Phononic crystal generative design method based on diffusion model and unstructured polygonal mesh

Abstract

A phononic crystal generative design method based on a diffusion model and an unstructured polygonal mesh belongs to the field of phononic crystal design, and comprises the following steps: firstly, generating phononic crystal unit cells, calculating a dispersion curve of the phononic crystal unit cells, and constructing a data set; secondly, constructing a diffusion model and designing a neural network to learn a reverse process of the diffusion model; inputting the training data into the neural network for training until a termination condition is met; and finally, extracting a dispersion curve from the test data set as a reverse design target, quickly generating a group of photonic crystals by using a neural network and a diffusion model sampling algorithm, calculating the dispersion curve corresponding to the generated photonic crystals, and calculating an error between the dispersion curve and the reverse design target. According to the method, the dependence of an existing generative design method on a rule design domain can be broken through, and the adaptability to irregular geometrical shapes and complex boundaries is improved; efficient learning of the complex mapping relation between the mechanical attributes and the geometric configuration is achieved; the method has the advantages of high design speed and high precision.

Description

Technical Field

[0001] The present invention belongs to the field of phononic crystal design, and relates to a generative design method of phononic crystals based on diffusion models and unstructured polygon meshes. Background Art

[0002] Vibration phenomena are widespread in the engineering field. Although they can be utilized in specific situations, in most engineering applications, they often have negative impacts, such as causing mechanical structure fatigue, reducing equipment performance, and even threatening system safety. Therefore, in the process of engineering design, manufacturing, and operation, how to effectively suppress and control vibration has become an important research topic. In recent years, phononic crystals, as a kind of periodically arranged artificial composite material or structure, with their periodic characteristics and material property differences, form band gaps in a specific frequency range to prevent the propagation of elastic waves, showing broad application prospects in the fields of vibration suppression and noise control. However, the dispersion curves and band gap characteristics of phononic crystals are highly affected by their structural design parameters, including the geometric shape of the unit structure, material properties, and arrangement patterns, etc. Therefore, how to achieve efficient vibration reduction and noise reduction effects through structural parameter optimization has become the core direction of current research.

[0003] Currently, the design methods of phononic crystals mainly include two strategies: forward design and inverse design. Forward design usually relies on empirical rules or heuristic methods, and tests different design schemes based on theoretical models or finite element numerical simulations, and conducts systematic scans within the preset parameter range. Although this method has achieved certain success to some extent, its design flexibility is limited, it is difficult to efficiently optimize specific functions, and it is difficult to adapt to the design of phononic crystals with complex requirements. In contrast, inverse design can directly search for the optimal structure that meets the requirements according to specific target responses (such as dispersion curves, band gap characteristics, etc.), thus significantly improving the design efficiency. Before the rise of machine learning technology, inverse phononic design mainly relied on gradient-based topology optimization methods (such as the variable density method, level set method, etc.) and gradient-free topology optimization methods. These methods optimize design parameters by minimizing or maximizing the objective function to achieve the target function. However, there are still problems such as low design efficiency and slow optimization convergence speed when facing complex design tasks with multiple constraint conditions. Therefore, developing efficient and intelligent inverse design methods is crucial for promoting the practical application of phononic metamaterials.

[0004] In recent years, with the rapid development of artificial intelligence, data-driven technologies, etc., they have been gradually applied to the design of phononic crystals. The data-driven metamaterial design methods can be classified into the following two types according to their design processes: iterative optimization design methods and non-iterative optimization methods. Iterative optimization design constructs a surrogate model using machine learning algorithms to quickly solve the complex structure response and solves the design problem through an optimization algorithm. However, as the degrees of freedom increase, the solution space expands significantly, and the number of iterations increases exponentially. Non-iterative optimization uses neural networks to realize the mapping from the target quantity of interest to the design scheme. In particular, the generative design method uses deep generative models (such as variational autoencoders, generative adversarial networks, or diffusion models) to generate novel topological structures. Through this method, the design process does not require complex iterative processes, can efficiently explore a vast design space, and achieve innovative designs. Especially when facing the diversity and complexity that are difficult to solve by traditional methods, the generative method shows great potential. However, the above-mentioned generative design methods mainly rely on dividing the design domain into structured uniform grids. However, in the face of complex design domains, directly discretizing the design domain into structured grids becomes challenging. In contrast, unstructured grids are more suitable for dealing with complex geometries and boundaries, and can flexibly handle irregular objects and surfaces. In addition, unstructured grids are easier to generate in complex geometries and have fewer imposed constraints.

[0005] For complex design domains, the present invention proposes a data-driven generative design method. This method combines a diffusion model with unstructured polygon grids, and is specifically used for design tasks dealing with irregular geometries and complex boundary conditions, significantly improving the geometric adaptability of generative design. First, to ensure geometric consistency and flexibly adapt to the boundaries of complex design domains, while avoiding the common node problems of triangular grids, DiffUPM discretizes the design domain using unstructured polygon grids, thereby providing higher geometric flexibility and numerical stability. In addition, since the discretization method of polygon grids breaks the regular arrangement of traditional structured grids, its generated variables no longer follow the conventional arrangement in Euclidean space, resulting in non-Euclidean characteristics of the design variables. For this reason, this study introduces a model system based on Diffusion Transformer (DiT) to learn the conditional diffusion model, thereby effectively modeling the structural distribution corresponding to specific mechanical properties and quickly mapping these properties to potential geometric configurations. This method breaks through the applicability limitations of existing generative design methods in irregular design domains, providing a new idea for the optimization and innovation of complex geometric structures. Summary of the Invention

[0006] Aiming at the problems existing in the prior art, the present invention proposes a phonon crystal design method combining a diffusion model and an unstructured polygonal mesh, aiming to solve the challenges of generative design in complex design domains, simplify the design process of phonon crystals, and provide a new solution for realizing more intelligent and customized inverse design of phonon crystals.

[0007] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0008] A generative design method of phonon crystals based on a diffusion model and an unstructured polygonal mesh, comprising the following steps:

[0009] In the first step, generate a phonon crystal unit cell and calculate its dispersion curve to construct a data set required for training a neural network. Specifically as follows:

[0010] Step 1.1: Divide a complex design domain containing irregular voids into an unstructured polygonal mesh based on Voronoi tessellation;

[0011] Step 1.2: Based on the unstructured polygonal mesh generated in Step 1.1, use a topology optimization method based on the variable density method and homogenization to generate as many phonon crystal unit cells with connectivity and p4m symmetry (plane symmetry, four-fold rotation axis, and mirror symmetry) as possible, and perform binarization processing on the generated phonon crystal unit cells, where 1 represents the presence of material and 0 represents the absence of material;

[0012] Step 1.3: Use finite element software to perform simulation on the phonon crystal unit cell obtained in Step 1.2. The reference material is a linear elastic material, and Floquet periodic boundary conditions are set. For the phonon crystal unit cell satisfying p4m symmetry, only consider the wave vector k located on the boundary of the irreducible Brillouin zone to obtain the dispersion curve corresponding to the phonon crystal unit cell;

[0013] Step 1.4: Expand the dispersion curve obtained in Step 1.4 into a one-dimensional vector and perform 0-1 normalization to eliminate the problem that the neural network is biased towards a certain feature due to the differences between the dimensions of the dispersion curve. Use the phonon crystal unit cell and its normalized dispersion curve as training data, and divide them into a training set and a validation set, where 90% of the data is used for training and the remaining 10% is used for validation;

[0014] In the second step, construct a diffusion model and design a neural network to learn the reverse process of the diffusion model. Input the training data into the neural network for training until the termination condition is met. Specifically as follows:

[0015] Step 2.1: The diffusion model includes a forward process and a reverse process. In the forward process, appropriate numbers of time steps and variance scheduling functions can be pre-selected; in the reverse process, a neural network model based on Diffusion Transformer needs to be constructed, and the reverse process is learned by minimizing the training objective as follows:

[0016]

[0017] where, ∥·∥1 represents the L1 norm; θ represents the neural network parameters; represents all mathematical expectations; represents the optimized design variables; is the design variable after adding noise; represents the set composed of seed points, N is the number of design variables, d is the spatial dimension; t is the diffusion time step, is the noise sampled from the normal distribution.

[0018] Step 2.2: Using the training data obtained in the first step, input it into the diffusion model for iterative training. A total of 500 epochs of iterative training are performed to obtain a trained neural network.

[0019] Third step, use the trained neural network for the inverse design of phononic crystals. Specifically as follows:

[0020] Step 3.1: Extract the dispersion curve from the test dataset obtained in Step 1.4 as the inverse design target. Use the neural network trained in Step 2.2, and quickly generate a group of phononic crystals through the diffusion model sampling algorithm. Calculate the dispersion curve corresponding to the generated phononic crystals using finite element software, and use the following formula to measure the error ε between the dispersion curve corresponding to the generated phononic crystals and the inverse design target r :

[0021]

[0022] where, |·| represents the calculation of the absolute value, and respectively represent the i-th feature of the target dispersion curve and the generated dispersion curve; K represents the dimension of the dispersion curve. For the 5000 groups of data in the test dataset, the average error is calculated to be 3.94%, and phononic crystals with the properties of the target dispersion curve can be generated.

[0023] Step 3.2: Verify the generalization ability and application potential of the present invention in bandgap regulation. Through three adjustment strategies of band compression (i.e., reducing the frequency proportionally), band stretching (i.e., increasing the frequency proportionally), and band translation, re-customize the existing dispersion curve, and use the re-customized dispersion curve as the target of inverse design to quickly generate a new set of phononic crystals. Then use finite element software to calculate the error between the dispersion curve corresponding to the generated phononic crystal and the re-customized dispersion curve. Through calculation, the obtained error is less than 5.5%, and phononic crystals with customized dispersion curve attributes can be generated.

[0024] The beneficial effects of the present invention are as follows:

[0025] (1) The design method proposed by the present invention can quickly generate phononic crystals with a target dispersion curve through the trained neural network and the diffusion model sampling algorithm under a given desired dispersion curve;

[0026] (2) The design method proposed by the present invention discretizes the complex design domain using unstructured polygon meshes, breaks through the dependence of existing generative design methods on regular design domains, and significantly improves the adaptability to irregular geometries and complex boundaries;

[0027] (3) The design method proposed by the present invention introduces a neural network based on Diffusion Transformer to process non-Euclidean geometry data, and realizes efficient learning of the complex mapping relationship between mechanical properties and geometric configurations;

[0028] In summary, the present invention has the advantages of fast design speed and high precision, and provides a new solution for realizing more intelligent and customized inverse design of phononic crystals under complex design domains. Description of the Drawings

[0029] Figure 1 Schematic diagram of a complex design domain and an unstructured polygon mesh; Figure 1 (a) Complex design domain; Figure 1 (b) Unstructured polygon mesh based on Voronoi tessellation;

[0030] Figure 2 Schematic diagram of different unit cells of phononic crystals;

[0031] Figure 3 For a square lattice, where blue represents the irreducible Brillouin zone, and the wave vector k is sampled along the edge M-Γ-X-M of the irreducible Brillouin zone;

[0032] Figure 4 Schematic diagram of some phononic crystal microstructures and dispersion curves in the training dataset;

[0033] Figure 5Schematic diagram of the modified Diffusion Transformer network architecture;

[0034] Figure 6 Variation diagram of the neural network training loss function; Figure 6 (a) shows the variation of the loss function during 500 training cycles; Figure 6 (b) is Figure 6 (a) Magnified view of the red box area, further showing the variation of the loss function during 460 - 500 cycle iterations;

[0035] Figure 7 Example 1 of the inverse design of the dispersion curve in the test set;

[0036] Figure 8 Example 2 of the inverse design of the dispersion curve in the test set;

[0037] Figure 9 Example 1 of the inverse design of the re - customized dispersion curve;

[0038] Figure 10 Example 2 of the inverse design of the re - customized dispersion curve;

[0039] Figure 11 Flowchart of the method of the present invention. Detailed implementation manners

[0040] In order to make the purpose, technical solutions and advantages of the present invention clearer and more understandable, the technical solutions of the present invention will be described in detail below in combination with theoretical formulas, drawings and specific implementation cases. The technical solutions adopted by the present invention are illustrated by the following examples:

[0041] A generative design method of phononic crystals based on a diffusion model and an unstructured polygon mesh, comprising the following steps:

[0042] Step 1.1:

[0043] As Figure 1 shown, the length and width of the design domain are both l = 10 mm, and it has p4m symmetry (plane symmetry, four - fold rotation axis and mirror symmetry), and its basic repeating unit accounts for one - eighth of the total image area. During the mesh generation process, first, an unstructured polygon mesh based on Voronoi tessellation is constructed within this basic repeating unit, and an initial division is generated using 1024 seed points P. Subsequently, the seed points are optimized by the Lloyd algorithm to gradually converge to a centroid Voronoi dissection, thereby improving the consistency and uniformity of the mesh (see Figure 1 ), and its mathematical expression is as follows:

[0044]

[0045] where p i is the i-th seed point; represents the set of seed points, N = 1024 is the number of seeds, d is the spatial dimension, and in this paper, for a two-dimensional plane, d = 2; represents the Lloyd algorithm acting on the seed points P. For each seed point p i the Lloyd algorithm on can be expressed as:

[0046]

[0047] where is the coordinate of point p i and μ(x) is a given density function defined on the domain Ω. The Lloyd algorithm starts from the initial point set P1 and in each iteration step k, updates the point set through the formula ; V i represents the area of the Voronoi cell corresponding to p i .

[0048] When the seed points P reach a steady state, that is, when they satisfy , these seed points P then form a set of centroid Voronoi tessellations.

[0049] Step 1.2:

[0050] To ensure the connectivity of the topological structure of the phononic crystal, a series of phononic crystal unit cells that can meet the requirements of the load transfer path are generated through the surrogate task of static analysis. During the design process, a topology optimization method based on SIMP is adopted and combined with the homogenization method to generate unit cells under periodic boundary conditions. The specific formula is as follows:

[0051] find: ρ = [ρ1 ρ2 … ρ N

[0052]

[0053] where ρ is the vector of all design variables, ρ e represents the e-th design variable, N represents all design variables; J is the objective function, which is defined by the combination of the homogenized elastic tensor ; K is the global stiffness matrix obtained by finite element discretization, F is the external load vector, and U is the displacement vector under periodic boundary conditions; V e represents the element volume, V represents the volume constraint of the structure, and its upper limit is V max .

[0054] The Young's modulus E of the artificial material e is represented by a power-law interpolation function based on the mapped material field: ​

[0055]

[0056] Among them, E0 is the Young's modulus of the fully solid material, p is the penalty factor, and E min is a small positive value used to avoid stiffness singularities in finite element analysis.

[0057] By adjusting the initial material layout and optimizing the objective function, and strictly enforcing the p4m symmetry constraint during the optimization iteration process, 50,000 phonon crystal unit cells with p4m symmetry were systematically generated. In addition, a threshold processing method was used to binarize these structures, Figure 2 showing some of the phonon crystal unit cells.

[0058] Step 1.3: In this embodiment, the commercial finite element software COMSOL 6.2 is used as the solver to perform wave analysis on the phonon crystal, and MATLAB is used for large-scale sample calculations. Specifically, the lattice constant a of the two-dimensional phonon crystal unit cell is 10 mm, and Floquet periodic boundary conditions are applied to accurately capture the wave propagation behavior within the periodic structure. The material model uses a linear elastic material with a Young's modulus E of 0.3 MPa, a Poisson's ratio of 0.49, and a mass density of 1050 kg / m 3 . For the phonon crystal unit cell with p4m symmetry, only the wave vectors k located on the boundary of the irreducible Brillouin zone are considered, and this region is Figure 3 marked in blue in Figure 4 . During the calculation, 30 wave vectors k are uniformly sampled along the M-Γ-X-M path, and their first 10 eigenfrequencies are calculated to derive the dispersion curve. Some of the phonon crystals and their corresponding dispersion curves are as

[0059] shown in

[0060] To avoid bias in neural network training caused by scale differences in different dimensions of the dispersion curve, the dispersion curve is unfolded into a one-dimensional vector and 0-1 normalization processing is performed. The formula is as follows:

[0061]

[0062] Among them, and are the maximum and minimum values of the i-th dimension of the dispersion curve, respectively. is the i-th dimension after normalization, and its value range is between [0,1]. represents the dispersion curve, represents the unique dispersion curve.

[0063] Furthermore, 90% of the data of the phononic crystal structure and its normalized dispersion curve are used for training to ensure that the neural network fully learns the mapping relationship between the phononic crystal and the dispersion curve; the remaining 10% of the data are used for verification to evaluate the generalization ability and prediction accuracy of the neural network.

[0064] Step 2.1:

[0065] For the forward process of the diffusion model, since there are no trainable parameters, set the number of diffusion steps T = 1000, and set the variance scheduling function to linearly increase from β1 = 10 -4 to β T = 0.02.

[0066]

[0067] where ρ0 is the unnoisy phononic crystal, ρ t is the noisy phononic crystal, t is the diffusion step, is the sampled noise, and β t is the variance at the t-th diffusion step.

[0068] For the reverse process of the diffusion model, use a neural network based on the improved Diffusion Transformer, as Figure 5 shown. This neural network architecture uses the self-attention mechanism to capture long-range dependencies and integrates conditional information through the adaptive layer normalization (adaLN) module, enabling the rapid generation of phononic crystals that satisfy the target dispersion curve. This model architecture consists of three key modules: the embedding module, the DiT encoder, and the decoder. The embedding module first maps the low-dimensional input to a high-dimensional space using multi-layer perceptrons (MLPs) Next, add the noise embedding ρ τ to the position embedding P τ to obtain the generated feature embedding Similarly, fuse the time embedding t τ with the dispersion curve embedding y τ to form the conditional embedding The DiT encoder consists of six DiT blocks for processing features and integrating mechanical information. Through the self-attention mechanism, these blocks establish global dependencies, enabling effective feature processing and extraction, and its mathematical representation is as follows:

[0069]

[0070] where W Q , W K and are learnable projection matrices; d represents the dimension after passing through the projection matrix; Q, K, and A vector representing the output after passing through the projection matrix.

[0071] In addition, the adaLN module is used to integrate conditional information. The c is processed through a multi-layer perceptron to dynamically generate normalization parameters (i.e., the scaling parameter γ and the offset parameter β) to enhance the adaptive learning ability, which is defined as follows: τ For the element-wise multiplication (also known as the Hadamard product), it is defined as follows:

[0072]

[0073] Where, and represent the feature embeddings of the i-th and (i + 1)-th layers respectively, represents the element-wise multiplication (also known as the Hadamard product).

[0074] Figure 5 The dimensional scaling parameter α in is also obtained by MLP regression and is applied immediately before any residual connection in the DiT block. All linear layers in adaLN are initialized to zero vectors. After the last DiT block, in this embodiment, a decoder is used to decode the noise features into the output noise prediction ∈ θ , ensuring that the shape of the output matches the input noisy unit cell ρ t . This neural network has a total of 6.3 million trainable parameters. The training process minimizes the optimization objective The mathematical definition is as follows:

[0075]

[0076] Where, ∥·∥1 represents the L1 norm; θ represents the neural network parameters; represents all mathematical expectations; represents the optimized design variables; is the noisy design variable; represents the set composed of seed points, N is the number of design variables, d is the space dimension; t is the diffusion time step, is the noise sampled from the normal distribution.

[0077] Step 2.2:

[0078] Use the Adam optimizer and set the initial learning rate to 2×10 -4 . In addition, a learning rate scheduler is adopted. Every 10 iterations, the learning rate decays to 0.98 of the previous value. The neural network is trained for a total of 500 iteration rounds. The convergence history of the loss function is as Figure 6 shown.

[0079] Step 3.1:

[0080] By selecting 5000 dispersion curves from the test set as the inverse design targets, using the trained neural network, and adopting the diffusion model sampling algorithm to generate the corresponding phononic crystals according to the design targets, and putting the generated phononic crystals into the commercial finite element software COMSOL6.2 for verification, with the error ε r used as a metric to quantitatively evaluate the generated dispersion curve (the dispersion curve corresponding to the generated phononic crystal) and the target dispersion curve:

[0081]

[0082] where |·| represents the calculation of the absolute value, and represent the i-th feature of the target dispersion curve and the generated dispersion curve respectively; K represents the dimension of the dispersion curve.

[0083] The average error ε r between the 5000 generated dispersion curves and the target dispersion curve is 3.94%, indicating a high degree of agreement between the generated dispersion curve and the target dispersion curve, and proving that the proposed method can successfully generate phononic crystals that match the given dispersion curve.

[0084] In addition, we selected two representative cases from the test set and detailedly showed the phononic crystals and dispersion curves corresponding to the minimum and median errors ε r in each case, as Figure 7 and Figure 8 shown. The proposed method can accurately generate structures with similar geometries, and the dispersion curves of these generated phononic crystals highly match the target dispersion curves in terms of frequency values, overall trends, and local features. Interestingly, the generated phononic crystals exhibit significant diversity, which further proves that the method can quickly design phononic crystals with different geometries according to the given dispersion curve.

[0085] Step 3.2:

[0086] Further explored the generalization ability and application potential of the proposed framework in bandgap regulation. For this purpose, the existing dispersion curves were re-customized and used as the inverse design targets. Since it is difficult to precisely control the frequency points in the dispersion curve, three adjustment strategies were adopted: band compression (decreasing the frequency proportionally), band stretching (increasing the frequency proportionally), and band translation to achieve the regulation of the bandgap. Specifically:

[0087] Re-customized the two dispersion curves in Step 3.1. As Figure 9As shown, 30% compression is applied to frequency bands 1 to 3, while frequency bands 4 to 10 are gradually shifted upward by 150 Hz. This adjustment effectively introduces a new bandgap between frequency bands 3 and 4, with a frequency range of 381 Hz to 878 Hz. Similarly, Figure 10 shows the expansion of the bandgap. In Figure 9 and 10 , the first column of each case shows the target dispersion curve, the second column shows the generated dispersion curve, and the third column shows the corresponding generated phononic crystal.

[0088] As Figure 9 and 10 shown, the error between the generated dispersion curve and the target dispersion curve is less than 8%. The results show that the proposed method can effectively explore new types of phononic crystals and quickly generate structures consistent with the target bandgap, whether it is the expansion or merger of the bandgap. Although the training conditions are based on the given dispersion curve rather than the bandgap, the generated dispersion curve has a high degree of consistency with the target dispersion curve in terms of the overall trend and bandgap range, improving the flexibility and efficiency of bandgap adjustment. In addition, as the bandgap gradually expands or merges, the generated phononic crystals exhibit obvious continuous and consistent changes, highlighting the stability and reliability of the method in precise adjustment and complex design scenarios. It should be noted that the dispersion curve customized by the method proposed in this paper is idealized and may exceed the design space or violate physical laws. Therefore, the network does not always generate feasible structures that meet our requirements.

[0089] The embodiments described above only represent the implementation modes of the present invention, but should not be construed as limiting the scope of the patent of the present invention. It should be pointed out that for those skilled in the art, without departing from the concept of the present invention, several deformations and improvements can still be made, and these all belong to the protection scope of the present invention.

Claims

1. A generative design method for phononic crystals based on diffusion models and unstructured polygonal meshes, characterized in that It includes the following steps: In the first step, generate a phonon crystal unit cell and calculate its dispersion curve to construct the dataset required for training the neural network. Specifically as follows: Divide the complex design domain into unstructured polygon meshes and generate multiple phonon crystal unit cells. Then, perform simulation on them to obtain the dispersion curves corresponding to the phonon crystal unit cells. Finally, after processing the dispersion curves, use them as training data and divide them into a training dataset and a validation dataset. In the second step, construct a diffusion model and design a neural network to learn the reverse process of the diffusion model. Input the training data into the neural network for training until the termination condition is met. Specifically as follows: Step 2.1: The diffusion model includes a forward process and a reverse process. For the forward process, appropriate time steps and variance scheduling functions can be pre-selected. For the reverse process, a neural network model based on Diffusion Transformer needs to be constructed, and the reverse process is learned by minimizing the training objective. Step 2.2: Use the training data obtained in the first step, input it into the diffusion model for iterative training to obtain a trained neural network. In the third step, use the trained neural network for inverse design of phonon crystals. Specifically as follows: Extract the dispersion curve from the test dataset in the first step as the inverse design target. Use the neural network trained in the second step and quickly generate a group of phonon crystals through the diffusion model sampling algorithm. Calculate the dispersion curve corresponding to the generated phonon crystals by finite element software, and through error verification, phonon crystals with the properties of the target dispersion curve can be generated.

2. The generative design method of phononic crystal based on diffusion model and unstructured polygon mesh according to claim 1, characterized in that, The specific content of the first step is as follows: Step 1.1: Divide the complex design domain with irregular voids into unstructured polygon meshes based on Voronoi tessellation. Step 1.2: Based on the unstructured polygon meshes generated in Step 1.1, use the topology optimization method based on the variable density method and homogenization to generate as many phonon crystal unit cells with connectivity and p4m symmetry (plane symmetry, four-fold rotation axis, and mirror symmetry) as possible, and perform binarization processing on the generated phonon crystal unit cells, where 1 represents the existence of materials and 0 represents the non-existence of materials. Step 1.3: Use finite element software to perform simulation on the phonon crystal unit cells obtained in Step 1.

2. The reference material is a linear elastic material, and the Floquet periodic boundary condition is set. For the phonon crystal unit cells with p4m symmetry, only consider the wave vector k located on the boundary of the irreducible Brillouin zone to obtain the dispersion curve corresponding to the phonon crystal unit cell. Step 1.4: Expand the dispersion curve obtained in Step 1.4 into a one-dimensional vector and perform 0-1 normalization. Use the phonon crystal unit cell and its normalized dispersion curve as training data, where 90% of the data is divided into the training dataset for training, and the remaining 10% is divided into the validation dataset for validation.

3. The generative design method of phononic crystals based on diffusion model and unstructured polygonal meshes according to claim 2, characterized in that, In Step 1.3, the p4m symmetry includes plane symmetry, four-fold rotation axis, and mirror symmetry.

4. A phonon crystal generative design method based on a diffusion model and an unstructured polygon mesh according to claim 2, characterized in that The mathematical formula of Step 2.1 is expressed as follows: Among them, ||·||1 represents the L1 norm; θ represents the neural network parameters; represents all mathematical expectations; represents the optimized design variables; is the design variable after adding noise; represents the set composed of seed points, N is the number of design variables, d is the space dimension; t is the diffusion time step, is the noise sampled from the normal distribution.

5. A phonon crystal generative design method based on a diffusion model and an unstructured polygon mesh according to claim 2, characterized in that In the third step, the error ε between the dispersion curve corresponding to the generated phononic crystal and the inverse design target is measured by formula (11). r : where |·| represents the calculation of the absolute value, and respectively represent the i-th feature of the target dispersion curve and the generated dispersion curve; K represents the dimension of the dispersion curve; Redefine the dispersion curve through three adjustment strategies, and use the redefined dispersion curve as the target of inverse design to generate a new set of phononic crystals. Then, use finite element software to calculate the error between the dispersion curve corresponding to the generated phononic crystals and the redefined dispersion curve. Through calculation, it is found that the error is less than 5.5%, indicating that phononic crystals with customized dispersion curve properties can be generated.

6. A phonon crystal generative design method based on a diffusion model and an unstructured polygon mesh according to claim 2, characterized in that The three adjustment strategies include band compression, band stretching, and band translation.

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