A method for constructing a dataset of double-connected three-dimensional microstructures with the same boundary

Abstract

The present invention belongs to the field of materials science and discloses a method for constructing a double-connected three-dimensional microstructure dataset with the same boundary. First, determine the boundary type and reconstruct the microstructure with the boundary type as the constraints for the six faces of the cube. Then, select relevant optimization objectives for each boundary type and create an initial dataset under the corresponding constraints. Next, use generative AI for structure generation to expand the attribute space of the microstructure, and use an active learning strategy for iterative expansion of the dataset. Finally, identify and retain the structures with double-connected features through the connected component labeling algorithm, and remove the disconnected structures, thereby constructing a dataset with the same boundary and double-connectedness. The LBBM dataset constructed by the present invention significantly improves the computational efficiency of multi-scale assembly, ensures the boundary connectivity between microstructures and the double-connectedness of the overall multi-scale structure. The material constructed based on this dataset has print manufacturing friendliness and can realize functions such as fluid flow, heat conduction, filtration, and gas exchange.

Description

Technical Field

[0001] The present invention belongs to the field of materials science, and more particularly relates to a three-dimensional microstructure dataset with the same boundaries and double connectivity for engineering applications. Background Art

[0002] Natural materials such as cancellous bone, wood, butterfly wings, and sponges exhibit highly interconnected pore networks with hierarchical and double-connected structural characteristics. In such double-connected microstructures, both the solid and void phases are continuously distributed, thus having important application values in fields such as aerospace, automotive manufacturing, biomedicine, and mechanical engineering, especially showing excellent performance in complex applications that require both high stiffness, large surface area, good air permeability, and adjustable permeability. Imitating structures in nature such as starfish and sea urchin spines, developing functional engineering materials often requires a large number of experiments and repeated adjustments, which are not only costly but also difficult to achieve structural diversity. Therefore, as an inverse design tool, topology optimization has become a key means for researching and developing such materials with its ability to efficiently explore the design space.

[0003] By combining spinodal structures with topology optimization, double-connected structures with parameter range limitations can be generated, but the related multi-scale structural optimization brings huge computational challenges. The three-periodic minimal surface (TPMS) structure has been widely used in the multi-scale design of bone implants, heat exchangers, and energy absorption devices due to its excellent mechanical properties. Although the TPMS structure has great application potential, insufficient boundary connection may also lead to structural failure. Common techniques for enhancing connectivity include optimization, interpolation, and deformation, but these methods also increase the corresponding computational cost. The difficulties in double-connected multi-scale design can be decomposed into the following three key problems: (1) a dataset with as wide an attribute coverage range as possible is needed; (2) the boundaries of each microstructure are the same; (3) all microstructures are double-connected. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for constructing a double-connected three-dimensional microstructure dataset with the same boundaries to solve the problems raised in the above background art.

[0005] To achieve the above purpose, the present invention provides the following technical solutions:

[0006] A method for constructing a double-connected three-dimensional microstructure dataset with the same boundaries specifically includes the following steps:

[0007] Step S1: Determine four boundary types through test analysis, and reconstruct the microstructure with the four boundary types as the constraints for the six faces of the cube;

[0008] Step S2: For each type of boundary, use the LIVE3D framework, and use maximizing the bulk modulus, maximizing the shear modulus, maximizing the Young's modulus, and the five-mode structure as the optimization objectives, and create an initial dataset according to the corresponding constraint conditions;

[0009] Step S3: Adopt generative AI for structure generation to expand the property space of the microstructure. The generative model uses a diffusion model, the network structure uses Unet, the loss function is the MSE loss. On this basis, use the initial dataset in S2 to train the Unet network, and adopt an active learning strategy for iterative expansion of the dataset;

[0010] Step S4: Identify and retain the structures with double-connected features through the connected component labeling algorithm, remove the unconnected structures, and construct a dataset with the same boundary and double-connectivity under the boundary type described in Step S1.

[0011] As a further solution of the present invention: In the objective function of the optimization objective described in Step S2, ρ is the density of the structure, C(ρ) is the compliance when the structure density is ρ, and the objective function of the optimization objective is as follows:

[0012] The objective function for maximizing the bulk modulus is: C 1111 and C 1122 are two components in the elastic tensor matrix C;

[0013] The objective function for maximizing the shear modulus is: f(C(ρ)) = C 1212 where C 1212 is a component in the elastic tensor matrix C;

[0014] The objective function for maximizing the Young's modulus is:

[0015] The objective function for the five-mode structure is: where the bulk modulus K is defined by the formula and the shear modulus G is defined by the formula G = C 1212 K0 and G0 are the initial values of the bulk modulus and the shear modulus calculated randomly respectively.

[0016] As a further solution of the present invention: The constraint conditions in Step S2 specifically refer to:

[0017] Equilibrium equation constraint: K(ρ)u = f, where K(ρ) is the stiffness matrix, u is the displacement vector, and f is the applied external force;

[0018] Volume fraction constraint: where M is the total number of discrete elements of the material, v e and ρ eare the volume fraction and density of the material at the e-th location, respectively, and |Ω| represents the volume of the microstructure design domain. This constraint ensures that the total volume of the material does not exceed V during the optimization process;

[0019] Boundary condition constraint: During the optimization process, ensure that the boundary b1 is as close as possible to the boundary b2, and ε is the allowable convergence error;

[0020] Isotropic constraint:

[0021] Poisson's ratio constraint: where v0 is the specified initial Poisson's ratio;

[0022] Density variable constraint: M is the total number of discretized elements of the material. That is, among all the discretized elements, for any arbitrarily selected element ρ e the material density is between [0, 1].

[0023] As a further solution of the present invention: The diffusion model in the step S3 includes a forward noise addition process and a reverse denoising process. The forward noise addition process adds noise to the real data x0 of x step by step through the equation t where ε t-1 is sampled from the standard Gaussian distribution, and the parameter α t is used to control the level of noise. The reverse denoising process starts from the data x t close to pure noise, and uses the neural network to predict the noise and indirectly recover x t-1 until the real data x0 is recovered.

[0024] As a further solution of the present invention: The loss function of the diffusion model in step S3 trains the network according to the following relational expression for guiding the neural network to learn to denoise the noisy data. In the relational expression, t represents the time step in the diffusion process, t ∼ [1, T] means that the time step t is uniformly sampled from the time interval [1, T], x0 represents a real data in the training data, q(x0) represents the probability distribution of the real data, x0 ∼ q(x0) means randomly sampling a data point x0 from q(x0), that is, selecting a sample from the training data as the input of the diffusion model, ε ∼ N(0, I) represents the noise sampled from the standard normal distribution, with a mean of 0 and a covariance matrix of the identity matrix I, x t is the noisy data at time t in the diffusion process, and the noisy data is generated by adding noise to the real data x0. x θ (x t , t) represents the predicted value of the neural network model with parameter θ.

[0025] As a further solution of the present invention: in the network training in step S3, all microstructures are represented by voxels, that is, each structure is represented as a three-dimensional binary matrix of n×n×n. Due to the cubic symmetry of the structure, only the upper left 1 / 8 of the structure needs to be considered as the input during training. At the same time, due to symmetry, noises with symmetry are also used during the forward and reverse diffusion processes.

[0026] As a further solution of the present invention: step S3 adopts a self-conditional strategy to improve the generation quality, that is, the content previously generated by the network is used as the condition for the next generation.

[0027] As a further solution of the present invention: in terms of the network structure in step S3, both the U-Net encoder and decoder contain 5 stages, and each stage is composed of a Resblock with an attention mechanism.

[0028] As a further solution of the present invention: to meet the requirement of constructing a large-scale micro-structure dataset in step S3, an active learning strategy is adopted for iterative training.

[0029] As a further solution of the present invention: the connected component labeling algorithm in step S4 specifically includes:

[0030] The first scan: for each pixel point p located at the coordinate (i, j), output the image label L(i, j), and the calculation process is as follows:

[0031]

[0032] where N(p) represents the neighboring pixels of p. If there are multiple labels in N(p), select the smallest label and record these labels as equivalence relations;

[0033] The second scan: traverse each pixel point p in the image again, and update the label to ensure that all equivalent labels are unified into one label: L(i, j) = find(L(i, j)), where the find function finds the smallest label equivalent to L(i, j) to achieve the unification of labels.

[0034] Compared with the prior art, the beneficial effects of the present invention are as follows: By using the constructed dataset of three-dimensional microstructures with the same boundary and double connectivity for multi-scale structural design, the elastic modulus of each unit is determined by optimizing the macroscopic structure, and then the corresponding microscopic structure is selected from the dataset. Since the dataset constructed by the present invention has a unified boundary and ensures double connectivity, the natural connectivity between microscopic structures during the multi-scale assembly process is ensured. The LBBM dataset constructed by the present invention significantly improves the computational efficiency of multi-scale assembly, ensures the boundary connectivity between microstructures and the double connectivity of the overall multi-scale structure. The material constructed based on this dataset has print manufacturing friendliness and can realize functions such as fluid flow, heat conduction, filtration, and gas exchange. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 Schematic diagrams of four boundary condition structures obtained from the test analysis of the present invention;

[0036] Figure 2 Attribute relationship diagram of the optimized dataset under four boundary conditions of the present invention;

[0037] Figure 3 Schematic diagram of the active learning network framework in the present invention;

[0038] Figure 4 Attribute relationship diagram of a large-scale, boundary-consistent and double-continuous microstructure (LBBM) dataset under four boundary conditions of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0039] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0040] In the method for constructing a dataset of double-connected three-dimensional microstructures with the same boundary in the embodiments of the present invention, for the mechanical properties of materials, during the homogenization process, complex microscopic structures are simplified into equivalent homogeneous materials, making macroscopic analysis more direct. The method in the embodiments of the present invention uses the representative volume element (RVE) of the microscopic structure to capture micro-scale changes to predict the overall mechanical behavior. All RVEs considered in this method exhibit cubic symmetry, which means that these microscopic structures remain unchanged under the symmetry operations of the cubic crystal system. This symmetry indicates that the material has the same physical properties along three mutually perpendicular axes. For cubic cubic symmetric materials, the elastic tensor matrix C is represented as follows:

[0041]

[0042] Among them, C 1111 , C 1122 , C 1212 are the elastic constants of the material.

[0043] The bulk modulus K and shear modulus G of a cubic symmetric material can be calculated from the elastic constants C 1111 , C 1122 , C 1212 and the calculation formulas are as follows:

[0044] The calculation formulas for Young's modulus E and Poisson's ratio v are as follows:

[0045]

[0046] The Zener ratio A, a parameter describing material anisotropy, has the following calculation formula:

[0047]

[0048] A Zener ratio A close to 1 indicates greater isotropy, and a deviation from 1 indicates greater anisotropy.

[0049] The method in the embodiments of the present invention for constructing a double-connected three-dimensional microstructure dataset with the same boundary. The four boundary conditions in this method are as Figure 1 shown. Taking these four boundary conditions as constraint conditions respectively, 4 datasets are constructed. Taking the first boundary mask1 as an example, the specific steps for constructing the dataset with mask1 as the boundary are as follows:

[0050] To design a double-connected microstructure with a wide range of property coverage and with mask1 as the boundary, the embodiments of the present invention designed 10 optimization models, including 5 isotropic models and 5 anisotropic models. The relevant parameters in the 10 optimization models are described as follows: ρ represents the design variable vector, which is the density of the material; f(C(ρ)) is the objective function, which is the negative value of the minimized bulk modulus here; u is the displacement field within the microstructure design domain; f is the applied external force; K(ρ) is the global stiffness matrix; overall, K(ρ)u = f represents the equilibrium equation constraint; M is the number of discrete elements in the design domain; ρ e is the density of the e-th unit of the material; v e is the volume of the e-th unit, and further represents the volume of the material; |Ω| represents the volume of the microstructure design domain; V represents the specified volume fraction; here the volume fraction constraint is that the total volume of the material does not exceed V compared to the volume of the design domain; b1 is the boundary of the material, b2 is the specified boundary, ε is the allowable convergence error, and the boundary constraint conditions Ensure that the material boundary b1 is as close as possible to the specified boundary b2; (A-1) 2 <ε is an isotropic constraint, that is, it is required that the Zener ratio A is as close as possible to 1 within the allowable error range; It is a density variable constraint, 0 indicates that the material is empty, and 1 indicates that the material is solid.

[0051] (1) Five isotropic optimization models

[0052] The Solid Isotropic Material with Penalty (SIMP) model is used, which is a commonly used material interpolation method in topology optimization. By introducing a penalty factor, the SIMP model guides the material distribution to a 0-1 distribution.

[0053] (1) Maximize the bulk modulus optimization model

[0054] Considering the isotropic and boundary constraints, the optimization formula for maximizing the bulk modulus is as follows:

[0055]

[0056] s.t. K(ρ)u = f

[0057]

[0058]

[0059] (A-1) 2 <ε

[0060]

[0061] (2) Maximize the shear modulus optimization model

[0062] To expand the dataset, the embodiment of the present invention also designs an optimization model for maximizing the shear modulus with fixed boundaries:

[0063]

[0064] s.t. K(ρ)u = f

[0065]

[0066]

[0067] (A-1) 2 <ε

[0068]

[0069] (3) Maximize the Young's modulus optimization model

[0070] Similarly, by modifying the objective function, an optimization model for maximizing Young's modulus with fixed boundaries is obtained:

[0071]

[0072] s.t. K(ρ)u = f

[0073]

[0074]

[0075] (A - 1) 2 <ε

[0076]

[0077] (4) Optimization model of five-mode structure with Poisson's ratio close to 0.5

[0078] For isotropic microstructures, the upper limit of Poisson's ratio is 0.5. To generate a five-mode structure with Poisson's ratio close to 0.5, the embodiments of the present invention design the following optimization model:

[0079]

[0080] s.t. K(ρ)u = f

[0081]

[0082] (A - 1) 2 <ε

[0083]

[0084] where the bulk modulus K and the shear modulus G are defined by the formulas and G = C 1212 respectively. To prevent singularities in the denominator, 0.05 is added as a correction in the denominator.

[0085] (5) Optimization model of negative Poisson's ratio structure

[0086] After using these four optimization models, the embodiments of the present invention design microstructures with Poisson's ratios of approximately 0.2 and close to 0.5. To design microstructures with a wider range of properties, reducing the Poisson's ratio of the material is a key factor. The embodiments of the present invention introduce a specific Poisson's ratio as a constraint condition into the optimization process and design a structure with negative Poisson's ratio and fixed boundaries. The specific optimization model is as follows:

[0087]

[0088] s.t. K(ρ)u = f

[0089]

[0090] (A-1) 2 <ε

[0091]

[0092]

[0093] Among them, is the constraint condition with a specified Poisson's ratio of v0.

[0094] (2) Five anisotropic optimization models

[0095] Anisotropic microstructures have unique physical properties such as negative refractive index, negative Poisson's ratio, and ultra-high elastic modulus, which are crucial in science and engineering. To enhance the dataset of the present invention, a batch of anisotropic material data has been optimized. (6) Optimization model for maximizing the bulk modulus:

[0096]

[0097] s.t. K(ρ)u = f

[0098]

[0099]

[0100]

[0101] (7) Optimization model for maximizing the shear modulus:

[0102]

[0103] s.t. K(ρ)u = f

[0104]

[0105] (8) Optimization model for maximizing the Young's modulus:

[0106]

[0107] s.t. K(ρ)u = f

[0108]

[0109] (9) Optimization model for a five-mode structure with a Poisson's ratio close to 0.5:

[0110]

[0111] s.t. K(ρ)u = f

[0112]

[0113] (10) Negative Poisson's ratio structure optimization model:

[0114] The optimization model for specifying Poisson's ratio is as follows:

[0115]

[0116] s.t. K(ρ)u = f

[0117]

[0118]

[0119] Through the above 10 optimization models with fixed boundaries, the embodiments of the present invention optimize the microstructure with the mask1 boundary. By screening out the structures with fully connected solid parts and good boundary connections, 21,620 high-quality microstructures (the solid parts are all connected, and the voids are not necessarily connected) are finally obtained.

[0120] Figure 2 Part a in shows the optimized data set containing 21,620 data points, which covers the microstructures with double connectivity and non-connected voids. The outermost six faces of all microstructures are the same as the mask1 image in the upper right corner. The figure shows the relationship between Young's modulus, Poisson's ratio, and Zener ratio. Each microstructure is represented in the form of a cube, and its size reflects the porosity, and the color represents the surface-to-volume ratio (S / V). The red cube highlights a specific microstructure instance indicated by the arrow. The bottom and right projection diagrams respectively show the relationship between Young's modulus and Poisson's ratio, and the relationship between Poisson's ratio and Zener ratio.

[0121] Figure 2 Part b in shows the relationship between Young's modulus, Poisson's ratio, and Zener ratio in the optimized data set under the condition of the second boundary (mask2). The optimized data set contains 28,767 data points; Figure 2 Part c in shows the relationship between Young's modulus, Poisson's ratio, and Zener ratio in the optimized data set under the condition of the third boundary (mask3). The optimized data set contains 26,532 data points; Figure 2 Part d in shows the relationship between Young's modulus, Poisson's ratio, and Zener ratio in the optimized data set under the condition of the fourth boundary (mask4). The optimized data set contains 14,930 data points.

[0122] To expand and enrich the data set, the present invention proposes an active learning strategy based on the diffusion model, as Figure 3 shown. To improve the quality of the generation by the diffusion model, an adaptive method is adopted, which uses a denoising function It takes forward estimation and the sample x at the current time step t t as inputs. During the training process, is set to f(x t ,0,t) with a probability of 50%, otherwise it is set to 0. Given that the microstructure of the present invention is a three-fold symmetric voxel structure, only the unit cell is used during the training and generation processes.

[0123] To generate a microstructure with specific boundaries and moduli, the present invention uses three independent components C 1111 , C 1122 , C 1212 in the boundary constraint and the elastic tensor matrix C as conditions, and introduces these conditions into each layer of the U-Net through the adaptive instance normalization technique.

[0124] During the training process, the AdamW optimizer is used to optimize the denoising loss function, with the learning rate set to 10 -4 , the batch size set to 24, and a total of 510 training epochs are performed. The loss calculation is carried out by minimizing the mean squared error between the model output and the target value over the noise and time steps, enabling the model to gradually learn to generate a microstructure with specified boundaries under given conditions.

[0125] During the generation process, the diffusion model generates diverse three-dimensional microstructures under the specified boundary and elastic tensor matrix conditions. To cover a wide range of modulus property values, the present invention calculates the signed distance function (SDF) values of each node on the Cartesian grid within the property space of the training dataset, samples in the regions where the SDF values are less than 0, ensuring that the sampling range extends beyond the coverage of the training dataset. These elastic tensor matrix components, together with the encoding of the target boundary, are input into the network, and 50-step DDPM sampling is used to generate the microstructures.

[0126] To obtain a large-scale bicontinuous dataset with consistent boundaries, the embodiments of the present invention adopt an active learning strategy. Initially, the adaptive diffusion model is trained using an optimized initial dataset, and then the generated bicontinuous microstructures are added to the dataset, and the spatial density is adjusted to ensure a uniform distribution of data points. Structures that do not meet the bicontinuity criteria and have low quality are iteratively removed. This process ultimately forms a large-scale, boundary-consistent, and bicontinuous microstructure (LBBM) dataset. As Figure 4As shown in part a), the LBBM dataset with mask1 as the boundary condition is presented, which contains 107,383 high-quality biconnected data. At the same time, the relationships among its Young's modulus, Poisson's ratio, and Zener ratio are shown. For the other three boundaries with boundaries of Mask2 - Mask4, the embodiments of the present invention sequentially use the active learning network framework and finally obtain 192,925, 75,660, and 79,744 high-quality biconnected microstructures respectively. The relationships among their Young's modulus, Poisson's ratio, and Zener ratio are shown in parts b), c), and d) of Figure 4 respectively.

[0127] For those skilled in the art, it is obvious that the present invention is not limited to the details of the above exemplary embodiments, and without departing from the spirit or basic characteristics of the present invention, the present invention can be implemented in other specific forms. Therefore, from any point of view, the embodiments should be regarded as exemplary and non-limiting. The scope of the present invention is defined by the appended claims rather than the above description. Therefore, all changes falling within the meaning and scope of the equivalent elements of the claims are intended to be encompassed within the present invention. Any reference signs in the claims should not be construed as limiting the claims involved.

[0128] In addition, it should be understood that although this specification is described according to embodiments, not every embodiment only contains an independent technical solution. This narrative way of the specification is only for clarity. Those skilled in the art should regard the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A method for constructing a dataset of double-connected three-dimensional microstructures with the same boundary, characterized in that, It includes the following steps: Step S1: Determine four boundary types through test analysis, and reconstruct the microstructure with the four boundary types as the constraints for the six faces of the cube; Step S2: For each boundary type, use the LIVE3D framework, use maximizing the bulk modulus, maximizing the shear modulus, maximizing the Young's modulus, and the five-mode structure as the optimization objectives, and create an initial data set according to the corresponding constraint conditions; Step S3: Adopt generative AI for structure generation to expand the property space of the microstructure. The generative model uses a diffusion model, the network structure uses Unet, the loss function is the MSE loss. On this basis, use the initial data set in S2 to train the Unet network, and adopt an active learning strategy for iterative expansion of the data set; Step S4: Identify and retain the structures with double-connected features through the connected component labeling algorithm, remove the unconnected structures, and construct a data set with the same boundary and double-connectivity under the boundary type described in Step S1; The diffusion model in step S3 includes a forward noise addition process and a reverse denoising process. The forward noise addition process adds noise to the true data x0 of x step by step through the equation where ε is sampled from a standard Gaussian distribution, and the parameter α is used to control the level of noise. The reverse denoising process starts from the data x t close to pure noise, uses a neural network to predict the noise and indirectly recover x t-1 until the true data x0 is recovered; t which is close to pure noise, and uses a neural network to predict the noise and indirectly recover x y starting from the data x t-1 until the true data x0 is recovered. The loss function of the diffusion model in Step S3 trains the network according to the following relational expression, which is used to guide the neural network to learn to denoise the noisy data: where tr represents the time step in the diffusion process, t ∼ [1, T] means that the time step t is uniformly sampled from the time interval [1, T], x0 represents a true data point in the training data, q(x0) represents the probability distribution of the true data, x0 ∼ q(x0) means randomly sampling a data point x0 from q(x0), that is, selecting a sample from the training data as the input of the diffusion model, ε ∼ N(0, I) represents the noise sampled from the standard normal distribution with a mean of 0 and a covariance matrix of the identity matrix I, x t is the noisy data at time t in the diffusion process, and this noisy data is generated by adding noise to the true data x0, x θ (x t , t) represents the predicted value of the neural network model with parameter θ; In terms of the network structure described in step S3, both the U-Net encoder and decoder consist of 5 stages, and each stage is composed of a Resblock with an attention mechanism; using three independent components C 1111 , C 1122 , C 1212 in the boundary constraint and the elastic tensor matrix C as conditions, and introducing these conditions into each layer of the U-Net through the adaptive instance normalization technique; Step S3 is to construct a large-scale microstructure data set and adopt an active learning strategy for iterative training.

2. The method for constructing a double-connected three-dimensional microstructure dataset with the same boundary according to claim 1, wherein In the objective function of the optimization objective in Step S2, ρ is the density of the structure, and C(ρ) is the compliance when the structure density is ρ. The specific objective function is as follows: The objective function for maximizing the bulk modulus is: C 1111 And C 1122 are two components in the elastic tensor matrix C; The objective function for maximizing the shear modulus is: f(C(ρ)) = C 1212 , where C 1212 is a component in the elastic tensor matrix C; The objective function for maximizing the Young's modulus is as follows: The objective function of the five-module structure is as follows: where the bulk modulus K is defined by the formula and the shear modulus G is defined by the formula G = C 1212 where K0 and G0 are the initial values of the bulk modulus and the shear modulus obtained by random initial calculation, respectively.

3. The method for constructing a double-connected three-dimensional microstructure dataset with the same boundary according to claim 1, characterized in that The constraint conditions in Step S2 specifically refer to: Equilibrium equation constraint: K(ρ)u = f, where K(ρ) is the stiffness matrix, u is the displacement vector, and f is the applied external force; Volume fraction constraint: where M is the total number of discretized elements of the material, v e and ρ e are the volume fraction and density of the material at the e-th location respectively, and |Ω| represents the volume of the microstructure design domain. This constraint ensures that the total volume of the material does not exceed V during the optimization process; Boundary condition constraint: During the optimization process, ensure that boundary b1 is close to boundary b2, where ε is the allowable convergence error; Isotropic constraint: Poisson ratio constraint: where v0 is the specified initial Poisson ratio; Density variable constraint: 0 ≤ ρ e ≤ 1, M is the total number of discretized units of the material, that is, among all the discretized units, the material density ρ of any selected unit e is between [0, 1].

4. The method for constructing a double-connected three-dimensional microstructure dataset with the same boundary according to claim 1, wherein In the network training in Step S3, all microstructures are represented by voxels, that is, each structure is represented as a three-dimensional binary matrix of n×n×n. Due to the cubic symmetry of the structure, only the upper left 1 / 8 of the structure needs to be considered as the input during training. At the same time, due to symmetry, symmetric noise is also used in the forward and reverse diffusion processes.

5. The method for constructing a double-connected three-dimensional microstructure dataset with the same boundary according to claim 1, wherein Step S3 adopts a self-conditional strategy to improve the generation quality, that is, using the content currently generated by the network as the condition for the next generation.

6. The method for constructing a double-connected three-dimensional microstructure dataset with the same boundary according to claim 1, characterized in that The connected component labeling algorithm described in Step S4 specifically includes: The first scan: For each pixel point p located at coordinates (i, j), output the image label L(i, j). The calculation process is as follows: where N(p) represents the neighboring pixels of p. If there are multiple labels in N(p), select the smallest label and record these labels as equivalence relations; The second scan: Traverse each pixel point p in the image again to update the label to ensure that all equivalent labels are unified into one label: L(i, j) = find(L(i, j)), where the find function finds the smallest label equivalent to L(i, j) to achieve label unification.

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