Rigidity-constrained sine compression-torsion metamaterial band gap design method

Abstract

A rigidity-constrained sine compression-torsion metamaterial band gap design method comprises the following steps: S100, aiming at a sine compression-torsion metamaterial unit, establishing an analytical model among geometric parameters, equivalent rigidity and band gap boundary frequency of the sine compression-torsion metamaterial unit, so as to obtain a constraint equation describing the geometric parameters and the equivalent rigidity and the band gap boundary frequency; s200, fusing the constraint equation generated in the step S100, and constructing and training a physical information neural network for realizing high-precision and high-efficiency forward mapping from unit geometric parameters to rigidity and band gap frequency; s300, constructing and training a neural network for realizing reverse mapping from target stiffness and band gap frequency to unit geometric parameters, and providing an initial scheme for rapid customized design; and S400, setting target rigidity and band gap conditions, and adopting an intelligent optimization algorithm to call the trained physical information neural network model as a performance evaluator to perform iterative optimization. The method aims at solving the problems that in the prior art, rigidity constraint and band gap performance cannot be effectively balanced, and design and calculation efficiency is low.

Description

Technical Field

[0001] This invention relates to the field of metamaterial design technology, and in particular to a stiffness-constrained sinusoidal compression-torsional metamaterial bandgap design method. Background Technology

[0002] Metamaterials have made significant achievements in the field of vibration suppression. Their core advantage lies in achieving bandgap engineering through periodic structural design, thereby precisely isolating the propagation of vibrations within a specific frequency range and providing an innovative solution to engineering vibration problems.

[0003] However, existing research still faces three major challenges that severely restrict the practical application of metamaterials for vibration isolation.

[0004] The first pain point is the neglect of structural load-bearing stiffness. Many studies focus on adjusting microstructure configurations through topology optimization or parametric scanning to optimize geometric parameters, aiming to widen the bandgap or lower the bandgap frequency. However, stiffness performance is often relegated to a secondary consideration, or even omitted from optimization objectives and constraints. This results in many metamaterials with theoretically excellent vibration suppression performance being difficult to apply in practical engineering scenarios due to insufficient stiffness. For example, some metamaterials that achieve wide low-frequency bandgap by significantly reducing structural density and adding flexible elements cannot withstand conventional loads in mechanical systems, rail transportation, and other scenarios, and are prone to excessive deformation or structural failure.

[0005] The second pain point is low design efficiency. To balance stiffness and bandgap performance, some studies have introduced multi-objective optimization algorithms (such as genetic algorithms and particle swarm optimization) for collaborative design. However, the performance evaluation stage relies heavily on time-consuming finite element simulations, resulting in an extremely long overall design cycle. In this iterative "optimization, iteration, simulation, and evaluation" cycle, the simulation evaluation of a single parameter combination can take several hours. If thousands of parameter combinations need to be verified, the simulation stage alone can take weeks or even months. This not only makes it difficult to adapt to the "rapid response to design requirements" scenario in engineering but also limits the exploration of large-scale design spaces, further delaying the engineering process of metamaterials.

[0006] The third pain point is the lack of physical information and reverse design capabilities. Existing machine learning methods introduced in metamaterial design are mostly simple data-driven surrogate models, merely fitting the mapping relationship between geometric parameters and performance (bandgap, stiffness) through a large number of samples, without incorporating the physical laws governing metamaterial mechanical behavior (such as compression-torsional coupling effects, elastic wave propagation equations, etc.). This "pure data-driven" model not only requires an extremely large number of training samples but may also produce "physically inconsistent" predictions, increasing the cost of subsequent verification and correction. More importantly, these models are mostly limited to "forward prediction" (input geometric parameters, output performance indicators), failing to achieve the "reverse design" urgently needed in engineering—that is, when engineers propose "a structure with stiffness X and bandgap Y," existing models struggle to directly output feasible geometric parameter solutions, often requiring repeated trial and error or traversal searches using forward models, which is inefficient and makes it difficult to find the globally optimal solution, failing to fully leverage the advantages of machine learning's "intelligent design."

[0007] In summary, the key technological bottleneck currently facing the field of metamaterial vibration suppression lies in the failure to construct an integrated design system that balances performance synergy (stiffness-bandgap synergy), design efficiency, and intelligent reverse engineering capabilities. Therefore, developing an efficient and intelligent design framework to systematically solve the bandgap optimization problem under stiffness constraints has become a pressing technical challenge in this field. Summary of the Invention

[0008] To address the aforementioned shortcomings, this invention aims to solve the technical problems existing in metamaterial design technology, such as the inability to effectively balance stiffness constraints and bandgap performance, low design calculation efficiency, and lack of intelligent reverse design capabilities.

[0009] To achieve this objective, the present invention adopts the following technical solution:

[0010] A stiffness-constrained design method for bandgap in sinusoidal compressive-torsional metamaterials includes the following steps:

[0011] S100: Physical modeling and constraint generation. For sinusoidal compression-torsion metamaterial elements, an analytical model is established based on mechanical theory to establish the relationship between their geometric parameters, equivalent stiffness, and bandgap boundary frequency, thereby obtaining constraint equations describing the intrinsic physical relationship between the two.

[0012] S200: Construct and train the PINN model, integrate the constraint equations generated in step S100, construct and train a physical information neural network, namely the PINN model, to achieve high-precision and high-efficiency forward mapping from element geometric parameters to stiffness and bandgap frequency;

[0013] S300: Construct and train an inverse NN model. Construct and train a standard neural network, namely an inverse NN model, to realize the inverse mapping from target stiffness and bandgap frequency to unit geometric parameters, providing an initial solution for rapid customized design.

[0014] S400: Performs constraint-based hybrid intelligent optimization, sets target stiffness and bandgap conditions, adopts intelligent optimization algorithms, calls a trained physical information neural network model as a performance evaluator, and iteratively optimizes geometric parameters.

[0015] Preferably, in step S100, the geometric parameters include the sine amplitude A, the number of periods n, the diameter of the curved beam d, the initial phase φ, the radial dimension R, the straight-line distance between the two ends of the curved beam l, and the diameter of the disk R. d The thickness t of the disk and the distance H between the upper and lower disks;

[0016] The metamaterial unit cell includes an upper disk and a curved beam mounted on the bottom surface of the upper disk. A compressive force F along the z-direction is applied to the upper disk. z At that time, the upper disk and the bending beam not only undergo compressive deformation It will also produce torsional deformation φ around the z-axis. z When a torque T along the z-direction is applied to the upper disk z At that time, the upper disk and the curved beam will also exhibit and φ z This compression-torsional coupling behavior can be mathematically described by the following equation:

[0017] Equation (1)

[0018] Where K l For compressive stiffness, K c For torsional stiffness, K t To determine the compressive-torsional coupling stiffness, the mechanical behavior of the unit cell was analyzed based on elasticity and beam theory, by applying a compressive force F in the z-direction to the disk. z The upper disk and the bent beam undergo compressive deformation. The equivalent axial stiffness K is derived. z The parsing expression:

[0019] Equation (2)

[0020] When a metamaterial unit cell exhibits an infinite structure along the z-direction, when subjected to excitation in the z-direction, i.e., axial compression or torsion, the two disks of the j-th staggered array show [something] in the z-direction. and Displacement and rotation angle about the z-axis and Based on Newton's second theorem and Bloch's theorem, a dynamic analysis of the j-th unit cell is performed, and the expression for its bandgap boundary frequency is derived:

[0021] Equation (3)

[0022] The above analytical expressions constitute the physical constraint equations describing the intrinsic relationship between geometric parameters and mechanical properties; where J is the moment of inertia, m is the mass of the disk, and f is the natural frequency.

[0023] These equations will be used as prior knowledge and embedded into subsequent neural network models to ensure that their predictions conform to physical laws.

[0024] Preferably, in step S200, a physical information neural network model is constructed based on the physical constraints obtained in step S100.

[0025] S210, determine the input and output layer parameters of the PINN model: The input layer has five nodes, corresponding to the five core geometric parameters of the sinusoidal compression-torsion metamaterial unit cell, namely, the radial position R of the curved beam, the diameter d of the curved beam, the sinusoidal amplitude A, the number of periods n, and the initial phase φ; the output layer has seven nodes, corresponding to the four types of stiffness parameters and three types of bandgap frequency parameters of the metamaterial, among which the stiffness parameters include the compression spring stiffness K. l Torsional spring stiffness K t Compression-torsional coupling stiffness K c Equivalent axial stiffness K z The bandgap frequency parameters include the lower boundary frequency f of the first bandgap. π1 The upper boundary frequency f of the first bandgap 03 The lower boundary frequency f of the second bandgap 04 ;

[0026] S220, Design the hidden layer structure of the PINN model: adopt a five-layer fully connected structure, with each layer node being 256×512×512×256×128 respectively;

[0027] S230, Generate training and validation datasets: Generate 2000 sets of sample data covering the above five core geometric parameters through Latin hypercube sampling, obtain the seven types of mechanical performance response parameters of the corresponding samples based on the three-dimensional modeling of metamaterial unit cells, and divide the dataset into training set and validation set in an 8:2 ratio.

[0028] S240, Define the loss function for the PINN model: The loss function consists of a physics-driven term and a data-fitting term, and its expression is:

[0029] Equation (4)

[0030] in It is a physics-driven term. This is the data fitting term. In the loss function, MAE represents the mean absolute error. and These represent the results of the prediction dataset; then... and The results are obtained by substituting the predicted data into formulas (3) and (4);

[0031] S250, and train the PINN model: the model activation function is the tanh function, the initial learning rate is set to 0.0003, the Adam optimizer is used, and an adaptive learning rate decay strategy is used during training. When the model training enters the loss plateau period, the optimizer step size is automatically reduced.

[0032] The model is trained on the training set partitioned in step S230. The model parameters are optimized using the loss function defined in step S240. The model performance is evaluated using the validation set. The weights of the loss components are adjusted, and the weights of each loss component are determined through Bayesian optimization until the model loss function converges and the average coefficient of determination (R²) of the validation set is obtained. 2 With a value not less than 0.998, a PINN forward surrogate model is obtained that can achieve high-precision mapping from metamaterial unit cell geometric parameters to mechanical properties.

[0033] Preferably, step S300 constructs a standard inverse neural network model, namely the NN model, which includes the following steps:

[0034] S310, Determine the input and output parameters of the NN inverse design model: The input layer is set with four nodes, corresponding to the four core performance indicators of the sinusoidal compression-torsion metamaterial, namely the equivalent axial stiffness K. z The lower boundary frequency f of the first bandgap π1 The upper boundary frequency f of the first bandgap 03 The lower boundary frequency f of the second bandgap 04 The output layer is set with four nodes, which correspond to the four core geometric parameters of the metamaterial unit cell, namely the radial position R of the curved beam, the diameter d of the curved beam, the sinusoidal amplitude A, and the initial phase φ.

[0035] S320, Constructing the network structure of the NN inverse design model: Three fully connected hidden layers are used, with the number of nodes in each hidden layer being 128×256×128, and the activation function is the tanh function;

[0036] S330, Generate training and validation datasets: Obtain 2000 sets of sample data covering the above four types of performance indicators and corresponding four types of geometric parameters. The sample data comes from the output results of the PINN forward surrogate model or a high-precision finite element simulation database. Divide the dataset into training set and validation set in a ratio of 8:2, and classify the dataset according to the value of the metamaterial unit cell period number n to form two sub-datasets.

[0037] S340, Define the training objective of the NN inverse design model: minimize the mean square error between the predicted geometric parameters of the model output and the true geometric parameters in the sample data.

[0038] S350, Training the NN inverse design model: Using a fixed learning rate of 0.0015, train NN inverse design models corresponding to n=1 and n=2 based on the two sub-datasets after classification. Update the model parameters through the training set and monitor the convergence status of the model using the validation set until the training loss and validation loss of the two models tend to be stable, thus obtaining the NN inverse design model that can deduce the initial scheme of geometric parameters from the target performance index.

[0039] Furthermore, step S400 specifically includes the following steps:

[0040] S410, Set optimization objectives and constraints: Define the preset stiffness constraints and target bandgap requirements of the metamaterial. The target bandgap is at least one of minimizing the initial bandgap frequency, maximizing the bandgap width, or achieving a preset bandgap frequency range of 100-1000Hz.

[0041] S420, configured with intelligent optimization algorithm parameters: the multi-island genetic algorithm is selected as the core optimization algorithm, and the algorithm parameters are set as follows: the number of islands is 5, the subpopulation size of each island is 100, the crossover probability is 0.8, the mutation probability is 0.3, the migration rate is 5%, the migration interval is 10 generations, and the inter-island individual migration adopts a circular topology structure.

[0042] S430, Determine the initial population of the optimization algorithm: Call the trained NN inverse design model, input the target stiffness and target bandgap parameters, obtain the initial geometric parameter scheme and use it as the elite individuals in the initial population. The proportion of these elite individuals shall not be less than 5%, and the remaining individuals shall be randomly generated within the feasible range of geometric parameter engineering to form a complete initial population.

[0043] S440, Performance Evaluation and Fitness Calculation: For the metamaterial unit cell geometric parameters corresponding to each individual in the initial population, the trained PINN forward surrogate model is called to quickly output the equivalent stiffness and bandgap frequency performance indicators, and the fitness value of each individual is calculated based on the preset stiffness constraints and target bandgap requirements.

[0044] S450, execute the iterative optimization process: perform genetic operations such as selection, crossover, and mutation within the island according to the rules of the intelligent optimization algorithm. When the migration interval is reached, select the top 5% of elite individuals in fitness on each island for inter-island migration; repeat the performance evaluation, genetic operations, and inter-island migration steps until the termination condition is met.

[0045] S460, Output the optimal design scheme: When the termination condition is met, extract the geometric parameter combination corresponding to the individual with the best fitness, which is the optimal design scheme of the metamaterial unit cell that satisfies the preset stiffness constraint and the target bandgap requirement; The termination condition is finding at least one of the following: finding the optimal solution that satisfies all constraints or reaching the preset maximum number of iterations.

[0046] S470, Array Construction and Verification: The optimal unit cells obtained in the steps are arranged in a preset layout to form a metamaterial array. The vibration suppression performance and stiffness of the array are verified through simulation analysis and experimental testing.

[0047] Furthermore, in step S410, the preset bandgap frequency range of 100 to 1000 Hz can be divided into multiple sub-bands, namely 100 to 400 Hz, 200 to 500 Hz, and 400 to 1000 Hz. Corresponding metamaterial unit cells are designed for different sub-bands, and then continuous vibration suppression across the entire target frequency band is achieved through array combination.

[0048] Further, in step S440, the specific rules for fitness calculation are as follows: if the equivalent stiffness of an individual does not meet the preset constraint, which is less than 5000 N / m, the fitness value is set to 0; if the equivalent stiffness meets the constraint, the fitness value is calculated by weighted summation based on the matching degree between the bandgap frequency and the target bandgap, the bandgap width, and other indicators, with the bandgap matching degree weight ≥ 0.6.

[0049] One of the above technical solutions includes the following beneficial effects: This solution systematically solves the problem of coordinating stiffness and bandgap design: by introducing stiffness as a hard constraint into the optimization process, the engineering practicality of the design results is ensured. Design efficiency is improved by orders of magnitude: the PINN model replaces most finite element calculations, enabling complex optimization designs to be completed within minutes. The model has strong prediction accuracy and generalization: PINN incorporates physical laws, ensuring the reliability of prediction results, and can provide reasonable predictions even in untrained data regions. True on-demand reverse design is achieved: through the reverse neural network model, initial design schemes can be quickly generated directly based on engineering performance requirements, greatly improving the intelligence and customization level of the design and shortening the design iteration cycle. Attached Figure Description

[0050] Figure 1 Overall flowchart of the design method of this invention;

[0051] Figure 2 (a) shows the structure of the sinusoidal compression-torsion metamaterial unit, and (b) shows a schematic diagram of the equivalent mass spring system.

[0052] Figure 3 (a) shows the structure of the sinusoidal compression-torsion metamaterial array, (b) shows the one-dimensional equivalent theoretical model, and (c) shows the band gap schematic diagram.

[0053] Figure 4 In the middle, (a) represents the forward and (b) represents the reverse training of the neural network model, and (c)-(e) represent the training loss function curves;

[0054] Figure 5 A flowchart illustrating the design of an intelligent optimization algorithm, using MIGA as an example;

[0055] Figure 6 (a) shows the experimental results, and (b) shows the simulated vibration test results.

[0056] Figure 7 (a) shows the optimized model, (b) shows the simulation structure, and (c) shows the experimental transfer rate comparison. Detailed Implementation

[0057] Embodiments of the present invention are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0058] like Figure 1 As shown, a method for designing the bandgap of a stiffness-constrained sinusoidal compressive-torsional metamaterial includes the following steps:

[0059] S100: Physical modeling and constraint generation. For sinusoidal compression-torsion metamaterial elements, an analytical model is established based on mechanical theory to establish the relationship between their geometric parameters, equivalent stiffness, and bandgap boundary frequency, thereby obtaining constraint equations describing the intrinsic physical relationship between the two.

[0060] S200: Construct and train the PINN model, integrate the constraint equations generated in step S100, construct and train a physical information neural network, namely the PINN model, to achieve high-precision and high-efficiency forward mapping from element geometric parameters to stiffness and bandgap frequency;

[0061] S300: Construct and train an inverse NN model. Construct and train a standard neural network, namely an inverse NN model, to realize the inverse mapping from target stiffness and bandgap frequency to unit geometric parameters, providing an initial solution for rapid customized design.

[0062] S400: Performs constraint-based hybrid intelligent optimization, sets target stiffness and bandgap conditions, adopts intelligent optimization algorithms, calls a trained physical information neural network model as a performance evaluator, and iteratively optimizes geometric parameters.

[0063] This solution systematically addresses the challenge of coordinating stiffness and bandgap design: by introducing stiffness as a rigid constraint into the optimization process, it ensures the engineering practicality of the design results. Design efficiency is improved by orders of magnitude: the PINN model replaces most finite element calculations, enabling complex optimization designs to be completed within minutes. The model boasts strong prediction accuracy and generalization: PINN incorporates physical laws, guaranteeing the reliability of prediction results, and providing reasonable predictions even in untrained data regions. It achieves true on-demand reverse design: through the inverse neural network model, initial design schemes can be quickly generated directly based on engineering performance requirements (such as target stiffness and bandgap), greatly improving the intelligence and customization level of the design and shortening the design iteration cycle.

[0064] In step S100, the geometric parameters include the sine amplitude A, the number of periods n, the diameter of the curved beam d, the initial phase φ, the radial dimension R, the straight-line distance between the two ends of the curved beam l, and the diameter of the disk R. d The thickness t of the disk and the distance H between the upper and lower disks;

[0065] The metamaterial unit cell includes an upper disk and a curved beam mounted on the bottom surface of the upper disk. A compressive force F along the z-direction is applied to the upper disk. z At that time, the upper disk and the bending beam not only undergo compressive deformation It will also produce torsional deformation φ around the z-axis. z When a torque T along the z-direction is applied to the upper disk z At that time, the upper disk and the curved beam will also exhibit and φ z This compression-torsional coupling behavior can be mathematically described by the following equation:

[0066] Equation (1)

[0067] Where K l For compressive stiffness, K c For torsional stiffness, K t To determine the compressive-torsional coupling stiffness, the mechanical behavior of the unit cell was analyzed based on elasticity and beam theory, by applying a compressive force F in the z-direction to the disk. z The upper disk and the bent beam undergo compressive deformation. The equivalent axial stiffness K is derived. z The parsing expression:

[0068] Equation (2)

[0069] When a metamaterial unit cell exhibits an infinite structure along the z-direction, when subjected to excitation in the z-direction, i.e., axial compression or torsion, the two disks of the j-th staggered array show [something] in the z-direction. and Displacement and rotation angle about the z-axis and Based on Newton's second theorem and Bloch's theorem, a dynamic analysis of the j-th unit cell is performed, and the expression for its bandgap boundary frequency is derived:

[0070] Equation (3)

[0071] The above analytical expressions constitute the physical constraint equations describing the intrinsic relationship between geometric parameters and mechanical properties; where J is the moment of inertia, m is the mass of the disk, and f is the natural frequency.

[0072] These equations will be used as prior knowledge and embedded into subsequent neural network models to ensure that their predictions conform to physical laws.

[0073] Physical modeling clarified the mechanical nature and parameter control laws of metamaterials, defining the physically feasible boundary for subsequent model training and optimization, and establishing the synergistic control relationship between stiffness and bandgap.

[0074] In step S200, a physical information neural network model is constructed based on the physical constraints obtained in step S100.

[0075] S210, determine the input and output layer parameters of the PINN model: The input layer has five nodes, corresponding to the five core geometric parameters of the sinusoidal compression-torsion metamaterial unit cell, namely, the radial position R of the curved beam, the diameter d of the curved beam, the sinusoidal amplitude A, the number of periods n, and the initial phase φ; the output layer has seven nodes, corresponding to the four types of stiffness parameters and three types of bandgap frequency parameters of the metamaterial, among which the stiffness parameters include the compression spring stiffness K. l Torsional spring stiffness K t Compression-torsional coupling stiffness K c Equivalent axial stiffness K z The bandgap frequency parameters include the lower boundary frequency f of the first bandgap. π1 The upper boundary frequency f of the first bandgap 03 The lower boundary frequency f of the second bandgap 04 ;

[0076] S220, Design the hidden layer structure of the PINN model: adopt a five-layer fully connected structure, with each layer node being 256×512×512×256×128 respectively;

[0077] S230, Generate training and validation datasets: Generate 2000 sets of sample data covering the above five core geometric parameters through Latin hypercube sampling, obtain the seven types of mechanical performance response parameters of the corresponding samples based on the three-dimensional modeling of metamaterial unit cells, and divide the dataset into training set and validation set in an 8:2 ratio.

[0078] S240, Define the loss function for the PINN model: The loss function consists of a physics-driven term and a data-fitting term, and its expression is:

[0079] Equation (4)

[0080] in It is a physics-driven term. This is the data fitting term. In the loss function, MAE represents the mean absolute error. and These represent the results of the prediction dataset; then... and The results are obtained by substituting the predicted data into formulas (3) and (4);

[0081] S250, and train the PINN model: the model activation function is the tanh function, the initial learning rate is set to 0.0003, the Adam optimizer is used, and an adaptive learning rate decay strategy is used during training. When the model training enters the loss plateau period, the optimizer step size is automatically reduced.

[0082] The model is trained on the training set partitioned in step S230. The model parameters are optimized using the loss function defined in step S240. The model performance is evaluated using the validation set. The weights of the loss components are adjusted, and the weights of each loss component are determined through Bayesian optimization until the model loss function converges and the average coefficient of determination (R²) of the validation set is obtained. 2 With a value not less than 0.998, a PINN forward surrogate model is obtained that can achieve high-precision mapping from metamaterial unit cell geometric parameters to mechanical properties.

[0083] The analytical expression for the equivalent axial stiffness of the metamaterial unit cell in step S240 is derived based on elasticity mechanics and beam theory, and is used to describe the stiffness characteristics of the unit cell under compressive force in the z-direction. The analytical expression for the bandgap boundary frequency is derived based on Newton's second law and Bloch's law, and is used to describe the elastic wave propagation bandgap boundary characteristics of the metamaterial array structure. The process of obtaining the response value through three-dimensional modeling in step S230 involves performing mechanical simulation analysis on the three-dimensional model of the metamaterial unit cell to obtain the stiffness parameters and bandgap frequency parameters under the corresponding geometric parameters, ensuring the accuracy of the dataset. The adaptive learning rate decay strategy in step S250 automatically reduces the step size of the optimizer when the model training enters the plateau period of the loss function, thereby improving the stability and convergence accuracy of the model training. By integrating physical constraints and neural networks, high efficiency, high accuracy, and physical consistency in performance prediction are achieved, which improves efficiency and reliability compared to traditional finite element simulation, and provides support for subsequent reverse design and optimization.

[0084] Figure 4 (c) shows the loss function curves for training and validation samples, as well as the learning rate changes throughout the training process. The graphs show that the loss value initially decreases significantly with increasing training epochs, stabilizing around 4000 epochs. An adaptive learning rate decay strategy is employed, automatically reducing the optimizer's step size when the model enters a plateau. The PINN model is trained using the Adam optimizer until the loss function converges, resulting in a high-precision, high-efficiency, and physically consistent positive surrogate model.

[0085] Step S300 constructs a standard inverse neural network model, namely the NN model, which includes the following steps:

[0086] S310, Determine the input and output parameters of the NN inverse design model: The input layer is set with four nodes, corresponding to the four core performance indicators of the sinusoidal compression-torsion metamaterial, namely the equivalent axial stiffness K. z The lower boundary frequency f of the first bandgap π1 The upper boundary frequency f of the first bandgap 03 The lower boundary frequency f of the second bandgap 04 The output layer is set with four nodes, which correspond to the four core geometric parameters of the metamaterial unit cell, namely the radial position R of the curved beam, the diameter d of the curved beam, the sinusoidal amplitude A, and the initial phase φ.

[0087] S320, Constructing the network structure of the NN inverse design model: Three fully connected hidden layers are used, with the number of nodes in each hidden layer being 128×256×128, and the activation function is the tanh function;

[0088] S330, Generate training and validation datasets: Obtain 2000 sets of sample data covering the above four types of performance indicators and corresponding four types of geometric parameters. The sample data comes from the output results of the PINN forward surrogate model or a high-precision finite element simulation database. Divide the dataset into training set and validation set in a ratio of 8:2, and classify the dataset according to the value of the metamaterial unit cell period number n to form two sub-datasets.

[0089] S340, Define the training objective of the NN inverse design model: minimize the mean square error between the predicted geometric parameters of the model output and the true geometric parameters in the sample data.

[0090] S350, Training the NN inverse design model: Using a fixed learning rate of 0.0015, train NN inverse design models corresponding to n=1 and n=2 based on the two sub-datasets after classification. Update the model parameters through the training set and monitor the convergence status of the model using the validation set until the training loss and validation loss of the two models tend to be stable, thus obtaining the NN inverse design model that can deduce the initial scheme of geometric parameters from the target performance index.

[0091] By constructing a reverse mapping model from performance to geometry, on-demand customized design of metamaterials was realized, solving the problems of low efficiency and poor accuracy in traditional reverse design, and providing a starting point for subsequent optimization. Figure 4 (d) and (e) present the training and validation loss curves for models with n=1 and n=2, respectively: During training, the loss values ​​of both models first decrease rapidly and then tend to stabilize, indicating that the models have converged effectively and have good robustness in learning the inverse mapping relationship. The predicted R by φ... 2 The value is relatively low because φ is less sensitive to changes in structural performance.

[0092] Specifically, step S400 includes the following steps:

[0093] S410, Set optimization objectives and constraints: Define the preset stiffness constraints and target bandgap requirements of the metamaterial. The target bandgap is at least one of minimizing the initial bandgap frequency, maximizing the bandgap width, or achieving a preset bandgap frequency range of 100-1000Hz.

[0094] S420, configured with intelligent optimization algorithm parameters: the multi-island genetic algorithm is selected as the core optimization algorithm, and the algorithm parameters are set as follows: the number of islands is 5, the subpopulation size of each island is 100, the crossover probability is 0.8, the mutation probability is 0.3, the migration rate is 5%, the migration interval is 10 generations, and the inter-island individual migration adopts a circular topology structure.

[0095] S430, Determine the initial population of the optimization algorithm: Call the trained NN inverse design model, input the target stiffness and target bandgap parameters, obtain the initial geometric parameter scheme and use it as the elite individuals in the initial population. The proportion of these elite individuals shall not be less than 5%, and the remaining individuals shall be randomly generated within the feasible range of geometric parameter engineering to form a complete initial population.

[0096] S440, Performance Evaluation and Fitness Calculation: For the metamaterial unit cell geometric parameters corresponding to each individual in the initial population, the trained PINN forward surrogate model is called to quickly output the equivalent stiffness and bandgap frequency performance indicators, and the fitness value of each individual is calculated based on the preset stiffness constraints and target bandgap requirements.

[0097] S450, execute the iterative optimization process: perform genetic operations such as selection, crossover, and mutation within the island according to the rules of the intelligent optimization algorithm. When the migration interval is reached, select the top 5% of elite individuals in fitness on each island for inter-island migration; repeat the performance evaluation, genetic operations, and inter-island migration steps until the termination condition is met.

[0098] S460, Output the optimal design scheme: When the termination condition is met, extract the geometric parameter combination corresponding to the individual with the best fitness, which is the optimal design scheme of the metamaterial unit cell that satisfies the preset stiffness constraint and the target bandgap requirement; The termination condition is finding at least one of the following: finding the optimal solution that satisfies all constraints or reaching the preset maximum number of iterations.

[0099] S470, Array Construction and Verification: The optimal unit cells obtained in the steps are arranged in a preset layout to form a metamaterial array. The vibration suppression performance and stiffness of the array are verified through simulation analysis and experimental testing.

[0100] By integrating the technologies of early physical constraints, forward prediction, and reverse design through a hybrid intelligent optimization framework, the problem of synergistic optimization of stiffness and bandgap is solved, and the design efficiency and optimization accuracy are optimized, enabling the solution to support customized and arrayed applications in multiple scenarios.

[0101] To verify the effectiveness of this hybrid optimization process, this study focuses on "obtaining a structure with customized vibration isolation performance under specified stiffness constraints" as the core objective, selecting "stiffness constraints of not less than 5000 N / m" and "..." With "frequency band vibration suppression" as the specific design requirement, the design and verification of a customized array solution were carried out. Initially, The area is divided into three sub-zones ( , and ), to customize the corresponding unit cell. For those targeting and The low-to-mid-frequency structure is optimized by adjusting the first bandgap to cover the desired range. Specifically, the lower boundary frequency (f) of the target first bandgap is... π1 The upper boundary frequency (f) is set to 100Hz and 200Hz. 03 The frequencies were set to 400Hz and 500Hz respectively. For... The intermediate frequency structure is optimized by adjusting the second bandgap to cover the required range, with the lower boundary frequency of the second bandgap set at 400Hz. The specific correspondence of the optimal geometric parameters obtained through the above process in this embodiment is as follows: [The text abruptly ends here, likely due to an incomplete sentence or missing information.] The geometric parameters of the low-to-medium frequency vibration isolation unit cell (OP-41) are: R=13.8mm, d=1.72mm, A=3.35mm, n=1, φ=52.8°; (This is for...) The geometric parameters of the mid-frequency vibration isolation unit cell (OP-42) are: R = 20 mm, d = 4.66 mm, A = 4.5 mm, n = 1, φ = 80°; (This is for...) The geometric parameters of the mid-frequency vibration isolation unit cell (OP-43) are: R = 8 mm, d = 7 mm, A = 3.62 mm, n = 1, φ = 90°, as follows: Figure 7 (a) shows a 1×5 array formed by arranging elements in a 2-1-2 layout, which can achieve... It provides continuous vibration suppression across the entire target frequency band, and the array axial stiffness reaches 5069 N / m, meeting the stiffness constraint requirement of not less than 5000 N / m.

[0102] Subsequently, a metamaterial model was created using 3D printing, and a vibration testing experimental platform was built. Figure 6 (a) and simulation test Figure 6 (b). Figure 6 (a) illustrates the experimental setup for measuring vibration transmissibility. Specifically, a vibration exciter (model DH40020) is securely bolted to the test platform, and a 1×5 array structure is horizontally suspended by four flexible ropes, one end of which is rigidly coupled to the output of the vibration exciter via nuts. Furthermore, accelerometers (model 1A312E) are rigidly mounted on both the input and output sides of the array structure to acquire input and output acceleration signals; a signal generator (model DH1301N) generates a swept-frequency sinusoidal harmonic excitation signal with a frequency range of 0–1800 Hz. The definition of vibration transmissibility in decibels (dB) is as follows:

[0103] Equation (5)

[0104] in and These represent the acceleration amplitudes of the output and input, respectively.

[0105] The transitivity simulation model of a 1×5 array is as follows: Figure 6 As shown in (b), the structure is subdivided into tetrahedral elements during simulation, with a size range of [missing information]. The material properties are defined as follows: elastic modulus E = 2.6 GPa and density ρ = 1.3 g / cm³. 3 During simulation, a harmonic sweep excitation with a frequency range from 0 to 1800 Hz is applied to one end of the structure. By monitoring the acceleration response at both ends, the transmission curve of the structure is calculated using equation (4).

[0106] Figure 7 (a) The geometric model of the optimized 1×5 array structure adopts a 2-1-2 layout form of “OP-41×2+OP-42×1+OP-43×2”; Figure 7 (b) is a 3D printed physical model of the array; Figure 7 In (c), the horizontal axis represents transmissibility, and the vertical axis represents frequency. Experimental and simulation results are compared with the prediction results of the hybrid optimization model. The transmissibility curve shows that the optimized array structure... Superior vibration suppression was achieved within the target range. These results validate the feasibility of custom low- to mid-frequency vibration isolation in practical applications of bandgap engineering. This fully demonstrates the effectiveness, accuracy, and engineering applicability of the method of this invention.

[0107] Those skilled in the art should understand that the scope of protection of the present invention is not limited to the above embodiments.

[0108] In addition, in step S410, the preset bandgap frequency range of 100 to 1000 Hz can be divided into multiple sub-bands, namely 100 to 400 Hz, 200 to 500 Hz, and 400 to 1000 Hz. For each sub-band, a corresponding metamaterial unit cell is designed, and then the array combination is used to achieve continuous vibration suppression across the entire target frequency band.

[0109] By dividing the preset bandgap frequency range of 100-1000Hz into three sub-bands of 100-400Hz, 200-500Hz, and 400-1000Hz, and designing a matching metamaterial unit cell for each sub-band, the problem of a single unit cell being unable to cover a wide range of vibration suppression is solved, and the suppression gap between different frequency bands is avoided.

[0110] Furthermore, in step S440, the specific rules for fitness calculation are as follows: if the equivalent stiffness of an individual does not meet the preset constraint, which is less than 5000 N / m, the fitness value is set to 0; if the equivalent stiffness meets the constraint, the fitness value is calculated by weighted summation based on the matching degree between the bandgap frequency and the target bandgap, the bandgap width, and other indicators, with the bandgap matching degree weight ≥ 0.6.

[0111] The fitness calculation logic enables the intelligent optimization algorithm to quickly distinguish between good and bad solutions: there is no need to perform subsequent bandgap analysis on solutions that do not meet the stiffness requirements, reducing invalid calculations; at the same time, the high-weighted bandgap matching degree guides the algorithm to focus on the solution space with qualified stiffness and optimal bandgap, reducing blind search, enabling complex designs to be completed within minutes, greatly improving optimization efficiency and accuracy, and shortening the design iteration cycle.

[0112] The technical principles of the present invention have been described above with reference to specific embodiments. These descriptions are merely for explaining the principles of the invention and should not be construed as limiting the scope of protection of the invention in any way. Based on this explanation, those skilled in the art can readily conceive of other specific embodiments of the invention without inventive effort, and these embodiments will all fall within the scope of protection of the present invention.

Claims

1. A method for designing the bandgap of a stiffness-constrained sinusoidal compressive-torsional metamaterial, characterized in that, Includes the following steps: S100: Physical modeling and constraint generation. For sinusoidal compression-torsion metamaterial elements, an analytical model is established based on mechanical theory to establish the relationship between their geometric parameters, equivalent stiffness, and bandgap boundary frequency, thereby obtaining constraint equations describing the intrinsic physical relationship between the two. S200: Construct and train the PINN model, integrate the constraint equations generated in step S100, construct and train a physical information neural network, namely the PINN model, to achieve high-precision and high-efficiency forward mapping from element geometric parameters to stiffness and bandgap frequency; S300: Construct and train an inverse NN model. Construct and train a standard neural network, namely an inverse NN model, to realize the inverse mapping from target stiffness and bandgap frequency to unit geometric parameters, providing an initial solution for rapid customized design. S400: Performs constraint-based hybrid intelligent optimization, sets target stiffness and bandgap conditions, adopts intelligent optimization algorithms, calls a trained physical information neural network model as a performance evaluator, and iteratively optimizes geometric parameters.

2. The method for designing the bandgap of a stiffness-constrained sinusoidal compression-torsional metamaterial according to claim 1, characterized in that, In step S100, the geometric parameters include the sine amplitude A, the number of periods n, the diameter of the curved beam d, the initial phase φ, the radial dimension R, the straight-line distance between the two ends of the curved beam l, and the diameter of the disk R. d The thickness t of the disk and the distance H between the upper and lower disks; The metamaterial unit cell includes an upper disk and a curved beam mounted on the bottom surface of the upper disk. A compressive force F along the z-direction is applied to the upper disk. z At that time, the upper disk and the bending beam not only undergo compressive deformation It will also produce torsional deformation φ around the z-axis. z When a torque T along the z-direction is applied to the upper disk z At that time, the upper disk and the curved beam will also exhibit and φ z This compression-torsional coupling behavior can be mathematically described by the following equation: Equation (1) Where K l For compressive stiffness, K c For torsional stiffness, K t To determine the compressive-torsional coupling stiffness, the mechanical behavior of the unit cell was analyzed based on elasticity and beam theory, by applying a compressive force F in the z-direction to the disk. z The upper disk and the bent beam undergo compressive deformation. The equivalent axial stiffness K is derived. z The parsing expression: Equation (2) When a metamaterial unit cell exhibits an infinite structure along the z-direction, when subjected to excitation in the z-direction, i.e., axial compression or torsion, the two disks of the j-th staggered array show [something] in the z-direction. and Displacement and rotation angle about the z-axis and Based on Newton's second theorem and Bloch's theorem, a dynamic analysis of the j-th unit cell is performed, and the expression for its bandgap boundary frequency is derived: Equation (3) The above analytical expressions constitute the physical constraint equations describing the intrinsic relationship between geometric parameters and mechanical properties; where J is the moment of inertia, m is the mass of the disk, and f is the natural frequency.

3. These equations will be used as prior knowledge and embedded into subsequent neural network models to ensure that their predictions conform to physical laws.

4. The method for designing the bandgap of a stiffness-constrained sinusoidal compression-torsional metamaterial according to claim 2, characterized in that, In step S200, a physical information neural network model is constructed based on the physical constraints obtained in step S100. S210, determine the input and output layer parameters of the PINN model: The input layer has five nodes, corresponding to the five core geometric parameters of the sinusoidal compression-torsion metamaterial unit cell, namely, the radial position R of the curved beam, the diameter d of the curved beam, the sinusoidal amplitude A, the number of periods n, and the initial phase φ; the output layer has seven nodes, corresponding to the four types of stiffness parameters and three types of bandgap frequency parameters of the metamaterial, among which the stiffness parameters include the compression spring stiffness K. l Torsional spring stiffness K t Compression-torsional coupling stiffness K c Equivalent axial stiffness K z The bandgap frequency parameters include the lower boundary frequency f of the first bandgap. π1 The upper boundary frequency f of the first bandgap 03 The lower boundary frequency f of the second bandgap 04 ; S220, Design the hidden layer structure of the PINN model: adopt a five-layer fully connected structure, with each layer node being 256×512×512×256×128 respectively; S230, Generate training and validation datasets: Generate 2000 sets of sample data covering the above five core geometric parameters through Latin hypercube sampling, obtain the seven types of mechanical performance response parameters of the corresponding samples based on the three-dimensional modeling of metamaterial unit cells, and divide the dataset into training set and validation set in an 8:2 ratio. S240, Define the loss function for the PINN model: The loss function consists of a physics-driven term and a data-fitting term, and its expression is: Equation (4) in It is a physics-driven term. This is the data fitting term. In the loss function, MAE represents the mean absolute error. and These represent the results of the prediction dataset; then... and The results are obtained by substituting the predicted data into formulas (3) and (4); S250, and train the PINN model: the model activation function is the tanh function, the initial learning rate is set to 0.0003, the Adam optimizer is used, and an adaptive learning rate decay strategy is used during training. When the model training enters the loss plateau period, the optimizer step size is automatically reduced. The model is trained on the training set partitioned in step S230. The model parameters are optimized using the loss function defined in step S240. The model performance is evaluated using the validation set. The weights of the loss components are adjusted, and the weights of each loss component are determined through Bayesian optimization until the model loss function converges and the average coefficient of determination (R²) of the validation set is obtained. 2 With a value not less than 0.998, a PINN forward surrogate model is obtained that can achieve high-precision mapping from metamaterial unit cell geometric parameters to mechanical properties.

5. The method for designing the bandgap of a stiffness-constrained sinusoidal compression-torsional metamaterial according to claim 3, characterized in that, Step S300 constructs a standard inverse neural network model, namely the NN model, which includes the following steps: S310, Determine the input and output parameters of the NN inverse design model: The input layer is set with four nodes, corresponding to the four core performance indicators of the sinusoidal compression-torsion metamaterial, namely the equivalent axial stiffness K. z The lower boundary frequency f of the first bandgap π1 The upper boundary frequency f of the first bandgap 03 The lower boundary frequency f of the second bandgap 04 The output layer is set with four nodes, which correspond to the four core geometric parameters of the metamaterial unit cell, namely the radial position R of the curved beam, the diameter d of the curved beam, the sinusoidal amplitude A, and the initial phase φ. S320, Constructing the network structure of the NN inverse design model: Three fully connected hidden layers are used, with the number of nodes in each hidden layer being 128×256×128, and the activation function is the tanh function; S330, Generate training and validation datasets: Obtain 2000 sets of sample data covering the above four types of performance indicators and corresponding four types of geometric parameters. The sample data comes from the output results of the PINN forward surrogate model or a high-precision finite element simulation database. Divide the dataset into training set and validation set in a ratio of 8:2, and classify the dataset according to the value of the metamaterial unit cell period number n to form two sub-datasets. S340, Define the training objective of the NN inverse design model: minimize the mean square error between the predicted geometric parameters of the model output and the true geometric parameters in the sample data. S350, Training the NN inverse design model: Using a fixed learning rate of 0.0015, train NN inverse design models corresponding to n=1 and n=2 based on the two sub-datasets after classification. Update the model parameters through the training set and monitor the convergence status of the model using the validation set until the training loss and validation loss of the two models tend to be stable, thus obtaining the NN inverse design model that can deduce the initial scheme of geometric parameters from the target performance index.

6. The method for designing the bandgap of a stiffness-constrained sinusoidal compression-torsional metamaterial according to claim 4, characterized in that, Step S400 specifically includes the following steps: S410, Set optimization objectives and constraints: Define the preset stiffness constraints and target bandgap requirements of the metamaterial. The target bandgap is at least one of minimizing the initial bandgap frequency, maximizing the bandgap width, or achieving a preset bandgap frequency range of 100-1000Hz. S420, configured with intelligent optimization algorithm parameters: the multi-island genetic algorithm is selected as the core optimization algorithm, and the algorithm parameters are set as follows: the number of islands is 5, the subpopulation size of each island is 100, the crossover probability is 0.8, the mutation probability is 0.3, the migration rate is 5%, the migration interval is 10 generations, and the inter-island individual migration adopts a circular topology structure. S430, Determine the initial population of the optimization algorithm: Call the trained NN inverse design model, input the target stiffness and target bandgap parameters, obtain the initial geometric parameter scheme and use it as the elite individuals in the initial population. The proportion of these elite individuals shall not be less than 5%, and the remaining individuals shall be randomly generated within the feasible range of geometric parameter engineering to form a complete initial population. S440, Performance Evaluation and Fitness Calculation: For the metamaterial unit cell geometric parameters corresponding to each individual in the initial population, the trained PINN forward surrogate model is called to quickly output the equivalent stiffness and bandgap frequency performance indicators, and the fitness value of each individual is calculated based on the preset stiffness constraints and target bandgap requirements. S450, execute the iterative optimization process: perform genetic operations such as selection, crossover, and mutation within the island according to the rules of the intelligent optimization algorithm. When the migration interval is reached, select the top 5% of elite individuals in fitness on each island for inter-island migration; repeat the performance evaluation, genetic operations, and inter-island migration steps until the termination condition is met. S460, Output the optimal design scheme: When the termination condition is met, extract the geometric parameter combination corresponding to the individual with the best fitness, which is the optimal design scheme of the metamaterial unit cell that satisfies the preset stiffness constraint and the target bandgap requirement; The termination condition is finding at least one of the following: finding the optimal solution that satisfies all constraints or reaching the preset maximum number of iterations. S470, Array Construction and Verification: The optimal unit cells obtained in the steps are arranged in a preset layout to form a metamaterial array. The vibration suppression performance and stiffness of the array are verified through simulation analysis and experimental testing.

7. The method for designing the bandgap of a stiffness-constrained sinusoidal compression-torsional metamaterial according to claim 5, characterized in that, In step S410, the preset bandgap frequency range of 100 to 1000 Hz can be divided into multiple sub-bands, namely 100 to 400 Hz, 200 to 500 Hz, and 400 to 1000 Hz. For each sub-band, a corresponding metamaterial unit cell is designed, and then the array combination is used to achieve continuous vibration suppression across the entire target frequency band.

8. The method for designing the bandgap of a stiffness-constrained sinusoidal compression-torsional metamaterial according to claim 6, characterized in that, In step S440, the specific rules for fitness calculation are as follows: if the equivalent stiffness of an individual does not meet the preset constraint, which is less than 5000 N / m, the fitness value is set to 0; if the equivalent stiffness meets the constraint, the fitness value is calculated by weighted summation based on the matching degree between the bandgap frequency and the target bandgap, the bandgap width, and other indicators, with the bandgap matching degree weight ≥ 0.6.

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