Method and device for calibrating intrinsic characteristic parameters of metamaterials

Abstract

The present invention provides a method and apparatus for calibrating intrinsic characteristic parameters of a metamaterial. The method comprises: obtaining classical equivalent performance parameters and size-dependent equivalent performance parameters of different lattices using classical homogenization techniques and non-classical homogenization techniques, respectively; constructing a training set using the geometric parameter descriptor of the metamaterial as input features and the results obtained by the homogenization technique as output targets; processing the training set based on a radial basis kernel function to construct a covariance matrix of the input features; optimizing the radial basis kernel function based on the covariance matrix to obtain a corresponding Gaussian process regression model; inputting the geometric parameter descriptor of the metamaterial to be calibrated into the Gaussian process regression model to obtain equivalent performance parameter predictions; and determining the intrinsic characteristic parameters of the metamaterial to be calibrated based on the equivalent performance parameter predictions and the non-classical homogenization model. The present invention can solve the problem of difficulty in accurately and efficiently calibrating the intrinsic characteristic parameters of metamaterials.

Description

Technical Field

[0001] The present invention relates to the technical field of metamaterial equivalent mechanical performance calculation, and in particular to a method and device for calibrating intrinsic characteristic parameters of a metamaterial. Background Art

[0002] The rapid development of 3D printing technology has revolutionized the design and manufacturing of metamaterials. By precisely manipulating microstructures (e.g., lattice, porous, and helical topologies), metamaterials can achieve anomalous mechanical properties (such as negative stiffness, negative Poisson's ratio, and quasi-zero stiffness) unattainable by traditional materials, as well as multifunctional coupling characteristics (coordinated mechanical, acoustic, and thermal control). These advantages, combined with lightweight construction, make them ideal for high-end applications such as aerospace and marine applications.

[0003] However, due to the process limitations of 3D printing technology, the macroscopic characteristic dimensions (such as length, width, and height) of metamaterial structures are often of similar magnitude to their internal lattice dimensions. This leads to significant coupling effects between macroscopic and microstructural deformation, which in turn triggers size-dependence of the structure. Numerical simulations, limited by the computational resources and time costs of finite element methods, present significant challenges in modeling metamaterials with fully resolved microstructures at full scale. Unlike classical elasticity theory, which is based on dimensionless strain metrics, generalized continuum mechanics effectively characterizes the size-dependent mechanical behavior of metamaterial structures by introducing a dimensionless intrinsic characteristic length parameter. This means that, in addition to traditional material parameters (such as elastic modulus and density), metamaterials possess a critical, physically meaningful intrinsic characteristic length parameter. Currently, the precise calibration of this parameter remains a technical challenge in the field. Summary of the Invention

[0004] In view of this, it is necessary to provide a method and device for calibrating the intrinsic characteristic parameters of metamaterials to solve the technical problem that the intrinsic characteristic parameters of metamaterials are difficult to accurately calibrate.

[0005] In order to solve the above problems, in a first aspect, the present invention provides a method for calibrating intrinsic characteristic parameters of a metamaterial, comprising:

[0006] Perform homogenization analysis on metamaterials to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of metamaterials;

[0007] A training set is constructed by using a geometric parameter descriptor of a metamaterial as an input feature and using a classical equivalent performance parameter of the metamaterial and the size-dependent equivalent performance parameter as output targets;

[0008] Processing the training set based on a radial basis kernel function to construct a covariance matrix of input features;

[0009] Based on the covariance matrix, optimizing the radial basis kernel function to obtain a corresponding Gaussian process regression model;

[0010] The geometric parameter descriptor of the metamaterial to be calibrated is input into the Gaussian process regression model to obtain the equivalent performance parameter prediction value, and the intrinsic characteristic parameters of the metamaterial to be calibrated are determined based on the equivalent performance parameter prediction value.

[0011] In a possible implementation, the equivalent performance parameter includes: equivalent Young's modulus;

[0012] Perform homogenization analysis on metamaterials to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of metamaterials, including:

[0013] When periodic boundary conditions are applied to boundaries of a representative volume unit of the metamaterial in three different directions, and when a macroscopic tensile strain is applied to a single direction boundary of the representative volume unit, a macroscopic equivalent stress is obtained based on the local stress field of the representative volume unit at a microscopic scale, and a classical equivalent performance parameter is obtained based on the macroscopic equivalent stress and the macroscopic tensile strain;

[0014] When the boundaries of the representative volume unit of the metamaterial are kept consistent with the macroscopic boundary conditions, the macroscopic equivalent stress is obtained based on the local stress field of the representative volume unit at the microscopic scale, and the size-dependent equivalent performance parameters are obtained based on the macroscopic equivalent stress and the macroscopic tensile strain.

[0015] In a possible implementation, the equivalent performance parameter prediction value includes: a classical equivalent performance parameter prediction value and a size-dependent equivalent performance parameter prediction value;

[0016] Determining the intrinsic characteristic parameters of the metamaterial to be calibrated based on the predicted equivalent performance parameters includes:

[0017] Based on the predicted values ​​of the classical equivalent performance parameters and the predicted values ​​of the size-dependent equivalent performance parameters, and the macroscopic thickness of the metamaterial structure, the intrinsic characteristic parameters of the metamaterial to be calibrated are determined.

[0018] In one possible implementation, the intrinsic characteristic parameters of the metamaterial to be calibrated are calculated using a non-classical homogenization model, where the non-classical homogenization model is:

[0019]

[0020] in, is the size-dependent prediction of the equivalent performance parameter, is the predicted value of the classical equivalent performance parameter, is the intrinsic characteristic parameter of the metamaterial to be calibrated, i.e., the non-local intrinsic characteristic length, H is the macroscopic thickness of the metamaterial structure.

[0021] In one possible implementation, the training set is processed based on a radial basis kernel function to construct a covariance matrix of input features, including:

[0022] Calculating the covariance between any two input features in the training set based on the radial basis kernel function;

[0023] Based on the covariance between any two input features in the training set, a covariance matrix of the input features is constructed.

[0024] In a possible implementation, the calculation formula of the radial basis kernel function is:

[0025]

[0026] in, is the covariance between any two input features, is the predicted value signal variance of the Gaussian process regression prediction model, It is d dimensional length scale, The input feature d dimensional features, X is the input feature.

[0027] In one possible implementation, when optimizing the parameters in the radial basis kernel function, the following objective function is used:

[0028]

[0029] Where Y is the output target in the training set, is the noise variance between the predicted value of the Gaussian process regression model and the output target, K is the covariance matrix, is a combination of hyperparameters, , T is the transpose symbol ， is the log marginal likelihood function ， I is the identity matrix.

[0030] In one possible implementation, a training set is constructed using a geometric parameter descriptor of a metamaterial as an input feature and using the classical equivalent performance parameters and the size-dependent equivalent performance parameters of the metamaterial as output targets, including:

[0031] A basic data set is constructed by taking the geometric parameter descriptor of the metamaterial as an input feature and taking the classical equivalent performance parameter of the metamaterial and the size-dependent equivalent performance parameter as output targets;

[0032] The training set is randomly divided into five parts, and each division produces a set of training sets and corresponding test sets;

[0033] Based on each test set, the performance index of the Gaussian process regression model obtained by training the corresponding training set is evaluated, and five sets of performance indexes are obtained;

[0034] Determining the performance index of the Gaussian process regression model based on the average of the five groups of performance indexes;

[0035] When the performance index of the Gaussian process regression model does not meet the preset requirements, the radial basis kernel function is re-optimized to update the Gaussian process regression model.

[0036] In a second aspect, the present invention provides a device for calibrating intrinsic characteristic parameters of a metamaterial, comprising:

[0037] Homogenization processing module, used to perform homogenization analysis on metamaterials to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of metamaterials;

[0038] A training set construction module is used to construct a training set using a geometric parameter descriptor of a metamaterial as an input feature and using a classical equivalent performance parameter of the metamaterial and the size-dependent equivalent performance parameter as output targets;

[0039] A covariance calculation module, configured to process the training set based on a radial basis kernel function to construct a covariance matrix of input features;

[0040] A model training module is used to optimize the radial basis kernel function based on the covariance matrix to obtain a corresponding Gaussian process regression model;

[0041] The intrinsic characteristic calculation module is used to input the geometric parameter descriptor of the metamaterial to be calibrated into the Gaussian process regression model to obtain equivalent performance parameter prediction values, and determine the intrinsic characteristic parameters of the metamaterial to be calibrated based on the equivalent performance parameter prediction values.

[0042] In a third aspect, the present invention provides a non-transitory computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the method for calibrating the intrinsic characteristic parameters of a metamaterial as described in any one of the above.

[0043] The beneficial effects of the above implementation are as follows: the method and apparatus for calibrating the intrinsic characteristic parameters of a metamaterial provided by the present invention obtain the metamaterial's classical equivalent performance parameters and size-dependent equivalent performance parameters by homogenizing the metamaterial. A Gaussian process regression model trained with a training set is constructed using the metamaterial's geometric parameter descriptor as input features and the two equivalent performance parameters as output targets. Simply inputting the geometric parameters of the metamaterial structure into the trained Gaussian process regression model can obtain corresponding equivalent performance predictions. The obtained equivalent performance predictions take into account the influence of the metamaterial structure's size, resulting in results that are more accurate than those obtained using classical homogenization methods and more efficient than non-classical homogenization methods. Consequently, the intrinsic characteristic parameters of the metamaterial to be calibrated are also more accurate and efficient, thereby resolving the technical problem of the difficulty in accurately and efficiently calibrating the intrinsic characteristic parameters of metamaterials. Therefore, the present invention combines data-driven (Gaussian process regression model) with physical models (homogenization theory) to maintain the efficiency of machine learning while ensuring that the prediction results conform to the laws of mechanics. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative work.

[0045] Figure 1 A flow chart of an embodiment of a method for calibrating intrinsic characteristic parameters of a metamaterial provided by the present invention;

[0046] Figure 2 A flowchart of another embodiment of the method for calibrating intrinsic characteristic parameters of a metamaterial provided by the present invention;

[0047] Figure 3 Schematic diagram of the prediction results of the Gyroid lattice using finite element homogenization, Gaussian process regression model and non-classical homogenization model provided by the present invention;

[0048] Figure 4 A principle block diagram of an embodiment of a device for calibrating intrinsic characteristic parameters of a metamaterial provided by the present invention;

[0049] Figure 5 This is a schematic structural diagram of an embodiment of the electronic device provided by the present invention. DETAILED DESCRIPTION

[0050] The following will provide a clear and complete description of the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative work are within the scope of protection of the present invention.

[0051] In the description of the embodiments of the present application, unless otherwise specified, “a plurality of” means two or more.

[0052] The terms "including" and "having" and any variations thereof in the embodiments of the present invention are intended to cover non-exclusive inclusions. For example, a process, method, apparatus, product or device that includes a series of steps or modules is not necessarily limited to those steps or modules explicitly listed, but may include other steps or modules that are not explicitly listed or are inherent to these processes, methods, products or devices.

[0053] The naming or numbering of the steps in the embodiments of the present invention does not mean that the steps in the method flow must be executed in the time / logical sequence indicated by the naming or numbering. The execution order of the named or numbered process steps can be changed according to the technical purpose to be achieved, as long as the same or similar technical effects can be achieved.

[0054] References herein to "embodiments" mean that a particular feature, structure, or characteristic described in connection with the embodiments may be included in at least one embodiment of the present invention. The appearance of this phrase in various places in the specification does not necessarily refer to the same embodiment, nor does it constitute a separate or alternative embodiment that is mutually exclusive of other embodiments. It is understood, both explicitly and implicitly, by those skilled in the art that the embodiments described herein may be combined with other embodiments.

[0055] The present invention provides a method and device for calibrating intrinsic characteristic parameters of a metamaterial, which are described below.

[0056] like Figure 1 As shown, the present invention provides a method for calibrating intrinsic characteristic parameters of a metamaterial, comprising:

[0057] S101. Perform homogenization analysis on the metamaterial to obtain classical equivalent performance parameters and size-dependent equivalent performance parameters of the metamaterial.

[0058] It is understood that the homogenization process here refers to the finite element homogenization process, and the size dependence can refer to thickness dependence or length dependence. The equivalent performance parameters can include equivalent Young's modulus or equivalent electromagnetic performance parameters (such as negative refractive index, equivalent dielectric constant and magnetic permeability). Taking the equivalent Young's modulus as an example, according to the metamaterial structure, a suitable representative volume element (RVE) is selected, and the representative volume element is obtained by Periodic boundary conditions are applied to the boundaries in three directions to realize the micro-macroscopic mechanical quantity transfer. The equivalent Young's modulus is the representative volume unit of the metamaterial. The modulus corresponding to the infinite array in three directions is also called the classical equivalent Young's modulus. This method is the classical homogenization technique.

[0059] In some directions, the number of representative volume units is less than five. In this case, the boundary deformation of the metamaterial structure will have a great influence on the deformation of the representative volume unit, resulting in the equivalent Young's modulus of the structure being inconsistent with the above prediction. In order to obtain the equivalent Young's modulus in this case, also called the size-dependent equivalent Young's modulus , the boundaries of the representative volume unit must be consistent with the macroscopic boundary conditions. That is, the boundary conditions in the direction where the lattice size approaches the macroscopic structure size are no longer periodic boundary conditions. This is the so-called non-classical homogenization technique. In this embodiment, different TPMS (Triply Periodic Minimal Surface) lattices can be selected as representative volume units.

[0060] S102 , constructing a training set using the geometric parameter descriptor of the metamaterial as input features and the classical equivalent performance parameters and the size-dependent equivalent performance parameters of the metamaterial as output targets.

[0061] It is understood that the metamaterial's geometric parameter descriptor, also known as a parameterized geometric description, can include the metamaterial's rod diameter, unit cell size, node radius, porosity, and topology type encoding. Using the metamaterial's geometric parameter descriptor as input features and the homogenization results as output targets, a basic dataset can be constructed and divided into a training set and a test set in an 8:2 ratio.

[0062] S103 : Processing the training set based on a radial basis kernel function to construct a covariance matrix of input features.

[0063] It is understood that the radial basis kernel function is also called the Gaussian kernel function or the square exponential kernel function. The similarity, i.e., the covariance, between any two input features in the training set is calculated based on the radial basis kernel function, and a covariance matrix is ​​constructed based on multiple covariances.

[0064] S104: Based on the covariance matrix, optimize the radial basis kernel function to obtain a corresponding Gaussian process regression model.

[0065] It is understood that the covariance matrix can be combined with the Bayesian optimization method to optimize the parameters of the kernel function in Gaussian process regression, namely the radial basis kernel function. The objective function in the optimization process adopts the logarithmic marginal likelihood function.

[0066] S105 , inputting the geometric parameter descriptor of the metamaterial to be calibrated into the Gaussian process regression model to obtain equivalent performance parameter prediction values, and determining the intrinsic characteristic parameters of the metamaterial to be calibrated based on the equivalent performance parameter prediction values ​​in combination with the non-classical homogenization model.

[0067] It is understood that the method provided by this invention is applicable to metamaterial structures of arbitrary topological configurations and non-classical homogenization models. A Gaussian process regression model trained on a basic dataset simply inputs the geometric parameters of the metamaterial structure to obtain equivalent performance. This equivalent performance, which accounts for the influence of structural dimensions, is more accurate than results obtained using traditional homogenization methods. Furthermore, by combining non-classical homogenization models, intrinsic characteristic parameters of the metamaterial can be obtained. Therefore, combining data-driven (GPR) with physical models (homogenization theory) preserves the efficiency of machine learning while ensuring that predictions conform to the laws of mechanics.

[0068] The method provided by the present invention can be implemented using a computer application, which can simulate the performance and structure of the metamaterial on the computer and obtain the corresponding equivalent performance parameters. Of course, the equivalent performance parameters can also be confirmed by the user and input into the computer for subsequent calculation.

[0069] In some embodiments, the equivalent performance parameters include: equivalent Young's modulus; performing homogenization analysis on the metamaterial to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of the metamaterial, including:

[0070] When periodic boundary conditions are applied to the boundaries of three different directions of a representative volume unit of the metamaterial, and when a macroscopic tensile strain is applied to a single direction boundary of the representative volume unit, a macroscopic equivalent stress is obtained based on the spatial volume occupied by the representative volume unit and the local stress field at the microscopic scale, and a classical equivalent performance parameter is obtained based on the macroscopic equivalent stress and the macroscopic tensile strain;

[0071] When the boundaries of the representative volume unit of the metamaterial are kept consistent with the macroscopic boundary conditions, the macroscopic equivalent stress is obtained based on the spatial volume occupied by the representative volume unit and the local stress field at the microscopic scale. Based on the macroscopic equivalent stress and the macroscopic tensile strain, the size-dependent equivalent performance parameters are obtained.

[0072] It is understood that the above single direction can be x Direction, according to the metamaterial structure, select the appropriate representative volume element (RVE), through the representative volume element Periodic boundary conditions are applied on the boundaries in three directions to realize the micro-macroscopic mechanical quantity transfer. Applied to the representative volume element in On the boundary of the direction, the macro equivalent stress can be obtained according to the following formula , and then the classical equivalent Young's modulus can be obtained.

[0073] In actual metamaterial structures, only a limited number of representative volume units can be filled. In fact, in some directions, the number of representative volume units is less than five. In this case, the boundary deformation of the metamaterial structure will have a great impact on the deformation of the representative volume unit, resulting in the equivalent Young's modulus of the structure being inconsistent with the above prediction. In order to obtain the equivalent Young's modulus in this case, also called the size-dependent equivalent Young's modulus , it is necessary to keep the boundaries of the representative volume unit consistent with the macroscopic boundary conditions, that is, the boundary conditions in the direction where the lattice size is close to the macroscopic structure size are no longer periodic boundary conditions.

[0074] In some embodiments, the equivalent performance parameter prediction value includes: a classical equivalent performance parameter prediction value and a size-dependent equivalent performance parameter prediction value;

[0075] Based on the predicted values ​​of the equivalent performance parameters and in combination with the non-classical homogenization model, the intrinsic characteristic parameters of the metamaterial to be calibrated are determined, including:

[0076] Based on the classical equivalent performance parameter prediction value and the size-dependent equivalent performance parameter prediction value, as well as the macroscopic thickness of the metamaterial structure and a geometric non-classical homogenization model, the intrinsic characteristic parameters of the metamaterial to be calibrated are determined.

[0077] The non-classical homogenization model of the intrinsic characteristic parameters of the metamaterial to be calibrated is:

[0078]

[0079] in, is the size-dependent prediction of the equivalent performance parameter, is the predicted value of the classical equivalent performance parameter, is the intrinsic characteristic parameter of the metamaterial to be calibrated, i.e., the non-local intrinsic characteristic length, H is the macroscopic thickness of the metamaterial structure.

[0080] It's understandable that the homogenization model based on generalized continuum mechanics can describe this phenomenon by introducing dimensionless intrinsic characteristic parameters. Taking the nonlocal homogenization model as an example, the homogenized macroscopic equivalent effect and macroscopic equivalent strain satisfy the nonlocal relation, allowing the above formula to be derived.

[0081] In some embodiments, processing the training set based on a radial basis kernel function to construct a covariance matrix of input features includes:

[0082] Calculating the covariance between any two input features in the training set based on the radial basis kernel function;

[0083] Based on the covariance between any two input features in the training set, a covariance matrix of the input features is constructed.

[0084] Furthermore, the calculation formula of the radial basis kernel function is:

[0085]

[0086] in, is the covariance between any two input features, is the predicted value signal variance of the Gaussian process regression prediction model, It is d dimensional length scale, The input feature d dimensional features, X is the input feature.

[0087] It can be understood that the signal variance is the output amplitude of the control function, that is, the overall "amplitude" or the size of the variation range of the output modulus of the signal variance control function. d The length scale of the dimension determines the range of influence of the parameter on the output. d Dimensional features can be rod diameter, unit cell size, etc. Indicates the smoothness of the control function, The larger the value is, the greater the range of the control function changes, and more drastic output values ​​may occur.

[0088] In some embodiments, when optimizing the parameters in the radial basis kernel function, the following objective function is used:

[0089]

[0090] Where Y is the output target in the training set, is the noise variance between the predicted value of the Gaussian process regression model and the output target, K is the covariance matrix, is a combination of hyperparameters, , T is the transpose symbol ， is the log marginal likelihood function ， I is the identity matrix.

[0091] It is understandable that by continuously adjusting the hyperparameter combination , so that the coefficient of determination (R²) of the model is maximized, thereby improving the model's prediction accuracy for the equivalent Young's modulus. Also known as observation noise variance, it represents the variance of Gaussian white noise in each observation value, and can also be regarded as the magnitude of measurement error or model error.

[0092] In some embodiments, a training set is constructed using a geometric parameter descriptor of a metamaterial as an input feature and using the classical equivalent performance parameter and the size-dependent equivalent performance parameter of the metamaterial as output targets, including:

[0093] A basic data set is constructed by taking the geometric parameter descriptor of the metamaterial as an input feature and taking the classical equivalent performance parameter of the metamaterial and the size-dependent equivalent performance parameter as output targets;

[0094] The training set is randomly divided into five parts, and each division produces a set of training sets and corresponding test sets;

[0095] Based on each test set, the performance index of the Gaussian process regression model obtained by training the corresponding training set is evaluated, and five sets of performance indexes are obtained;

[0096] Determining the performance index of the Gaussian process regression model based on the average of the five groups of performance indexes;

[0097] When the performance index of the Gaussian process regression model does not meet the preset requirements, the radial basis kernel function is re-optimized to update the Gaussian process regression model.

[0098] It is understandable that 5-fold cross validation is used to evaluate the generalization ability of the model, avoid overfitting, and ensure prediction stability. This allows for an accurate mapping of the metamaterial's parameterized geometry to equivalent mechanical performance, providing reliable basic data for characteristic length calibration. The 5-fold cross validation is:

[0099] The entire dataset was randomly divided into five subsets of equal (or nearly equal) size;

[0100] Select one fold as the validation set: Select one fold from the five folds as the validation set;

[0101] The remaining four folds are used as training sets: The remaining four folds of data are used to train the model;

[0102] Training model: Use training set data to train the model;

[0103] Validate the model: Evaluate the model performance on the validation set (e.g., calculate accuracy, F1 score, etc.);

[0104] Record results: save the performance indicators of this verification;

[0105] The final evaluation result of the model is the average of the five validation results (such as accuracy and error). This step can effectively reduce the accidental influence caused by data partitioning, thereby more comprehensively reflecting the generalization ability of the model.

[0106] In some embodiments, as Figure 2 As shown, the present invention provides a method for calibrating intrinsic characteristic parameters of a metamaterial, which specifically includes the following steps:

[0107] 1. Construct basic data sets based on finite element homogenization technology.

[0108] According to the metamaterial structure, a suitable representative volume element (RVE) is selected. Periodic boundary conditions are applied on the boundaries in three directions to realize the micro-macroscopic mechanical quantity transfer. Applied to the representative volume element in On the boundary of the direction, the macro equivalent stress can be obtained according to the following formula :

[0109]

[0110] Where, is the volume of space occupied by the representative volume unit, is the local stress field at the microscopic scale, and the stress components in other directions are 0. Therefore, the equivalent Young's modulus can be obtained according to the following formula :

[0111]

[0112] It should be noted that the equivalent Young's modulus at this time is the representative volume unit of the metamaterial. The modulus corresponding to the infinite array in three directions is also called the classical equivalent Young's modulus, and the corresponding method is called the classical homogenization technique. However, in actual metamaterial structures, only a finite number of representative volume units can be filled. Even in some directions, the number of representative volume units is less than five. At this time, the boundary deformation of the metamaterial structure will have a great influence on the deformation of the representative volume unit, resulting in the equivalent Young's modulus of the structure being inconsistent with the above prediction. In order to obtain the equivalent Young's modulus in this case, also called the size-dependent equivalent Young's modulus , it is necessary to keep the boundary of the representative volume unit consistent with the macro boundary conditions, that is, the boundary conditions in the direction where the lattice size is close to the macro structure size are no longer periodic boundary conditions. Apply macroscopic tensile strain to the boundary of the direction , according to formula (1) to obtain the macro equivalent stress , and then the size-dependent Young's modulus is obtained according to formula (2) , the corresponding method is called non-classical homogenization technology.

[0113] A large number of representative volume units are parameterized geometrically (e.g., porosity, surface area, volume, unit cell size, etc.), and then homogenized using the above-mentioned classical homogenization techniques and non-classical homogenization techniques to obtain the classical equivalent Young's modulus and size-dependent equivalent Young's modulus .

[0114] 2. Obtain a Gaussian process regression model based on basic data set training.

[0115] Takes geometric parameter descriptors (including but not limited to member diameter, unit cell size, node radius, porosity and topology type code) as input features , the equivalent Young's modulus calculated by the finite element homogenization method is used as the output target , construct the basic data set. The basic data set is divided into training set and test set in the ratio of 8:2 to ensure the independence of model training and validation. The radial basis kernel function is used to process mixed-type input parameters to capture the nonlinear relationship between continuous parameters, that is, any two input vectors The relationship between is calculated by the radial basis kernel function similarity:

[0116] (3)

[0117] in, is the signal variance (control function output amplitude); is the length scale of the dth dimension (determines the extent to which this parameter affects the output); The d-th dimension of the input vector (such as rod diameter, unit cell size, etc.). Based on covariance Construct the covariance matrix The parameters of the Gaussian process regression (GPR) kernel function (3) are optimized using Bayesian optimization based on the covariance matrix, where the optimized objective function adopts the logarithmic marginal likelihood function, namely:

[0118] (4)

[0119] in, is the observation noise variance. By continuously adjusting the hyperparameter combination , maximizing the model's coefficient of determination (R²), thereby improving the model's accuracy in predicting the equivalent Young's modulus. Five-fold cross validation was used to assess the model's generalization ability, avoid overfitting, and ensure predictive stability. This allows for a precise mapping of the metamaterial's parameterized geometry to equivalent mechanical properties, providing reliable foundational data for characteristic length calibration.

[0120] 3. Calibration of intrinsic characteristic length.

[0121] The method provided by the present invention integrates data-driven modeling (Gaussian process regression model) and non-classical mechanics homogenization model. The mapping relationship between the geometric parameters of the metamaterial and the equivalent mechanical properties is established through the Gaussian process regression (GPR) model. The equivalent mechanical properties are dependent on the characteristic dimensions (length, width and height) of the metamaterial structure. For example, the equivalent Young's modulus of the metamaterial structure increases with the increase of the structure width. Homogenization based on generalized continuum mechanics can describe this phenomenon by introducing dimensioned intrinsic characteristic parameters. Taking the non-local homogenization model as an example, the macroscopic equivalent stress and macroscopic equivalent strain after homogenization satisfy the non-local relationship, that is:

[0122] (5)

[0123] in, is the longitudinal strain applied to the representative volume element, is the nonlocal intrinsic characteristic length, H is the macroscopic thickness of the metamaterial structure. The Gaussian kernel function in the formula has the following error characteristics:

[0124] (6)

[0125] Where erf() is the Gaussian error function, formula (5) can be converted to:

[0126] (7)

[0127] Integrating the Gaussian error function over [0,H] yields:

[0128] (8)

[0129] Where erfc()=1-erf() is the Gaussian error complementary function. Substituting formula (8) into formula (7), the non-classical homogenization model can be expressed as:

[0130] (9)

[0131] Similarly, the relationship between the classical equivalent Young's modulus prediction value and the size-dependent equivalent Young's modulus prediction value also satisfies formula (9). Then the relationship between the classical equivalent Young's modulus prediction value and the size-dependent equivalent Young's modulus prediction value is as follows:

[0132] (10)

[0133] in, is the size-dependent prediction of the equivalent Young's modulus, is the classical equivalent Young's modulus prediction, is the intrinsic characteristic parameter of the metamaterial to be calibrated, i.e., the non-local intrinsic characteristic length, H is the macroscopic thickness of the metamaterial structure.

[0134] In other embodiments, the intrinsic characteristic parameters of the Gyroid lattice metamaterial are calibrated using the above method, and the corresponding prediction results are as follows: Figure 3 As shown, from Figure 3 It can be seen that the results of finite element homogenization are in good agreement with the results predicted by the Gaussian process regression model. Moreover, the curves obtained by the non-classical homogenization model based on the finite element homogenization technique and the prediction results of the Gaussian process regression model are in good agreement.

[0135] It should be noted that for different generalized continuum mechanics, the intrinsic characteristic parameters are different, such as the strain gradient parameter in the strain gradient theory, the torsion characteristic length and the bending characteristic length in the micropolar theory, etc. Based on the Gaussian process regression model, the metamaterial is realized to have equivalent elastic properties. and The intrinsic characteristic parameters are obtained by fitting the parameters of the non-classical mechanics homogenization model.

[0136] The method provided by this invention is applicable to metamaterial structures of arbitrary topological configurations and non-classical homogenization models. A Gaussian process regression model trained on a basic dataset simply inputs the geometric parameters of the metamaterial structure to obtain equivalent performance. This equivalent performance, which accounts for the influence of structural dimensions, is more accurate than results obtained using traditional homogenization methods. Therefore, combining data-driven (GPR) with physical models (homogenization theory) preserves the efficiency of machine learning while ensuring that predictions conform to the laws of mechanics.

[0137] It should be noted that different non-classical theories will introduce different intrinsic characteristic lengths, such as non-local theory that introduces non-local characteristic lengths, strain gradient theory that introduces strain gradient parameters, micropolar theory that introduces torsional characteristic lengths and bending characteristic lengths, and so on.

[0138] The method provided by the present invention has the following beneficial effects: First, the present invention uses finite element homogenization technology to predict the equivalent performance of different representative volume units and construct a basic data set; second, based on a training set of the basic data set, an optimal Gaussian process regression model is obtained; finally, the Gaussian process regression model is used to achieve online prediction of the equivalent performance of different metamaterials, and the corresponding intrinsic characteristic parameters are given based on a physically consistent non-classical homogenization model. This method realizes the calibration of the intrinsic characteristic length of metamaterials for the first time, and improves the efficiency and accuracy of equivalent performance prediction of metamaterial structures.

[0139] like Figure 4 As shown, the present invention further provides a calibration device 400 for intrinsic characteristic parameters of a metamaterial, comprising:

[0140] A homogenization processing module 401 is used to perform homogenization analysis on the metamaterial to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of the metamaterial;

[0141] A training set construction module 402 is configured to construct a training set using the geometric parameter descriptor of the metamaterial as an input feature and the classical equivalent performance parameter and the size-dependent equivalent performance parameter of the metamaterial as output targets;

[0142] A covariance calculation module 403 is used to process the training set based on a radial basis kernel function to construct a covariance matrix of the input features;

[0143] A model training module 404 is configured to optimize the radial basis kernel function based on the covariance matrix to obtain a corresponding Gaussian process regression model;

[0144] The intrinsic characteristic calculation module 405 is used to input the geometric parameter descriptor of the metamaterial to be calibrated into the Gaussian process regression model to obtain equivalent performance parameter prediction values, and determine the intrinsic characteristic parameters of the metamaterial to be calibrated based on the equivalent performance parameter prediction values ​​in combination with the non-classical homogenization model.

[0145] In other embodiments, the present invention further provides a device for calibrating intrinsic characteristic parameters of a metamaterial, comprising:

[0146] Offline Dataset Construction Module: This module uses 3D modeling software to build representative volumes of different metamaterials and finite element software to calculate the equivalent mechanical properties of these representative volume units. The geometric descriptors and equivalent mechanical properties of these representative volume units are stored in a fixed location according to specific specifications, ready for use by the Gaussian process regression model.

[0147] Gaussian Process Regression Prediction Module: This module primarily implements two functions: 1) Automatically optimizes the hyperparameters of the Gaussian Process Regression model based on an offline dataset, aiming to maximize the coefficient of determination (R²); and 2) It uses the module to predict the equivalent mechanical performance of a specified representative volume unit online.

[0148] Intrinsic characteristic length calibration module: Based on the prediction results of the Gaussian regression model, different intrinsic characteristic parameters are given for different non-classical homogenization models.

[0149] The device for calibrating the intrinsic characteristic parameters of metamaterials provided in the above embodiment can implement the technical solution described in the embodiment of the method for calibrating the intrinsic characteristic parameters of metamaterials. The specific implementation principles of the above modules or units can be found in the corresponding contents of the embodiment of the method for calibrating the intrinsic characteristic parameters of metamaterials, which will not be repeated here.

[0150] like Figure 5 As shown, the present invention also provides an electronic device 500. The electronic device 500 includes a processor 501, a memory 502 and a display 503. Figure 5 Only some of the components of the electronic device 500 are shown, but it should be understood that implementation of all of the shown components is not required, and more or fewer components may be implemented instead.

[0151] In some embodiments, the memory 502 may be an internal storage unit of the electronic device 500, such as a hard disk or memory of the electronic device 500. In other embodiments, the memory 502 may also be an external storage device of the electronic device 500, such as a plug-in hard disk, a smart media card (SMC), a secure digital (SD) card, a flash card, etc. equipped on the electronic device 500.

[0152] Furthermore, the memory 502 may include both an internal storage unit of the electronic device 500 and an external storage device. The memory 502 is used to store application software installed in the electronic device 500 and various data.

[0153] In some embodiments, the processor 501 may be a central processing unit (CPU), a microprocessor, or other data processing chip, configured to execute program codes or process data stored in the memory 502, such as the method for calibrating intrinsic characteristic parameters of metamaterials in the present invention.

[0154] In some embodiments, display 503 can be an LED display, a liquid crystal display, a touch-sensitive liquid crystal display, or an OLED (Organic Light-Emitting Diode) touchscreen. Display 503 is used to display information on electronic device 500 and to display a visual user interface. Components 501-503 of electronic device 500 communicate with each other via a system bus.

[0155] In some embodiments of the present invention, when the processor 501 executes the calibration program of the intrinsic characteristic parameters of the metamaterial in the memory 502, the following steps may be implemented:

[0156] Perform homogenization analysis on metamaterials to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of metamaterials;

[0157] A training set is constructed by using a geometric parameter descriptor of a metamaterial as an input feature and using a classical equivalent performance parameter of the metamaterial and the size-dependent equivalent performance parameter as output targets;

[0158] Processing the training set based on a radial basis kernel function to construct a covariance matrix of input features;

[0159] Based on the covariance matrix, optimizing the radial basis kernel function to obtain a corresponding Gaussian process regression model;

[0160] The geometric parameter descriptor of the metamaterial to be calibrated is input into the Gaussian process regression model to obtain equivalent performance parameter prediction values, and based on the equivalent performance parameter prediction values ​​and combined with the non-classical homogenization model, the intrinsic characteristic parameters of the metamaterial to be calibrated are determined.

[0161] It should be understood that, when the processor 501 executes the calibration program of the intrinsic characteristic parameters of the metamaterial in the memory 502 , in addition to the above functions, it can also implement other functions. For details, please refer to the description of the corresponding method embodiment above.

[0162] Furthermore, the embodiments of the present invention do not specifically limit the type of electronic device 500 mentioned. The electronic device 500 may be a portable electronic device such as a mobile phone, tablet computer, personal digital assistant (PDA), wearable device, or laptop computer. Exemplary embodiments of portable electronic devices include, but are not limited to, portable electronic devices running iOS, Android, Microsoft, or other operating systems. The portable electronic devices mentioned above may also be other portable electronic devices, such as a laptop computer with a touch-sensitive surface (e.g., a touch panel). It should also be understood that in other embodiments of the present invention, the electronic device 500 may not be a portable electronic device, but rather a desktop computer with a touch-sensitive surface (e.g., a touch panel).

[0163] In another aspect, the present invention further provides a non-transitory computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the method for calibrating the intrinsic characteristic parameters of the metamaterial provided by the above methods, the method comprising:

[0164] Perform homogenization analysis on metamaterials to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of metamaterials;

[0165] A training set is constructed by using a geometric parameter descriptor of a metamaterial as an input feature and using a classical equivalent performance parameter of the metamaterial and the size-dependent equivalent performance parameter as output targets;

[0166] Processing the training set based on a radial basis kernel function to construct a covariance matrix of input features;

[0167] Based on the covariance matrix, optimizing the radial basis kernel function to obtain a corresponding Gaussian process regression model;

[0168] The geometric parameter descriptor of the metamaterial to be calibrated is input into the Gaussian process regression model to obtain equivalent performance parameter prediction values, and based on the equivalent performance parameter prediction values ​​and combined with the non-classical homogenization model, the intrinsic characteristic parameters of the metamaterial to be calibrated are determined.

[0169] Those skilled in the art will appreciate that all or part of the process steps of the above-described embodiments can be implemented by instructing related hardware through a computer program, and the program can be stored in a computer-readable storage medium, such as a magnetic disk, an optical disk, a read-only memory, or a random access memory.

[0170] The above is a detailed introduction to the method and device for calibrating the intrinsic characteristic parameters of the metamaterial provided by the present invention. Specific examples are used herein to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only intended to help understand the method and core ideas of the present invention. At the same time, for those skilled in the art, based on the ideas of the present invention, there may be changes in the specific implementation methods and application scopes. In summary, the contents of this specification should not be understood as limiting the present invention.

Claims

1. A method for calibrating intrinsic characteristic parameters of metamaterials, characterized in that: include: Perform homogenization analysis on metamaterials to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of metamaterials; A training set is constructed by using a geometric parameter descriptor of a metamaterial as an input feature and using a classical equivalent performance parameter of the metamaterial and the size-dependent equivalent performance parameter as output targets; Processing the training set based on a radial basis kernel function to construct a covariance matrix of input features; Based on the covariance matrix, optimizing the radial basis kernel function to obtain a corresponding Gaussian process regression model; Inputting the geometric parameter descriptor of the metamaterial to be calibrated into the Gaussian process regression model to obtain equivalent performance parameter prediction values, and determining the intrinsic characteristic parameters of the metamaterial to be calibrated based on the equivalent performance parameter prediction values; The equivalent performance parameter prediction values ​​include: classic equivalent performance parameter prediction values ​​and size-dependent equivalent performance parameter prediction values; Determining the intrinsic characteristic parameters of the metamaterial to be calibrated based on the predicted equivalent performance parameters includes: Determining intrinsic characteristic parameters of the metamaterial to be calibrated based on the predicted values ​​of the classical equivalent performance parameters and the predicted values ​​of the size-dependent equivalent performance parameters, as well as the macroscopic thickness of the metamaterial structure; The intrinsic characteristic parameters of the metamaterial to be calibrated are calculated using a non-classical homogenization model, which is: in, is the size-dependent prediction of the equivalent performance parameter, is the predicted value of the classical equivalent performance parameter, is the intrinsic characteristic parameter of the metamaterial to be calibrated, i.e., the non-local intrinsic characteristic length, H is the macroscopic thickness of the metamaterial structure.

2. The method for calibrating the intrinsic characteristic parameters of metamaterials according to claim 1, characterized in that: The equivalent performance parameters include: equivalent Young's modulus; Perform homogenization analysis on metamaterials to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of metamaterials, including: When periodic boundary conditions are applied to boundaries of a representative volume unit of the metamaterial in three different directions, and when a macroscopic tensile strain is applied to a single direction boundary of the representative volume unit, a macroscopic equivalent stress is obtained based on the local stress field of the representative volume unit at a microscopic scale, and a classical equivalent performance parameter is obtained based on the macroscopic equivalent stress and the macroscopic tensile strain; When the boundaries of the representative volume unit of the metamaterial are kept consistent with the macroscopic boundary conditions, the macroscopic equivalent stress is obtained based on the local stress field of the representative volume unit at the microscopic scale, and the size-dependent equivalent performance parameters are obtained based on the macroscopic equivalent stress and the macroscopic tensile strain.

3. The method for calibrating intrinsic characteristic parameters of metamaterials according to claim 1, characterized in that: The training set is processed based on a radial basis kernel function to construct a covariance matrix of the input features, including: Calculating the covariance between any two input features in the training set based on the radial basis kernel function; Based on the covariance between any two input features in the training set, a covariance matrix of the input features is constructed.

4. The method for calibrating intrinsic characteristic parameters of metamaterials according to claim 3, characterized in that: The calculation formula of the radial basis kernel function is: in, is the covariance between any two input features, is the predicted value signal variance of the Gaussian process regression prediction model, It is d dimensional length scale, The input feature d dimensional features, X is the input feature.

5. The method for calibrating intrinsic characteristic parameters of metamaterials according to claim 4, characterized in that: When optimizing the parameters in the radial basis kernel function, the following objective function is used: Where Y is the output target in the training set, is the noise variance between the predicted value of the Gaussian process regression model and the output target, K is the covariance matrix, is a combination of hyperparameters, , T is the transpose symbol ， is the log marginal likelihood function ， I is the identity matrix.

6. The method for calibrating intrinsic characteristic parameters of a metamaterial according to any one of claims 1 to 5, characterized in that: A training set is constructed using the geometric parameter descriptor of the metamaterial as an input feature and the classical equivalent performance parameter and the size-dependent equivalent performance parameter of the metamaterial as output targets, including: A basic data set is constructed by taking the geometric parameter descriptor of the metamaterial as an input feature and taking the classical equivalent performance parameter of the metamaterial and the size-dependent equivalent performance parameter as output targets; The training set is randomly divided into five parts, and each division produces a set of training sets and corresponding test sets; Based on each test set, the performance index of the Gaussian process regression model obtained by training the corresponding training set is evaluated, and five sets of performance indexes are obtained; Determining the performance index of the Gaussian process regression model based on the average of the five groups of performance indexes; When the performance index of the Gaussian process regression model does not meet the preset requirements, the radial basis kernel function is re-optimized to update the Gaussian process regression model.

7. A device for calibrating intrinsic characteristic parameters of metamaterials, characterized in that: include: Homogenization processing module, used to perform homogenization analysis on metamaterials to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of metamaterials; A training set construction module is used to construct a training set using a geometric parameter descriptor of a metamaterial as an input feature and using a classical equivalent performance parameter of the metamaterial and the size-dependent equivalent performance parameter as output targets; A covariance calculation module, configured to process the training set based on a radial basis kernel function to construct a covariance matrix of input features; A model training module is used to optimize the radial basis kernel function based on the covariance matrix to obtain a corresponding Gaussian process regression model; an intrinsic characteristic calculation module, configured to input the geometric parameter descriptor of the metamaterial to be calibrated into the Gaussian process regression model to obtain equivalent performance parameter prediction values, and determine the intrinsic characteristic parameters of the metamaterial to be calibrated based on the equivalent performance parameter prediction values; The equivalent performance parameter prediction values ​​include: classic equivalent performance parameter prediction values ​​and size-dependent equivalent performance parameter prediction values; Determining the intrinsic characteristic parameters of the metamaterial to be calibrated based on the predicted equivalent performance parameters includes: Determining intrinsic characteristic parameters of the metamaterial to be calibrated based on the predicted values ​​of the classical equivalent performance parameters and the predicted values ​​of the size-dependent equivalent performance parameters, as well as the macroscopic thickness of the metamaterial structure; The intrinsic characteristic parameters of the metamaterial to be calibrated are calculated using a non-classical homogenization model, which is: in, is the size-dependent prediction of the equivalent performance parameter, is the predicted value of the classical equivalent performance parameter, is the intrinsic characteristic parameter of the metamaterial to be calibrated, i.e., the non-local intrinsic characteristic length, H is the macroscopic thickness of the metamaterial structure.

8. A non-transitory computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method for calibrating the intrinsic characteristic parameters of a metamaterial are implemented.

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