Inverse design method for energy-absorbing metamaterials based on diffusion model and parameterized level set

Abstract

This invention belongs to the field of energy-absorbing metamaterial design, specifically to a method for inverse design of energy-absorbing metamaterials based on a diffusion model and parameterized level sets. The invention generates a metamaterial given a desired stress-strain curve. The diffusion model learns the conditional distribution of metamaterials with given properties, achieving a one-to-many mapping from properties to geometry. The use of a level set representation improves the precision and design flexibility of the metamaterial topology, ensuring that the generated metamaterial possesses high-quality geometry. This invention offers the advantages of fast design speed and high precision, providing a new solution path for more intelligent and customized metamaterial design.

Description

Technical Field

[0001] The present invention belongs to the field of energy-absorbing metamaterial design, and in particular relates to an energy-absorbing metamaterial inverse design method based on a diffusion model and a parameterized level set. Background Art

[0002] Energy-absorbing materials and structures are widely used in aerospace, transportation, and personal protection due to their excellent impact resistance. When designing the ideal energy-absorbing metamaterial, collision avoidance designers consider multiple criteria, including not only high energy absorption performance but also structural stability and safety. With the desired energy absorption curve as the design goal and the base material determined, the distribution of the material within the structure becomes a key design variable. Material distribution plays a crucial role in optimizing the structure's load-displacement response and energy absorption characteristics. By carefully manipulating material distribution, designers can tailor structural behavior and enhance energy absorption performance to achieve specific design goals and meet desired performance standards.

[0003] Forward design methods for energy-absorbing structures primarily use empirical or biomimetic configurations to achieve varying energy absorption performance. Classic energy-absorbing structures include sandwich structures, plate structures, honeycomb structures, and foam structures. Bionic design, on the other hand, mimics the internal structure of living organisms to create structures with energy-absorbing properties. For example, the design of tubular nested geometries inspired by the Haversian bone structure can be used. To improve a structure's energy absorption capacity, geometric parameters are modified to develop highly efficient energy-absorbing structures.

[0004] Inverse design methods for energy-absorbing structures can be categorized into gradient-based and gradient-free methods. Gradient-based topology optimization design methods effectively meet design objectives without strict material requirements. This approach simplifies the design process and improves efficiency. This mature field encompasses a variety of methods, including the solid isotropic material penalty method (SIMP), the level set method (LSM), the parameterized level set method (PLSM), the bidirectional evolutionary structural optimization method (BESO), and the mobile deformable component method (MMC). These methods require rigorous sensitivity derivation and fast convergence based on finite element analysis. Because the topology optimization of energy-absorbing metamaterials must account for geometric, material, and contact nonlinearities, gradient-based topology optimization methods face significant challenges in this area. In contrast, gradient-free design methods based on intelligent algorithms have demonstrated promising performance in solving complex multi-objective, non-convex, or time-varying optimization problems that are difficult for sensitivity-based methods. The flexible model representation of intelligent algorithms improves convergence. However, due to the lack of sensitivity guidance, the optimization efficiency of these algorithms is often lower than that of gradient-based algorithms, especially as the number of design variables increases.

[0005] With the rapid development of artificial intelligence (AI) technology, AI-generated content (AIGC) has had a significant impact on multiple industries, including entertainment, media, and education. Well-known AIGC tools include ChatGPT and Sora. AIGC has attracted widespread attention beyond computer science. In the field of materials design, advances in the data-driven paradigm have greatly accelerated the exploration of inverse materials design methods. From the perspective of data representation, data-driven methods for inverse design of mechanical materials can be divided into two major categories. The first category uses parametric models to represent materials and establishes surrogate models to explore design parameters and achieve desired goals. In this category, the material structure is typically converted into a set of topological parameters, enabling efficient exploration of design parameters through the optimization capabilities of machine learning algorithms. The second category of data-driven design methods focuses on designing within a high-dimensional topological space rather than directly designing topological parameters. A common strategy is to partition the topological space into a series of pixels. However, these works discretize the design domain into variable pixel representations, resulting in jagged edges on the final topological structure, requiring further smoothing for additive manufacturing. Furthermore, the application of AI generation to the design of energy-absorbing metamaterials has not yet been considered. Furthermore, these methods have limited experimental validation, which affects their effectiveness and practicality in real-world applications.

[0006] This paper proposes a data-driven inverse design method for energy-absorbing metamaterials based on a diffusion model and a parameterized level set method. This method can generate metamaterials with smooth, well-defined boundaries based on a desired stress-strain curve. The core of the method is the diffusion model, which is adapted to learn the conditional distribution of metamaterials corresponding to the target stress-strain curve. This model allows for a one-to-many mapping capability, translating the desired mechanical properties into a set of different potential metamaterial topologies. This method, combined with a parameterized level set function, generates topological structures with clear boundaries. This method ensures the accuracy and efficiency of energy-absorbing metamaterial design and highlights the broad implications of integrating AI-generated methods into material and structural design. Summary of the Invention

[0007] The purpose of the present invention is to propose a design method for energy-absorbing metamaterials based on a diffusion model and parameterized level sets, aiming to overcome the difficulty of traditional methods in solving strong nonlinear problems, simplify the design process of energy-absorbing metamaterials, and provide a new solution path for achieving more intelligent and customized metamaterial design.

[0008] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0009] The reverse design method of energy-absorbing metamaterials based on diffusion model and parameterized level set is as follows:

[0010] Step 1.1: Generate as many metamaterials as possible using a topology optimization method based on homogenization and parameterized level set methods. Use a density threshold method to identify boundaries and use the Ramer-Douglas-Peucker (RDP) algorithm to effectively reduce the detected boundary data points to obtain a clear and accurate geometric representation for finite element simulation.

[0011] Step 1.2: To ensure the accuracy of the simulation results, a uniaxial tensile test of 304 stainless steel is conducted for constitutive parameter calibration. In combination with the theory of plasticity, the least squares method is used to identify the material parameters.

[0012] Step 1.3: Use finite element software to simulate the metamaterial obtained through topology optimization. The metamaterial is placed between two rigid plates. To prevent interpenetration, the interaction between the plates and the structure, as well as the self-interaction within the structure, is considered. A vertical displacement is set in the y-direction to simulate the compression process in a quasi-static manner, and the stress-strain curve of the structure is obtained.

[0013] Step 1.4: Use principal component analysis to reduce the dimensionality of the simulated stress-strain curves. By finding the principal components, the data complexity is reduced. The reduced data is then normalized to eliminate the problem of the model favoring a particular feature due to differences between features. The metamaterial and its reduced energy absorption curves are used as training data, with 90% of the generated data used for training and the remaining 10% for validation.

[0014] Step 2: Construct a diffusion model. The neural network architecture uses the U-Net network, initialize the network parameters, and use the metamaterial and its dimensionality-reduced energy absorption curve as training data for iterative training in the diffusion model.

[0015] Step 3: Given a target stress-strain curve for a trained diffusion model, a family of metamaterials with target mechanical properties can be quickly generated based on the complex nonlinear mapping relationships learned by the diffusion model.

[0016] The beneficial effects of the present invention are:

[0017] This paper proposes a design method for energy-absorbing metamaterials based on a diffusion model and a parameterized level set method. This method generates metamaterials given a desired stress-strain curve. The diffusion model learns the conditional distribution of metamaterials with given properties, achieving a one-to-many mapping from properties to geometry.

[0018] 2. The level set representation is used to improve the accuracy and design flexibility of metamaterial topology, ensuring that the generated metamaterial has high-quality geometry.

[0019] 3. In summary, the present invention has the advantages of fast design speed and high precision, and provides a new solution path for achieving more intelligent and customized metamaterial design. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1(a) shows the compactly supported radial basis function of C2 continuity;

[0021] Figure 1(b) shows the partial derivatives of the C2-continuous compactly supported radial basis functions along the x-direction;

[0022] Figure 2(a) shows the design variable α;

[0023] Figure 2(b) shows the zero level set Φ of the two-dimensional level set function;

[0024] Figure 2(c) shows the clear geometric boundary obtained by the density contour method;

[0025] Figure 3(a) is a drawing of the test piece;

[0026] Figure 3(b) shows the test loading conditions;

[0027] Figure 3(c) is the true stress-strain curve.

[0028] Figure 4(a) shows the load condition;

[0029] Figure 4(b) shows the finite element simulation;

[0030] Figure 4(c) shows the nominal stress-strain curve;

[0031] Figure 5(a) shows the cumulative percentage of the principal components;

[0032] Figure 5(b) shows the distribution of stress data in the first and second principal component projections.

[0033] Figure 6 Schematic diagram of the diffusion model neural network;

[0034] Figure 7(a) is the test curve;

[0035] Figure 7(b) shows the error between the test curve and the generated curve Box plot of ;

[0036] Figure 8 (a) is test condition 1; Figure 8 (b) is test condition 2; Figure 8 (c) is test condition 3; Figure 8 (d) is test condition 4;

[0037] Figure 9(a) shows the interpolation curve;

[0038] Figure 9(b) shows the error between the interpolation curve and the generated curve Box plot of ;

[0039] Figure 10 (a) is the interpolation curve 1; Figure 10 Middle (b) is interpolation curve 2; Figure 10 Middle (c) is interpolation curve 3; Figure 10 Middle (d) is interpolation curve 4;

[0040] Figure 11 (a) is the interpolation curve 1; Figure 11 Middle (b) is interpolation curve 2; Figure 11 Middle (c) is interpolation curve 3; Figure 11 Middle (d) is interpolation curve 4. DETAILED DESCRIPTION

[0041] In order to make the purpose, technical solution and advantages of the present invention more clear, the technical solution of the present invention is described in detail below with reference to theoretical formulas, drawings and specific implementation cases. The technical solution adopted by the present invention is described below by examples:

[0042] Step 1.1: Introduce the process of generating the metamaterial topology dataset for subsequent finite element simulation. The mathematical expression of the topology optimization method based on homogenization and parameterized level set method is as follows:

[0043] find:α=[α1 α2…α N ]

[0044]

[0045] Among them, α is the design variable, α L and α U are the lower and upper bounds of the design variables, respectively, and J is the homogenized elastic tensor E H The objective function is defined by the combination of , V is the volume constraint, and its upper limit is V max Φ is a parameterized level set function, which is approximated using a set of radial basis functions (RBFs) at each node in the entire design domain and is expressed as:

[0046]

[0047] The radial basis function sequence at each node is represented as a vector and is given as follows:

[0048]

[0049] In the present invention, a compactly supported radial basis function with C2 continuity is used, as shown in Figure 1(a) and Figure 1(b).

[0050] As shown, the formula is as follows:

[0051] φ(r)=(max(0,1-r)) 4 …(4r+1)

[0052] Where r is the size of the domain of influence of the basis function at the node in the design domain. Therefore, the parameterized level set function Φ is represented by the design variable α.

[0053] H is the Heaviside function, which is expressed as:

[0054]

[0055] Where η is a small positive number introduced in the numerical process to avoid singularities, and the width of the numerical approximation of Δ is used. The elastic equilibrium equation is expressed in weak variational form, where the bilinear energy term a(u, v, Φ) and the linear load term l(v, Φ) can be expressed as:

[0056] a(u,v)=∫ Ω H(Φ)σ(u):ε(v)dΩ

[0057]

[0058] Where f represents body force, t represents surface force, and virtual displacement v belongs to the allowed displacement space of motion.

[0059] By varying the initial material topology and volume constraints, we systematically generated 30,000 metamaterials with volume fractions ranging from 28% to 52%. In this study, the design variable α is uniformly distributed in the two-dimensional design domain, with 64 design variables in both the horizontal and vertical dimensions. Figure 2(a) shows the value of the design variable a at each node. Figure 2(b) shows the zero-level set corresponding to the level set function Φ. Boundary detection is performed using the density contour method, and the Ramer-Douglas-Peucker (RDP) algorithm is used to effectively reduce the detected boundary data points for finite element geometry modeling, as shown in Figure 2(c).

[0060] Step 1.2: To determine the mechanical properties of 304 stainless steel (304SS), a tensile specimen was designed according to GB / T 228.1-2021

[59] . The tensile diagram of the specimen is shown in Figure 3(a), in mm.

[0061] The tensile test was carried out using a tensile testing machine with the chuck displacement speed set at 2 mm / min (strain rate 6.67×10 -5 s -1 ), which is consistent with the quasi-static tensile process (the strain rate is roughly 10 -5 s -1 ~10 -1 s -1The specimen installation method is shown in Figure 3(b) to ensure accurate force-displacement data for the 304SS material during the test. Throughout the tensile test, a high-precision extensometer was used to measure the axial displacement of the specimen. The collected force-displacement data was processed to obtain the true stress-strain curve:

[0062] ε=ln(1+ε N )

[0063] σ=σ N (1+ε N )

[0064] Where, ε N and σ N where ε and σ represent the normal strain and stress, respectively. The true stress-strain relationship is shown by the blue line in Figure 3(c). The elastic modulus E is determined by the slope of the elastic phase curve shown in Figure 3(c) and can be approximately 185 GPa. The Poisson's ratio is 0.3. The present invention employs the isotropic hardening von Mises yield function and the associated flow law. The hardening law takes the following form:

[0065] σ y =A+B(ε p ) n

[0066] where σ y is the yield stress, ε p is the equivalent plastic strain. Where A is the initial yield stress, B is the hardening constant, and m is the dimensionless hardening exponent. According to the ASTM E8M-04 standard, the initial yield strength A was determined to be 344 MPa using the bias method (bias value is 0.2%). The least squares method was used to obtain B as 654 MPa and n as 0.61. The fitted stress-strain curve is shown in the red line in Figure 3(c). For dynamic simulation, the density was selected as 7.85×10 -9 tonne / mm 3 .

[0067] Table 1 lists all the material parameters used in the simulation

[0068]

[0069] Step 1.3: To minimize computational costs and fully demonstrate the innovative nature of this invention, the finite element simulations were performed using a single unit cell. The finite element simulation process is shown in Figures 4(a), 4(b), and 4(c). A 10 mm × 10 mm two-dimensional metamaterial was placed between two rigid plates. A key metric for evaluating the effectiveness of the energy absorber requires simulating the full compressive deformation behavior of the metamaterial. In this study, quasi-static compressive strains up to 60% were applied in the perpendicular direction. The deformation behavior of the two-dimensional metamaterial under quasi-static compression was simulated using the ABAQUS explicit software package. Under plane strain conditions, the solid portion of the material was discretized using CPE6M elements, while each rigid plate was discretized using R2D2 elements. A global seed size of 0.15 mm was set to ensure adequate resolution and mesh size independence across the entire simulation domain. To prevent interpenetration, interactions between the steel plate and the structure, as well as interactions within the structure, were considered. For metallic materials, plastic yielding, rather than elastic buckling, is the primary mechanism leading to initial structural damage. Therefore, buckling-induced damage was not considered in the simulations. Stress values ​​corresponding to uniformly sampled strain points were obtained from the compression simulations.

[0070] Step 1.4: The obtained topological structure and energy absorption curve are processed to facilitate deep neural network training. The energy absorption curve is reduced in dimensionality using principal component analysis. The complexity of the data is reduced by finding the principal components. As shown in Figure 5(a) and Figure 5(b), it is obvious that the first 15 principal components are capable of capturing 99.98% of the data variance. This shows that even if the dimension is significantly reduced, most of the pivot information inherent in the data is still retained. Therefore, the original stress data is projected into a space with only 15 dimensions. Furthermore, in order to eliminate the problem of the model being biased towards a certain feature due to differences between features, the data after dimensionality reduction is normalized, which is defined as follows:

[0071]

[0072] Where Y i is the i-th value of the strain-stress curve, Y i,max and Y i,min are the i-th maximum and minimum values ​​in all strain-stress curves, respectively.

[0073] Step 1.5: Since the generated metamaterial has mirror symmetry in both the horizontal and vertical directions, to fully utilize this mirror symmetry and reduce unnecessary redundant data, we selected a quarter of the original images, that is, 32×32 pixels of metamaterials, for model training. Ultimately, we divided the 30,000 metamaterials and their stress-strain curves into training and validation sets, with 90% of the generated data used for training and the remaining 10% for validation.

[0074] Step 2: Construct a diffusion model. The neural network architecture uses a U-Net network with attention mechanism and group normalization, as shown in Figure 6 As shown, U-Net consists of three main parts: an encoder, a decoder, and a skip connection layer. The encoder compresses the input image into a low-dimensional latent space through a convolutional layer. The decoder accepts the encoded input and maps it to the original resolution. The skip connection layer connects the encoder and decoder, forming a bottleneck in the latent space, emphasizing key features, and promoting bidirectional information flow. Our model mapping maps the metamaterial from 64×64 to a 4×4 image. At the same time, a cross-attention mechanism is used to integrate the energy absorption curve to control the deep generative model to obtain the corresponding metamaterial according to the target energy absorption curve. Initialize the network parameters, and use the metamaterial and its energy absorption curve as training data for iterative training in the diffusion model. The selected diffusion model has 4.21 million trainable parameters, and the loss function formula is as follows

[0075]

[0076] Where ∈ is the added noise, ∈ θ Is a trainable diffusion model that determines the amount of noise that should be removed in the current denoising step. Adam is used as the optimizer with a learning rate of 10 -4 , learning rate scheduling is implemented, and the learning rate decays by 0.98 every 50 epochs, for a total of 2000 epochs.

[0077] Step 3: Since the proposed method can learn the conditional probability distribution under given mechanical conditions, a set of metamaterials that closely match the target mechanical properties can be generated by sampling different initial noises. Four stress-strain curves in the test set are selected as generation conditions, as shown in Figure 7(a), and are put into the trained diffusion model to generate metamaterials. Subsequently, the generated metamaterials are imported into Abaqus for quasi-static loading to verify the consistency of the stress-strain curves corresponding to the generated metamaterials (referred to as generated curves) with the target curves. We use as a metric to quantify the error.

[0078]

[0079] Among them, Y g Indicates the generated stress-strain curve, Y t is the target stress-strain curve, ‖·‖1 represents the L1 norm. Figure 7(b) shows the error between the generated curve and the target curve The median error of all four target curves is less than 9%, indicating that the proposed method can accurately generate metamaterials that closely match the desired stress-strain curves.

[0080] Figure 8 The metamaterial and stress-strain curves are shown in detail. Each row represents the results generated for different working conditions. In the first column, the target curve is shown in black, the curve with the smallest error in the training dataset is shown in red, and the error in the generated curve is shown in red. The smallest curve is shown in blue, generating the mean error of the curve The median curve is shown in green. The second column shows the metamaterial corresponding to the target curve. The third column shows the metamaterial corresponding to the curve with the smallest error in the training curve. The fourth and fifth columns show the metamaterials in the generated curve. The curves with the smallest and median errors correspond to the metamaterials. Figure 8 The first column clearly shows the minimum error between the target curve and the generated curve. Less than 5.2%. Figure 8 (a) and Figure 8 In (b), the minimum error in the generated curve Less than the minimum error from the training dataset curve exist Figure 8 (c) and Figure 8 In (d), the minimum error of the generated curve Minimum error with all curves from the training dataset This indicates that the proposed method can discover superior metamaterials within the design space. Figure 8 The generated metamaterials in the fourth and fifth columns are significantly different from the target metamaterials in the second column and the training metamaterials in the third column, which proves the powerful generation ability of the proposed method, enabling it to generate metamaterials beyond the training set. Figure 8 From the fourth and fifth columns of , we can see that although the curves corresponding to the generated metamaterials are similar, their metamaterial topologies are different. This observation proves that the proposed method learns the conditional distribution of metamaterials given specific mechanical properties, realizes a one-to-many mapping from properties to geometry, and generates various geometric metamaterials with the same mechanical properties.

[0081] Step 3.2: To further evaluate the generation capability of the proposed method, we select two stress-strain curves from the training dataset and obtain a family of target curves by linear interpolation between the two curves. In Figure 9(a), the solid line represents the two curves selected from the training set, and the dotted line represents the interpolated curve (target curve). Figure 9(b) shows the error between the generated curve and the interpolated target curve. Box plot of the median error between the generated curve and the interpolated curve This result highlights the excellent generation capability and accuracy of the proposed method, which can effectively generate metamaterials that closely match the target curve even in complex interpolation scenarios.

[0082] Figure 10 The metamaterial and its stress-strain curve under interpolation conditions are shown in detail. Each row represents the generated result of each interpolation curve. In the first column, the black line represents the target curve (interpolation curve), the blue line and the green line represent the generated curve. The minimum and median curves, represented by the green line The generated curve of the median value, the red line represents the generated The second column is the training data set. The metamaterial corresponding to the minimum value. The third column shows the error in generating the metamaterial. Metamaterials corresponding to the minimum and median values. Figure 10 The first column shows, in some cases, the minimum error of the training curve Exceeded the minimum error for generating the curve At the same time, the metamaterials corresponding to the second and third columns are significantly different. This finding emphasizes the excellent generative ability of the proposed method in generating appropriate metamaterials, achieving an accurate mapping from mechanical properties to metamaterial geometry by learning and understanding the characteristics of the target curve.

[0083] By comparison Figure 10 In the metamaterials generated in

[15] , it can be seen that adjacent metamaterials exhibit smooth, gradual transitions, rather than abrupt ones, as the interpolated curve changes. This demonstrates the proposed method's ability to maintain continuity and consistency when interpolating the evolution of the target stress-strain curve. This capability reflects a deep understanding of the complex relationship between metamaterials and mechanical properties and demonstrates the reliability and stability of the conditional generation. Furthermore, the smooth transitions produced by the proposed method facilitate the exploration of a broader design space, potentially leading to new metamaterial topologies with superior or unique properties and opening up new possibilities and innovations in the field of metamaterial design.

[0084] Step 3.3: This section presents a series of experiments to verify the performance and accuracy of the proposed method. First, experimental samples of the metamaterial were prepared by mechanical processing. Quasi-static compression tests were performed using a tensile testing machine with a constant loading rate of 1 mm / min.

[0085] We select the error from the interpolation case The metamaterial corresponding to the minimum value is tested, and the experimental results are as follows Figure 11 (a) Figure 11 Middle (b), Figure 11 (c) and Figure 11(d) The error between the target and experimental curves is less than 12%. This experiment further validates the effectiveness and reliability of the proposed method and highlights its broad potential in various applications. These experiments pave the way for integrating the model into real-world design and manufacturing processes, promoting the development and innovation of advanced materials and structures.

Claims

1. A reverse design method for energy-absorbing metamaterials based on diffusion model and parameterized level set, characterized by: Here are the steps: Step 1.1: Generate the metamaterial using a topology optimization method based on homogenization and parameterized level set methods. Use density thresholding to identify boundaries and the Ramer-Douglas-Peucker algorithm to effectively reduce the detected boundary data points to obtain a geometric representation for finite element simulation. Step 1.2: Conduct uniaxial tensile tests on 304 stainless steel for constitutive parameter calibration, and use the least squares method to identify material parameters in combination with plasticity theory. Step 1.3: Use finite element software to simulate the metamaterial obtained by topology optimization. The metamaterial is placed between two rigid plates. A vertical displacement is set along the y direction to simulate the compression process in a quasi-static manner and obtain the stress-strain curve of the structure during compression; Step 1.4: Use principal component analysis to reduce the dimensionality of the simulated stress-strain curves. By finding the principal components, the complexity of the data is reduced. The reduced dimensionality data is further normalized to eliminate the problem of the model being biased towards a certain feature due to differences between features. Finally, the metamaterial and its reduced dimensionality energy absorption curves are used as training data, with 90% of the generated data used for training and the remaining 10% for validation. Step 2: Construct a diffusion model. The neural network architecture uses the U-Net network, initialize the network parameters, and use the metamaterial and its dimensionality-reduced energy absorption curve as training data for iterative training in the diffusion model. Step 3: Given a target stress-strain curve for a trained diffusion model, a family of metamaterials with target mechanical properties can be quickly generated based on the complex nonlinear mapping relationships learned by the diffusion model. The specific operations of step 1.1 are as follows: The mathematical expression of the topology optimization method based on homogenization and parameterized level set method is as follows: find:α=[α1 α2 … α N ] Among them, α is the design variable, α L and α U are the lower and upper bounds of the design variables, respectively, and J is the homogenized elastic tensor E H The objective function is defined by the combination of , V is the volume constraint, and its upper limit is V max Φ is a parameterized level set function, which is approximated by a set of radial basis functions at each node in the entire design domain and is expressed as: The radial basis function sequence at each node is represented as a vector and is given as follows: A compactly supported radial basis function with c2 continuity is used, and the formula is as follows: φ(r)=(max(0.1-r)) 4 …(4r+1) Where r is the size of the influence domain of the basis function at the node in the design domain; therefore, the parameterized level set function Φ is represented by the design variable α; H is the Heaviside function, which is expressed as: Where η is a small positive number introduced in the numerical process to avoid singularities, and the width of the numerical approximation of Δ is used. The elastic equilibrium equation is expressed in weak variational form, where the bilinear energy term a(u, v, Φ) and the linear load term l(v, Φ) can be expressed as: a(u,v)=∫ Ω H(Φ)σ(u):ε(v)dΩ Where f represents body force, t represents surface force, and virtual displacement v belongs to the allowed displacement space of motion.

2. The inverse design method for energy-absorbing metamaterials based on a diffusion model and parameterized level sets according to claim 1, wherein: The specific operations of step 1.2 are as follows: A high-precision extensometer is used to measure the axial displacement of the specimen. The collected force-displacement data is processed to obtain the true stress-strain curve: ε=ln(1+ε N ) s = s N (1+e N ) Where, ε N and σ N are normal strain and stress, respectively; ε and σ represent true strain and true stress, respectively.

3. The inverse design method for energy-absorbing metamaterials based on a diffusion model and parameterized level sets according to claim 2, wherein: In step 1.2, the isotropic hardening von Mises yield function is used, the associated flow law is used, and the hardening law is in the following form: s y =A+B(e p ) n where σ y is the yield stress, ε p is the equivalent plastic strain; A is the initial yield stress, B is the hardening constant, and n is the dimensionless hardening exponent.

4. The inverse design method for energy-absorbing metamaterials based on a diffusion model and parameterized level sets according to claim 1 or 2, wherein: In step 1.4, the original stress data is projected into a space with only 15 dimensions, and the reduced-dimensional data is normalized, which is defined as follows: Where Y i is the i-th value of the strain-stress curve, Y i,max and Y i,min are the i-th maximum and minimum values ​​in all strain-stress curves, respectively.

5. The inverse design method for energy-absorbing metamaterials based on a diffusion model and parameterized level sets according to claim 1 or 2, characterized in that: In step 2, the neural network architecture adopts a U-Net network with an attention mechanism and group normalization. The U-Net consists of three main parts: an encoder, a decoder, and a skip connection layer. The encoder compresses the input image into a low-dimensional latent space through a convolutional layer. The decoder accepts the encoded input and maps it to the original resolution. The skip connection layer connects the encoder and decoder, forming a bottleneck in the latent space, initializes the network parameters, and uses the metamaterial and its energy absorption curve as training data for iterative training in the diffusion model. The loss function formula is as follows: Where ∈ is the added noise, ∈ θ It is a trainable diffusion model that determines the amount of noise that should be removed in the current denoising step and uses Adam as the optimizer.

Images