic structures.

[0095] (a) Macroscopic material field series expansion

[0096] The cantilever beam is discretized into several finite element elements. The position coordinates of the center of each finite element element of the macroscopic cantilever beam are assembled into a coordinate matrix of the observation points of the macroscopic material field. Based on the correlation function expression (2), the correlation between each point in the coordinate matrix of the macroscopic material field observation points is calculated, and the correlation matrix C is constructed. mac Then, eigenvalue decomposition is performed to obtain the truncated macroscopic material field function, and the expansion coefficients of the material field function are taken as the macroscopic design variable η. mac .

[0097] (b) Field series expansion of micromaterials

[0098] Based on the zoning method of macroscopic cantilever beam structures, different microstructures are placed in the microscopic Ω according to their connectivity. mic Different regions (e.g.) Figure 3 As shown), a large microscopic material field is constructed. Construct the correlation matrix C mic Then, eigenvalue decomposition of the micromaterial field is performed to obtain the design variable η of the micromaterial field. mic .

[0099] The third step is to establish a topology optimization model.

[0100] In this example, compliance, the most commonly used property in topology optimization, is considered as the objective function, with constraints including macroscopic and maximum volume fraction limits for each microstructure.

[0101] (a) Objective function: Minimize the overall flexibility of the two-scale structure;

[0102] (b) Constraint: The total volume of the macrostructure and each microstructure must not exceed 50%;

[0103] (c) Design variables: Design variables for macroscopic and microscopic material fields [η] mac ,η mic ].

[0104] The fourth step is to perform finite element analysis and sensitivity analysis to update the design variables for the dual-material field.

[0105] After expanding the two material field series, the micro-homogenization and macro-finite element equations are solved, the objective function and sensitivity analysis are calculated, and the design variables are updated.

[0106] (a) Update the microstructure configuration according to different regions of the micromaterial field, obtain the equivalent elastic tensor through homogenization calculation, and obtain the stiffness matrix of different regions through stiffness matrix integration;

[0107] (b) Based on the stiffness matrix and macroscopic material field distribution of different regions of the cantilever beam structure, assemble the overall stiffness matrix, solve the macroscopic finite element method, obtain the objective function, and perform sensitivity analysis to obtain the characteristic function sensitivity of the macroscopic material field and the microscopic material field.

[0108] (c) Update the characteristic function coefficients of the two material fields using the sensitivity information, and repeat (a)-(b) until the objective function converges or the maximum number of iterations is reached.

[0109] The fifth step is to extract the optimization results and reconstruct the cantilever beam dual-scale structure.

[0110] Through this design method, optimized configurations of both macroscopic and microstructures can be obtained after a certain number of iterative steps, such as... Figure 5 As shown in (a), based on the distribution of macroscopic and microscopic structures, the optimized cantilever beam dual-scale structure can be reconstructed, such as... Figure 5 As shown in (b), the optimization results have good connectivity and can be easily applied in actual production.

[0111] The results show that, compared with the traditional two-scale reconstruction configuration that does not consider connectivity, the dual-scale reconstruction configuration obtained by the method of this invention can increase the stiffness of the structure by 18.08% under the same volume. Furthermore, the strain energy distribution obtained by the method of this invention is more uniform and reasonable, as shown in the strain energy distribution diagram below. Figure 6 As shown in (a), to ensure a fairer comparison, a simplified connection method is adopted for this multi-scale problem. This involves introducing thin walls between each subdomain to ensure the connectivity of the microstructure, while independently optimizing the microstructure within each subdomain. The strain energy distribution diagram of the optimized structure is shown in [image missing]. Figure 6 As shown in (b), compared with the optimized structure of the proposed method, not only does the volume fraction increase, but the stiffness of the reconstructed structure also decreases by 24.26%. This shows that our method can well guarantee the accuracy of dual-scale prediction, does not depend on the initial layout, and has good optimization performance.

[0112] The essence of this invention is to establish a novel dual-scale structural topology representation method that significantly reduces design variables while ensuring the connectivity between microstructures, and simultaneously solves two thorny problems existing in dual-scale parallel optimization. This method can be applied to the design of highly complex structures and specific functional structures in aerospace to further improve the structural load-bearing capacity in the aerospace field. Modifications to the optimization models, methods, and schemes described in the foregoing embodiments (such as more partitions, more complex partition forms, different unit cell configurations, etc.), or equivalent replacements of some or all of the method features (such as using other similar topology optimization methods, changing the objective function or the specific form of constraints, etc.), do not cause the essence of the corresponding methods and schemes to deviate from the scope of the methods and schemes of the embodiments of this invention.

[0113] The above-described embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. A two-scale connectable topology optimization method based on material field series expansion, characterized in that, In terms of dual-scale structure characterization, the aforementioned topology optimization method introduces two material field functions to characterize the macroscopic distribution of microstructures and the topological configurations of different microstructures, respectively. By adjusting the correlation lengths of different material fields, dual-scale topological configurations with different structures can be obtained. The microstructures continuously evolve during the optimization process, and the optimization results do not depend on the initial guess, thus obtaining a dual-scale topology optimization design with connectivity. The first step is to predefine the number of macro-partitions and micro-structures; The second step is to expand the material field series at both the macroscopic and microscopic scales. The third step involves calculating different elastic tensors in different regions of the microscopic material field using homogenization theory at the microscopic level, and then assembling the overall stiffness matrix using the microstructural elastic tensors from different regions at the macroscopic level for macroscopic finite element analysis, as detailed below: The projection function that maps the material field to the structural topology is discontinuous. Therefore, a generalized sigmoid projection function is introduced to suppress numerical problems, denoted as: (9) in, These are projection parameters that affect the steepness of the projection function. As the size increases, the sigmoid projection function will obtain a clearer topological form, which will improve numerical stability. The value gradually increases during the optimization process; Macroeconomic Or microscopic material field ; 3.1) Calculation of microscale homogenization For the dual-scale parallel topology optimization problem, the commonly used solid isotropic SIMP interpolation method is adopted as the material interpolation scheme to push the material distribution to 0 / 1; therefore, the constitutive matrix of each microscopic element is expressed as follows: (12) in, To avoid the lower density limit of finite element singularities, we take... , This represents the substrate equivalent elastic tensor at the microscopic level. Subsequently, the homogenization theory was used to calculate the first... i The equivalent elastic tensor matrix of each microstructure It can be written as: （13） in, Indicates the first i The position of a single cell in a microscopic material field Indicates the first i The volume of each microstructure Represents a given macroscopic prestrain; 3.2) Macroscale Finite Element Analysis Based on the SIMP interpolation method used, the macroscopic material stiffness matrix can be denoted as: (14) in, Represents the strain constant matrix. To avoid the lower density limit of finite element singularities, we take... , The first one is calculated by homogenization theory. i The equivalent elastic tensor matrix of each microstructure; After obtaining the stiffness matrix of each element, the overall stiffness matrix is ​​assembled, and a macroscopic finite element analysis is performed to solve for the macroscopic displacement response. The fourth step is to perform topology optimization, solving for the sensitivity of the two material field design variables separately using gradient-based methods, as detailed below: (4.1) Based on the distribution forms of macroscopic and microscopic material fields, a finite element analysis model is constructed, and finite element solutions and sensitivity calculations are performed. The sensitivity of the design variables is derived according to the specific forms of the objective function and constraint function, and the chain rule is used for the derivation of the sensitivity. (4.2) The gradient-based optimization algorithm is used to solve the optimization problem. In addition, when the gradient method is not applicable, non-gradient methods can be used to solve the problem. (4.3) Check if the convergence criterion is met. If it is met, proceed to step 5; otherwise, proceed to step 3. Step 5: Extract the final optimized topology of the dual-scale structure. Based on the optimized material field design variables, the dual material field layout is projected to obtain the final optimized dual-scale topology. The optimized dual-scale structure STL patch file is extracted, and the model is automatically reconstructed in CAD software. This yields an optimized and connectable dual-scale structure layout that does not rely on initial guesses.

2. The dual-scale connectable topology optimization method based on material field series expansion according to claim 1, characterized in that, Includes the following steps: In dual-scale structural design, the structure is divided into different regions according to actual needs. Each region has a different microstructure and bears different types of loads. The structure is predefined with partitions and the number of microstructures, where the subscript mac represents the macroscopic level and the subscript mic represents the microscopic level. First, determine the design domain of topology optimization based on the design object. Then, set the number of pre-defined macro-level partitions to define the design domain at the macro-scale. Pre-defined subdomains: Each partition has a different microstructure, denoted as: (1) in, and They represent the first i The and the first j According to the above formula, there should be no overlap between different subdomains, and the union of different subdomains should be the entire macroscopic design domain for topology optimization. ; To effectively utilize the spatial correlation function in the material field method, microstructures from different partitions are assembled into a microscopic material field based on their spatial connectivity. (2) In the formula, Indicates the first i The material field region occupied by each microstructure This indicates the region where the microscopic material field unfolds, assembled from different microstructures. Therefore, through Define microscopic material fields on a regional scale At the micro level, a microscopic material field is used. Different regions represent different microstructures; macroscopically, through Define macroscopic material fields in a region Using macroscopic material field Characterizes whether the microstructure exists at the corresponding location; The macroscopic structure and multiple microstructures are each composed of two material fields. , Characterization; 2.1) Macroscopic material field series expansion 1. Calculation of the material field correlation matrix First, the material field series is expanded on a macroscopic scale, introducing a spatially correlated material field function. To describe the macro design domain The topology in, that is: (3) in, This indicates a macroscopic region where microstructures exist, while This indicates an empty region where no microstructure exists. For macroscopic material fields Macro design domain Any two observation points and The correlation can be introduced using a correlation function that monotonically decreases with spatial distance, i.e. (4) in, It is a 2-norm. Used to describe the material field in the macroscopic design domain The degree of fluctuation; In practice, macro-design domain Many observation points are evenly distributed ,in According to the definition of the relevant function (4), the material field The correlation between any two points can be obtained by formula (4), and it is called the correlation matrix. ,Right now: (5) 2. Material field series expansion Use eigenvalue expansion to obtain the correlation matrix. eigenvectors and eigenvalues eigenvectors It can be used as a basis function to characterize the entire material field, through the correlation matrix Solving the eigenvalue problem, the eigenvalues ,in and the corresponding normalized eigenvectors ,in Arranged in descending order, the material field can be re-represented as a series sum: (6) in, This indicates the segmentation of the correlation matrix; Design variables representing macroscopic material fields; Indicates the number of observation points. and They represent the first k 1 eigenvalue and eigenvector; 3. Material field level reduction To improve the computational efficiency of the material field series expansion method, a truncation error is introduced. The design variables in the material field series expansion method are reduced to the first... M The coefficients of the characteristic functions, i.e. (7) in, To truncate errors, avoid excessive information loss, and simultaneously achieve a significant reduction in design variables, we take... ; The macroscopic material field can then be expressed in a reduced series expansion form as follows: (8) in, Represents the design variables for the macroscopic material field; Represents a composition of eigenvalues M×M diagonal matrix; Represents a structure composed of eigenvectors Matrix; By discarding several modal terms with less influence in the material field series expansion method, design variables Dimensions M Dimensions much smaller than the observation point ; 2.2) Field series expansion of micromaterials Various microstructures are placed in a micro-region based on their spatial connectivity. Materials field Series expansion; then in the microscopic... The above also introduces material field functions with certain spatial correlation. To describe the micro-design domain The topology in, that is: (9) in, This indicates the existence of microscopic regions within the material, while This indicates an empty region where no material exists; Numerous observation points are evenly distributed throughout the entire microscopic region. ,in Then, the correlation between different observation points is calculated, and the micro-correlation matrix is ​​calculated. Then, the material field series expansion method is used to expand and reduce the micro-region, and the reduced expression of the micro-material field is obtained as follows: (10) in, Represents the design variables of the microscopic material field; This represents the eigenvalues ​​of the micro-correlation matrix. M×M diagonal matrix; This represents the eigenvectors composed of the micro-correlation matrix. matrix; After obtaining the material field with microstructure, the microstructure of different macroscopic regions can be obtained from different regions of the micromaterial field. This can be achieved through a design variable during the optimization process. This allows for synchronized updates, thereby resolving connectivity issues between different microstructures.

3. The dual-scale connectable topology optimization method based on material field series expansion according to claim 2, characterized in that, In step (4.1) of the fourth step, the optimized model is as follows: (a) Objective function: Minimize the overall flexibility of the two-scale structure; (b) Constraint: The total volume of the macrostructure and each microstructure must not exceed a given proportion; (c) Design variables: Design variables for the two material fields.

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