A Material-Structure Integrated Topology Optimization Method Based on Ordered MFSE Interpolation

Abstract

This invention provides an integrated topology optimization method for materials and structures based on ordered MFSE interpolation, belonging to the field of integrated material and structure design for high-performance launch vehicles. The optimization method mainly comprises four parts: generation of arbitrary structure regular hexahedral meshes, macroscopic ordered material field series expansion multi-material interpolation method, microscopic correlation function-driven connectivity method, and multi-scale concurrent topology optimization. This invention does not introduce additional constraints during the optimization process, enabling connectivity between microstructures while significantly reducing the number of design variables, thus greatly improving the efficiency of topology optimization. This invention is expected to be applied to the integrated topology optimization design of materials and structures for small aerospace components and can be extended to high-complexity problems widely faced in the aerospace field, such as thermo-structure coupling and impact energy absorption.

Description

Technical Field

[0001] This invention belongs to the field of integrated material and structure design for high-performance launch vehicles, and relates to an integrated topology optimization method for materials and structures based on ordered material field series expansion (MFSE) interpolation for arbitrarily complex structures. Background Technology

[0002] Multiscale structures, widely existing in nature, exhibit outstanding properties such as high thermal conductivity, high buckling strength, high specific modulus, and resistance to accidental failure, attracting widespread attention from academia and engineering. Inspired by nature, many structures with excellent performance have been proposed and applied in various fields such as impact energy absorption, medical devices, and transportation equipment. However, with the advancement of modern technology, traditional methods of nature-inspired and experience-inspired approaches are increasingly insufficient to achieve innovative configurations, placing higher demands on structural design capabilities. As a numerical optimization method, topology optimization can eliminate human guesswork and find unexpectedly excellent structural configurations. Therefore, applying topology optimization methods to multiscale design, simultaneously designing the macroscopic distribution and specific material composition of microstructures—that is, integrated material-structure design—has become an effective way to improve the multifaceted performance of small aerospace components and enhance the overall performance of modern transportation equipment.

[0003] The method of material-structure integrated design based on homogenization theory and topology optimization falls under the research scope of Multiscale Concurrent Topology Optimization (MCTO). Conventional methods introduce only one type of microstructure during the optimization process. It should be noted that introducing more types of microstructures in multiscale concurrent topology optimization design will further expand the design space. To introduce more types of microstructures, two description methods can be used: methods based on predefined regions and methods based on multiple materials. However, due to the scale separation assumption in homogenization theory, the connectivity (compatibility) between different microstructures is poor, leading to a serious deviation between multiscale predictions and full-scale evaluations of structural performance, severely hindering the application of multiscale structures in engineering practice. Meanwhile, to improve the performance of multiscale structures, compared with methods based on predefined regions, using methods based on multiple materials improves the structural performance under multiscale evaluation, but more pronounced connectivity problems arise.

[0004] To date, research on MCTO methods based on multi-material interpolation has focused on the independent exploration of multi-material interpolation schemes and the discussion of microstructure connectivity, without in-depth exploration of simultaneously considering both within the optimization model. It should be noted that multi-scale structures optimized through different multi-material interpolation schemes exhibit different characteristics. Clearly, integrating the characteristics of the interpolation scheme optimization results into the optimization model and proposing corresponding connectivity methods can significantly reduce the performance degradation caused by ensuring connectivity. Furthermore, MCTO methods based on multi-material interpolation introduce a large number of design variables, severely impacting optimization efficiency. Moreover, due to their complex numerical implementation and low computational efficiency, existing MCTO methods are mostly limited to regular structure designs, resulting in limited appeal and application prospects in practical engineering.

[0005] Therefore, it is necessary to propose a topological characterization method that can effectively combine the characteristics of multi-material interpolation schemes to realize the connectivity of microstructures, and to propose a multi-scale concurrent topology optimization framework for arbitrarily complex design domains. The proposal of this framework will greatly improve the application prospects of multi-scale concurrent topology optimization methods in practical engineering, and under the guidance of lightweight design and multi-functional design of transportation equipment, it will help promote its subsequent development and application. Summary of the Invention

[0006] This invention primarily addresses the dual-scale structural design problem of small components in transportation equipment, proposing a material-structure integrated topology optimization method based on ordered interpolation material field series expansion. This method generates hexahedral meshes by voxelizing arbitrary geometric files, simultaneously achieving microstructure connectivity and topology dimensionality reduction. Through a single-field-table multiphase material field series expansion method and a connectivity-driven method based on correlation functions, a single material field function is introduced to describe the macroscopic distribution of multiple microstructures. A novel microstructure representation strategy is then defined to ensure the connectivity of different microstructures, enabling the design of connectable microstructures in dual-scale structures. This invention does not introduce additional constraints during the optimization process, naturally achieving connectivity between microstructures. Simultaneously, the number of design variables is significantly reduced, greatly improving the efficiency of topology optimization, with the majority of time spent on finite element analysis. This invention enables integrated material-structure design of arbitrary structures.

[0007] To achieve the above objectives, the specific technical solution of the present invention is as follows:

[0008] An integrated topology optimization method for materials and structures based on ordered MFSE interpolation is proposed. This method mainly comprises four parts: generation of arbitrary structure regular hexahedral meshes, macroscopic ordered material field series expansion multi-material interpolation method, microscopic correlation function-driven connectivity method, and multi-scale concurrent topology optimization. For ease of description, the superscripts "Mac" and "Mic" are used to represent the macroscopic and microscopic scales, respectively. The specific steps are as follows:

[0009] The first step is to generate a regular hexahedral mesh using voxelization for any structure.

[0010] 1.1) Obtain the spatial background mesh size of the structural design domain:

[0011] Generally, geometric information for any complex design domain can be provided in the form of a generic geometric model file, such as IGES or OBJ, to obtain the geometric model. Existing commercial CAD software is then used to convert the geometric model into an STL patch format file containing multiple triangular patch information, such as SOLIDWORKS, CATIA, or UG. For each triangular patch in the STL format file, the center point of each triangular patch is calculated using the following formula:

[0012]

[0013] in, This represents the coordinates of the center point of the i-th triangle. This represents the coordinates of the j-th vertex belonging to the i-th triangle.

[0014] Subsequently, based on the center point information of all the triangular pieces, the geometric dimensions of the complex design domain are calculated using the following formula:

[0015]

[0016] Among them, L min L represents the minimum coordinate value in the x-direction. max W represents the maximum coordinate value in the x-direction. min W represents the minimum coordinate value in the y-direction. max H represents the maximum coordinate value in the y-direction. min H represents the minimum coordinate value in the z-direction. max This represents the maximum coordinate value in the z-direction; Represents the minimum x-coordinate value of the center point of all triangular patches; This represents the maximum x-coordinate value of the center point of all triangular patches; This represents the minimum y-coordinate value of the center point of all triangular facets; This represents the maximum y-coordinate value of the center point of all triangular facets; This represents the minimum z-coordinate value of the center point of all triangular facets; This represents the maximum z-coordinate value of the center point of all triangular facets.

[0017] Therefore, the spatial background size of the structural design domain is obtained as Ω=[L min ,L max ]∪[W min Wmax ]∪[H min H max ].

[0018] 1.2) Generate a regular hexahedral mesh.

[0019] Subsequently, the element size of the hexahedral mesh is specified, and the spatial background size is divided according to its element size. The complex design domain is projected into a regularly divided spatial background hexahedral mesh. Then, it is checked whether the elements of each triangle and the divided hexahedral mesh overlap, constructing the boundary information of the design domain: if they overlap, the overlapping elements are marked as 1, indicating that the element will be part of the finite element analysis mesh; if they do not overlap, they are marked as 0, indicating that the element will be removed in the finite element analysis. After constructing the boundary information of the design domain, a flood filling algorithm is used to fill the interior of the design domain using the boundary information, and all internal elements are marked as 1, indicating that these elements will be part of the finite element analysis mesh. Subsequently, the voxels marked as 0 are deleted. This generates a regular hexahedral mesh for any structure, allowing subsequent multi-scale finite element calculations to be performed using homogenization theory.

[0020] The second step involves a macroscopically ordered material field series expansion (ordered-MFSE) multi-material interpolation method.

[0021] 2.1) After constructing the regular hexahedral mesh of the structural design domain, calculate the center point of each element and set it as the observation point of the material field. Use a correlation function to calculate the spatial correlation between several observation points. The correlation function is defined as:

[0022]

[0023] Where, x i =(x i ,y i ,z i ), x j =(x j ,y j ,z j ) represent the spatial locations of the two observation points i and j, respectively; l cx ,l cy , and l cz These are the correlation lengths along the three coordinate axes. Generally speaking, the correlation lengths in each direction are determined by the size of the sub-partition and can be defined independently, hence they are also called anisotropic correlation lengths.

[0024] Subsequently, the correlation between any two observation points in the material field is calculated using formula (3), and a correlation matrix is ​​constructed, i.e. in Where x iThis represents the coordinates of the selected i-th observation point. This represents each observation point, and Np represents the number of observation points.

[0025] 2.2) After calculating the correlation matrix, since the material field can be considered as a random field, the optimal linear estimation theory of random fields is used to perform eigenvalue decomposition on the correlation matrix and truncate the eigenmode terms with smaller eigenvalues. The material field is then expressed in a reduced basic mode superposition form, i.e.:

[0026]

[0027] Where, η k These are the modal coefficients corresponding to the k-th eigenvalues, set as design variables in the topology optimization process. Represents the material field function; l k (x) represents orthogonal characteristic basis functions, which can be represented by eigenvalues ​​λ. k and normalized mode ψ k The expression is as follows: M represents the design variable of the i-th material field; λ represents the design variable of the i-th material field. k ψ represents the eigenvalues ​​arranged in descending order of the k-th order; k Λ represents the modal matrix arranged in descending order of order k; η represents the material field design variables in the topology optimization process; Λ = diag(λ1, λ2, ..., λ M ) represents a diagonal matrix composed of eigenvalues; Φ = {ψ1,ψ2,…,ψ M} represents the basic mode matrix obtained after eigenvalue expansion; through mode expansion and truncation operations, the number of design variables will be greatly reduced, which will significantly improve optimization efficiency; at the same time, this expansion process is only performed once before optimization.

[0028] 2.3) Subsequently, assuming there are m types of microstructures at the microscopic level, in order to describe the macroscopic distribution of these microstructures, the macroscopic material field described by formula (4) is defined in the interval [-1, 1], and it is divided into m continuous transition regions according to the number of microstructures m, that is:

[0029]

[0030] Among them, Ω j (j=1,...,m) represents the transition region from the (j-1)th microstructure to the jth microstructure (where the 0th microstructure represents a hole); m represents the number of different microstructures manually specified before optimization; represents the material field located in [-1,1] introduced on a macroscopic scale; x represents a point located within the design domain.

[0031] Subsequently, the macroscopic material field located in the j-th transformation region is linearly scaled to [-1, 1], using the following scaling method: if in, This represents a macroscopic material field that has been linearly scaled.

[0032] After linear scaling, the macroscopic material field is projected using the Extended Piecewise Heaviside Projection function (EP-HP) to obtain the projected material field. The projection expression used is as follows:

[0033]

[0034] Where, β Mac It is a macroscopic projection parameter that affects the overall steepness of the projection. In the optimization process, a small initial value is usually used, and the value is gradually increased to the maximum value using an extension strategy.

[0035] Subsequently, all projected material fields are further scaled to the [0,1] interval to facilitate the use of an ordered interpolation method. This linear scaling method is as follows:

[0036]

[0037] in, Represents the material field obtained by linear scaling; a j =(j-1) / m, b j =j / m represent the upper and lower limits of the j-th conversion region, respectively; This represents the projected material field obtained through formula (6).

[0038] Subsequently, the stiffness matrix based on the ordered-MFSE multi-material interpolation scheme can be interpolated in the following form:

[0039]

[0040] in, Represents the elasticity tensor of the i-th macroscopic unit; represents the material field obtained by linear scaling of equation (7); p is the penalty coefficient for penalizing the contribution of intermediate stiffness, which is usually set to p = 3; and These are the elastic tensors of the j-th and (j+1)-th microstructures.

[0041] By using the multi-material interpolation method based on the series expansion of macroscopic ordered material fields, a clear macroscopic distribution of multiple microstructures can be obtained, and the scale of design variables can be greatly reduced.

[0042] The third step is the connectivity method driven by micro-correlation functions.

[0043] It is worth noting that, due to the spatial correlation of the material field, the optimization results obtained in the second step based on the ordered-MFSE multi-material interpolation method exhibit a gradual transition of materials. This means that the connectivity between microstructures can be guaranteed one by one, resulting in a well-connected full-scale structure. Based on the above findings, the specific steps of the correlation function-driven connectivity method are described below:

[0044] 3.1) First, specify the thickness of the connection region and divide all microstructures into two regions: the connection region and the internal region. Then, further divide the connection region according to the spatial dimension of the microstructure: the connection region of a two-dimensional microstructure is divided into four parts, and the connection region of a three-dimensional microstructure is divided into six parts.

[0045] 3.2) Since the processing procedures for two-dimensional and three-dimensional microstructures are similar, we will take a two-dimensional microstructure as an example. As described in 3.1), the two-dimensional microstructure is divided into five parts, one of which is the internal region Ω. intl The remaining four parts are four independent connection areas. Assuming A and B are neighboring cells in a sequence interpolation scheme, to ensure their connectivity, the connecting regions between the microstructures are assembled. in, This represents the second connection region of unit cell A; This represents the first connection region of the B-cell, where a connection material field is introduced. The material field is named the connecting material field of the connecting region; and the interior of the microstructure is also characterized by a material field, named the internal material field.

[0046] 3.3) After obtaining the connecting material field of the connecting region through step 3.2), according to the assembly relationship of the connecting region in step 3.2), the connecting material field is divided into two parts and distributed into two adjacent microstructures. The microstructure configuration can be obtained in the following form.

[0047]

[0048] in, This represents the material field in the intermediate region of the i-th microstructure. Let the material field of the j-th region representing the i-th microstructure be composed of segments connecting the material fields, i.e., Meanwhile, the notation H(·) in the above equation denotes the microscopic Heaviside projection function, which is given in sigmoid form as follows:

[0049]

[0050] Where, β Mic These are microscopic projection parameters that control the overall steepness of the microscopic projection. After obtaining the configurations of different microstructures, periodic boundary conditions are applied, and homogenization calculations are performed to obtain the equivalent elastic tensors of different microstructures. Then, using formula (8), the equivalent elastic tensor of each macroscopic unit can be calculated.

[0051] The micro-correlation function-driven connectivity method proposed in the third step of this invention can realize flexible connection forms between adjacent microstructures without introducing connectivity constraints of microstructures into the optimization model, and can obtain multi-scale structures with excellent performance.

[0052] The fourth step is multi-scale concurrent topology optimization of arbitrarily complex spatial structures.

[0053] 4.1) After converting the arbitrary geometric model into a regular hexahedral mesh in the first step, the objective function and constraint form are determined according to the characteristics of the optimization problem and the actual engineering requirements. For example, the most commonly used flexibility minimization in topology optimization is adopted as the objective function, and the upper limit of the total integral of the structure is used as the constraint function to establish a multi-scale concurrent topology optimization model of the structure.

[0054] 4.2) Based on the specific forms of the given objective function and constraint function, the sensitivity of the design variables is derived using the chain rule.

[0055] 4.3) The correlation function-driven connectivity strategy proposed in step 3 is used to expand the microscopic material field. Based on the given initial solution, the design variables (characteristic function coefficients) of multiple material fields (including macroscopic and microscopic) are initialized.

[0056] 4.4) Based on the internal material field and the connecting material field, calculate the configuration of multiple microstructures through step 3.3); then use the multi-material interpolation relationship in formula (8) to describe the distribution of multiple microstructures on a macroscopic scale, and calculate the equivalent elastic tensor and stiffness matrix of each macroscopic unit.

[0057] 4.5) Assemble the overall stiffness matrix, perform macroscopic finite element analysis, and calculate the sensitivity of the material field characteristic function coefficients.

[0058] 4.6) Update the design variables using a gradient solver and check the termination criteria. If the termination criteria are met, end the optimization process and generate a geometric file of the optimization results; otherwise, proceed to step 4.4). The termination criteria are generally selected as reaching a preset maximum number of iterations or the relative change of the objective function in five consecutive iterations being less than 0.1%.

[0059] The beneficial effects of this invention are as follows:

[0060] (1) While inheriting the inherent advantages of the original material field series expansion method, such as fewer design variables, smooth boundaries, and no need for additional sensitivity and density filtering operations, this invention proposes a microstructure topology characterization method driven by correlation functions. This method effectively utilizes the characteristics of the interpolation model and reduces the performance loss caused by ensuring connectivity. At the same time, it has fewer design variables, high optimization efficiency, and simple numerical implementation.

[0061] (2) Based on the arbitrary structure hexahedral voxelization route method proposed in this invention, this invention can be applied to multi-scale concurrent topology optimization design of arbitrarily complex structures in engineering, and is expected to become a powerful multi-scale concurrent topology optimization method with strong scalability and wide applicability. At the same time, this invention is a novel topology representation method that does not depend on the specific PDE solution method used, such as finite element method, finite volume method, and boundary element method, which can be effectively combined with this method.

[0062] In summary, this invention is expected to be applied to the integrated topology optimization design of materials and structures for small aerospace components, and can be extended to high-complexity problems such as thermo-structure coupling and impact energy absorption that are widely faced in the aerospace field. Attached Figure Description

[0063] Figure 1 The flowchart illustrates the implementation of an integrated topology optimization method for materials and structures based on ordered MFSE interpolation, as provided in this invention.

[0064] Figure 2 A schematic diagram of the microstructure parameterization framework provided by the present invention.

[0065] Figure 3 The complex three-dimensional armadillo design domain provided for the examples of this invention. Figure 3 (a) is a geometry file; Figure 3 (b) shows the regular hexahedral finite element model and boundary conditions after voxelization.

[0066] Figure 4 The dual-scale concurrent optimization results with five microstructures are provided for the examples of this invention. Figure 4 (a) is for optimizing the structure; Figure 4 (b) To optimize the microstructure.

[0067] Figure 5 The microstructure connectivity and macroscopic distribution of different microstructures are optimized results for examples of this invention. Figure 5 (a) represents the connectivity of the microstructure; Figure 5 (b) shows the macroscopic distribution of different microstructures.

[0068] Figure 6 This is the full-scale model obtained from an example of the present invention. Detailed Implementation

[0069] To make the problems addressed, the methods proposed, and the effects achieved by this invention clearer, the following detailed description, in conjunction with the technical solutions and accompanying drawings, further illustrates this invention. The algorithm flowchart of the proposed method is shown below. Figure 1 As shown, it should also be noted that, for ease of description, the accompanying drawings only list the parts related to the present invention, not all of them. The specific implementation steps are as follows:

[0070] Example: Integrated design of complex three-dimensional armadillo material structures under multiple loads

[0071] The first step is to voxelize the three-dimensional armadillo structure to generate a regular hexahedral mesh.

[0072] 1.1) Determine the design domain based on the actual situation of the structure, and convert it into an STL format patch file using commercial software, such as... Figure 3 As shown in (a), the center point information of each facet is calculated, and the length, width, and height information of the armadillo design domain are obtained through equation (2), and the unit size of the hexahedron is defined.

[0073] 1.2) Map the armadillo structure onto a regularly divided spatial background mesh, check the overlap between triangles and the divided hexahedral elements, and fill the shape of the structure. Then, use a flood filling algorithm to identify the hexahedral elements corresponding to the internal pores that are not connected to the outside. After identification, delete the external elements.

[0074] 1.3) After generating the regular hexahedral mesh, set the boundary conditions. In this example, apply a uniformly distributed downward load to the armadillo's two hands, while constraining its two feet. Then, set the design domain and non-design domain for the topology optimization process. In this example, for aesthetic and practical purposes, all boundary conditions are set to the non-design domain, and the remaining parts are set to the design domain, such as... Figure 3 As shown in (b).

[0075] The second step involves a multi-material interpolation method based on the series expansion of macroscopically ordered material fields.

[0076] The center point of each hexahedron is calculated, the anisotropic correlation length is specified, and the correlation between all observation points is calculated using the anisotropic correlation function to obtain the symmetric positive definite correlation matrix of the macroscopic material field. The correlation matrix is ​​then decomposed into eigenvalues ​​and the error is truncated to obtain the reduced series expansion form of the macroscopic material field.

[0077] The third step is the connectivity method driven by micro-correlation functions.

[0078] 3.1) Specify the number of microstructures as m = 5, with maximum volume fractions of 30%, 35%, 40%, 45%, and 50%, respectively. Sort the microstructures according to a multi-material interpolation scheme based on the sequence material field order expansion. Discretize each microstructure into 50×50×50 elements, specify the thickness of the connecting region as 4 elements, and divide the microstructure into several internal regions and connecting regions, such as... Figure 2 As shown in (a)

[0079] 3.2) Assemble the connection regions between adjacent microstructures, and introduce a connection material field into each of the different unit cell connection regions after assembly, such as... Figure 2 As shown in (b).

[0080] 3.2) Perform a material field series expansion on the internal and connecting material fields of all unit cells. Different microstructures are composed of the distribution of connecting material fields and internal material fields, i.e. The spatial correlation defined in the material field series expansion method drives the good connections between microstructures, and the overall assembly idea of ​​the microstructures is as follows: Figure 2 As shown in (c).

[0081] The fourth step is to establish a multi-scale concurrent topology optimization model for the structure and perform structural optimization design.

[0082] 4.1) For simplicity, in this example, we consider compliance minimization, which is the most commonly used topology optimization, as the objective function, and the constraints are the overall volume fraction and the maximum volume fraction limit for each microstructure.

[0083] (a) Objective function: Minimize the overall flexibility of the two-scale structure;

[0084] (b) Constraints: The total volume of the entire structure is less than 10%, and there is an upper limit constraint on the volume fraction of each microstructure;

[0085] (c) Design variables: Characteristic function coefficients of macroscopic and microscopic material fields [η] mac ,η mic ].

[0086] 4.2) After that, multiple initial solutions for microstructures and macrostructures are set based on information such as volume fraction, and the design variables are initialized.

[0087] 4.3) Based on the topological characterization method described above, the configuration of multiple microstructures and their equivalent elastic tensors are calculated through the microscopic material field. Then, their distribution on the macroscopic level is calculated through the macroscopic material field to obtain information such as the equivalent elastic tensor and stiffness matrix of each macroscopic unit.

[0088] 4.4) Assemble the overall stiffness matrix, perform macroscopic finite element analysis, and calculate the sensitivity of the objective function and constraints to the coefficients of the material field characteristic function based on the derived sensitivity.

[0089] 4.5) Repeat 4.3) and 4.4) until the set termination criterion is met; in this example, the termination criterion is set to the relative change of the objective function for two consecutive iterations being less than 0.1% or the maximum number of iterations being reached.

[0090] 4.6) After the above optimization process, the optimization result of the three-dimensional armadillo structure is as follows: Figure 4 As shown, the macroscopic distribution of the microstructure is given ( Figure 4 (a)) and the configuration of microstructures ( Figure 4 (b)). To visualize the connectivity between microstructures and the macroscopic distribution of different microstructures, Figure 5 (a) A schematic diagram of the connectivity between microstructures is given. Figure 5 (b) The macroscopic distribution of different microstructures is presented. The optimization results are extracted, and the dual-scale structure is reconstructed. The integrated material and structural design results of the optimized armadillo structure are shown below. Figure 6 As shown, the microstructure exhibits good connectivity.

[0091] The essence of this invention is to establish a new microstructure topology characterization method based on the optimized structural characteristics of the integrated material structure design method under multi-material interpolation description. This method naturally ensures the connectivity between microstructures during the optimization process and simultaneously solves three thorny problems existing in dual-scale parallel optimization: difficulty in voxelizing hexahedral meshes of complex structures, poor connectivity between microstructures, and a large number of design variables. This invention is an effective supplement to existing dual-scale concurrent topology optimization methods and is expected to be applied to the integrated material structure design of highly complex small structures in the fields of aerospace and other transportation equipment, to further obtain multi-functional multi-scale structures and improve the overall structural performance of transportation equipment. Modifications to the optimization models, methods, and schemes described in the foregoing embodiments (such as different unit cell configurations, different numbers of microstructures, different projection functions, etc.), or equivalent substitutions of some or all of the method features (such as changing the objective function or the specific form of constraints, changing the optimizer, 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.

[0092] 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 material structure integrated topology optimization method based on ordered MFSE interpolation, characterized in that, The optimization method comprises four parts: arbitrary structure regular hexahedral mesh generation, macroscopic ordered material field series expansion multi-material interpolation method, microscopic correlation function driven connectivity method, and structural multi-scale concurrent topology optimization. The integrated topology optimization method described above uses superscripts Mac and Mic to represent the macroscopic and microscopic scales, respectively. The specific steps are as follows: The first step is to generate a regular hexahedral mesh using voxelization for any structure. Step 1.1) Obtain the spatial background mesh size of the structural design domain: A general geometric model file provides geometric information for any complex design domain, resulting in a geometric model. This geometric model is then converted into an STL patch format file containing information for multiple triangles, and the center point of each triangle is calculated. Based on the center point information of all triangles, the geometric dimensions of the complex design domain are calculated using the following formula: （2） in, express x Minimum coordinate value of the direction; express x Maximum coordinate value of the direction; express y Minimum coordinate value of the direction; express y Maximum coordinate value of the direction; express z Minimum coordinate value of the direction; express z Maximum coordinate value of the direction; Represents the center point of all triangular facets x Minimum coordinate value of direction; Represents the center point of all triangular facets x Maximum coordinate value in direction; Represents the center point of all triangular facets y Minimum coordinate value of direction; Represents the center point of all triangular facets y Maximum coordinate value in direction; Represents the center point of all triangular facets z Minimum coordinate value of direction; This represents the maximum z-coordinate value of the center point of all triangular patches; Therefore, the spatial background size of the structural design domain is obtained as follows: ; Step 1.2) Generate a regular hexahedral mesh. The cell size of the hexahedral mesh is specified, and the spatial background is divided according to its cell size. The complex design domain is projected into a regularized hexahedral mesh of the spatial background. The overlap between each triangle and the cells of the divided hexahedral mesh is checked to construct the boundary information of the design domain: if they overlap, the overlapping cells are marked as 1, indicating that the cell will be part of the finite element analysis mesh; if they do not overlap, they are marked as 0, indicating that the cell will be removed from the finite element analysis. After constructing the boundary information of the design domain, the interior of the design domain is filled using the boundary information, and all interior cells are marked as 1, indicating that these cells will be part of the finite element analysis mesh. Voxels marked as 0 are deleted. Finally, a regular hexahedral mesh is generated for any structure, enabling subsequent multi-scale finite element calculations using homogenization theory. The second step involves using a macroscopically ordered material field series expansion multimaterial interpolation method to obtain a clear macroscopic distribution of multiple microstructures. Step 2.1) After constructing the regular hexahedral mesh of the structural design domain, calculate the center point of each element and set it as the observation point of the material field. Use the correlation function to calculate the spatial correlation between several observation points. Then, calculate the correlation between any two points of the material field observation points and construct the correlation matrix. Step 2.2) Using the optimal linear estimation theory of random fields, the correlation matrix is ​​decomposed into eigenvalues, and the eigenmode terms with smaller eigenvalues ​​are truncated. The material field is then expressed in a reduced basic mode superposition form, i.e.: （4） in, It is the first k The modal coefficients corresponding to the first-order eigenvalues ​​are set as design variables in the topology optimization process. Represents the material field function; Describe orthogonal characteristic basis functions, using eigenvalues and normalized modes To represent; Indicates the first i Design variables for each material field; Indicates the first k Eigenvalues ​​arranged in descending order; Indicates the first k The modal matrix arranged in descending order; This represents the material field design variables in the topology optimization process; This represents a diagonal matrix composed of eigenvalues; This represents the basis mode matrix obtained after eigenvalue expansion; the expansion process is performed only once before optimization. Step 2.3) Obtain a clear macroscopic distribution of multiple microstructures, as detailed below: 2.3.1) Assuming that microscopic phenomena exist m To describe the distribution of microstructures on a macroscopic scale, the macroscopic material field described by formula (4) is defined in the interval [-1, 1], and is further defined according to the number of microstructures. m Divide into segments. m A continuous transition region, namely: （5） in, Representing the ( j- 1) microstructure to the first j The transition region of the microstructures, where the 0th microstructure represents a pore; m This indicates the number of different microstructures specified manually before optimization; This represents the material field located in [-1,1] introduced on a macroscopic scale; This represents a point located within the design domain; 2.3.2) will be located in the first j The macroscopic material field of this portion of the transformation region is linearly scaled to [-1, 1]; 2.3.3) After linear scaling, the macroscopic material field is projected using the extended piecewise interpolation Heaviside projection function EP-HP to obtain the projected material field; 2.3.4) All projected material fields are further scaled to the [0,1] interval and interpolated using an ordered format; 2.3.5) Based on the ordered-MFSE method of sequential material field series expansion, the stiffness matrix of the multi-material interpolation scheme is interpolated in the following form: （8） in, Indicates the macro-level i The elastic tensor of each unit; This represents the material field obtained by linear scaling using equation (7); p It is a penalty coefficient used to penalize the contribution of intermediate value stiffness; and It is the first j The and the first j+ The elastic tensor of a microstructure; The third step is the connectivity method driven by micro-correlation functions. Due to the spatial correlation of the material field, the optimization results obtained in the second step based on the ordered-MFSE multi-material interpolation method exhibit the characteristic of gradual material transition. Therefore, the connectivity between microstructures is ensured one by one, and then a well-connected full-scale structure is obtained. The specific correlation function-driven connectivity method is as follows: Step 3.1) First, specify the thickness of the connection region and divide all microstructures into two regions: the connection region and the internal region. Then, further divide the connection region according to the spatial dimension of the microstructure: the connection region of the two-dimensional microstructure is divided into four parts, and the connection region of the three-dimensional microstructure is divided into six parts. Step 3.2) Since the processing procedures for two-dimensional and three-dimensional microstructures are similar, we will use two-dimensional microstructures as an example. As described in Step 3.1), the two-dimensional microstructure is divided into five parts, one of which is the internal region. The remaining four parts are four independent connection regions. , , , Assuming A and B are neighboring cells in a sequence interpolation scheme, to ensure their connectivity, the connecting regions between the microstructures are assembled. ,in, This represents the second connection region of unit cell A; This represents the first connection region of the B-cell, where a connection material field is introduced. The material field is named the connecting material field of the connecting region; and the interior of the microstructure is also characterized by a material field, named the internal material field. Step 3.3) After obtaining the connecting material field of the connecting region through step 3.2), according to the assembly relationship of the connecting region in step 3.2), the connecting material field is divided into two parts and distributed into two adjacent microstructures. The microstructure configuration is obtained in the following form. （9） in, Indicates the first i The material field in the intermediate region of each microstructure Indicates the first i The first microstructure j Each region of the material field is composed of segments connecting the material fields, that is, Meanwhile, the notation in the above formula The microscopic Heaviside projection function is used to represent the configuration of different microstructures. After obtaining the configuration of different microstructures, periodic boundary conditions are applied and homogenization calculation is performed to obtain the equivalent elastic tensor of different microstructures. Then, the equivalent elastic tensor of each macroscopic unit is calculated using formula (8). By using the connectivity method driven by micro-correlation functions, flexible connection forms between adjacent microstructures can be realized without introducing connectivity constraints of microstructures into the optimization model, and multi-scale structures with excellent performance can be obtained. The fourth step is multi-scale concurrent topology optimization of arbitrarily complex spatial structures. Step 4.1) After converting the arbitrary geometric model into a regular hexahedral mesh in the first step, determine the objective function and constraint form according to the characteristics of the optimization problem and the actual engineering requirements, and establish a multi-scale concurrent topology optimization model for the structure. Step 4.2) Based on the specific forms of the given objective function and constraint function, derive the sensitivity of the design variables using the chain rule; Step 4.3) Use the correlation function driven connectivity strategy proposed in step 3 to expand the microscopic material field, and initialize the design variables of multiple material fields according to the given initial solution; Step 4.4) Based on the internal material field and the connecting material field, calculate the configuration of multiple microstructures through step 3.3); use the multi-material interpolation relationship in formula (8) to describe the distribution of multiple microstructures on a macroscopic scale, and calculate the equivalent elastic tensor and stiffness matrix of each macroscopic unit; Step 4.5) Assemble the overall stiffness matrix, perform macroscopic finite element analysis, and calculate the sensitivity of the material field characteristic function coefficients; Step 4.6) Update the design variables using a gradient-based solver and check the termination criteria. If the termination criteria are met, end the optimization process and generate the geometry file of the optimization results; otherwise, go to step 4.

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2. The integrated topology optimization method for materials and structures based on ordered MFSE interpolation according to claim 1, characterized in that, In step 1.1), the formula for calculating the center point of each triangular piece is as follows: （1） in, Indicates the first i The coordinates of the center point of each triangle Indicates belonging to the first i The first triangular piece j Vertex coordinates; In step 1.2), the flood filling algorithm is used to fill the interior of the design domain using the boundary information of the design domain.

3. The integrated topology optimization method for materials and structures based on ordered MFSE interpolation according to claim 1, characterized in that, In step 2.1): The relevant function is defined as: （3） in, , These are two observation points. i and j Spatial location; , ,and These are the correlation lengths along the three coordinate axes. The correlation length in each direction is determined by the size of the sub-partition and can be defined independently, hence it is also called anisotropic correlation length. The correlation between any two observation points in the material field is calculated using formula (3), and a correlation matrix is ​​constructed, i.e. ,in ,in Indicates the selected number i Coordinates of each observation point , , , Indicates each observation point, Np This indicates the number of observation points.

4. The integrated topology optimization method for materials and structures based on ordered MFSE interpolation according to claim 1, characterized in that, In step 2.3): In section 2.3.3), the scaling method used is as follows: ;in, This represents a macroscopic material field that has been linearly scaled. In step 2.3.3), the projection expression used is as follows: （6） in, These are macroscopic projection parameters that affect the overall steepness of the projection. In the optimization process, a smaller initial value is used, and an extension strategy is adopted to gradually increase it to the maximum value. In section 2.3.4), the linear scaling method is as follows: （7） in, This represents the material field obtained by linear scaling; , Representing the first j Upper and lower limits of each conversion region; This represents the projected material field obtained through formula (6); In section 2.3.5), the penalty coefficient p =3.

5. The integrated topology optimization method for materials and structures based on ordered MFSE interpolation according to claim 1, characterized in that, In step 3.3), the microscopic Heaviside projection function is given in sigmoid form as follows: （10） in, These are microscopic projection parameters that control the overall steepness of the microscopic projection.

6. The integrated topology optimization method for materials and structures based on ordered MFSE interpolation according to claim 1, characterized in that, In step 4.6), the termination criterion is to reach the preset maximum number of iterations or for the relative change of the objective function in five consecutive iterations to be less than 0.1%.

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