Calibration method and device for intrinsic characteristic parameters of metamaterial

Abstract

The invention provides a metamaterial intrinsic characteristic parameter calibration method and device. The method comprises the following steps: respectively obtaining classical equivalent performance parameters and size-dependent equivalent performance parameters of different lattices through a classical homogenization technology and a non-classical homogenization technology; constructing a training set by taking a geometric parameter descriptor of the metamaterial as an input feature and taking a result obtained by a homogenization technology as an output target; processing the training set based on a radial basis kernel function to construct a covariance matrix of the input features; based on the covariance matrix, optimizing a radial basis kernel function to obtain a corresponding Gaussian process regression model; and inputting the geometric parameter descriptors of the to-be-calibrated metamaterial into the Gaussian process regression model to obtain an equivalent performance parameter predicted value, and determining the intrinsic characteristic parameters of the to-be-calibrated metamaterial based on the equivalent performance parameter predicted value in combination with the non-classical homogenization model. The problem that the intrinsic characteristic parameters of the metamaterial are difficult to accurately and efficiently calibrate can be solved.

Description

Technical Field

[0001] The present invention relates to the technical field of calculating the equivalent mechanical properties of metamaterials, and particularly to a method and device for calibrating the intrinsic characteristic parameters of metamaterials. Background Art

[0002] Due to the rapid development of 3D printing technology, the design and manufacturing of metamaterials have achieved revolutionary breakthroughs. By precisely regulating the microstructure (such as lattice, porous, helical and other topological configurations), metamaterials can achieve abnormal mechanical properties (such as negative stiffness, negative Poisson's ratio, quasi-zero stiffness, etc.) and multifunctional coupling characteristics (synergistic regulation of mechanics - acoustics - thermotics) that traditional materials do not possess, and at the same time have the advantage of lightweight, becoming an ideal material in high-end fields such as aerospace and navigation.

[0003] However, due to the process limitations of 3D printing technology, the macroscopic characteristic dimensions (such as length, width, height, etc.) of the metamaterial structure are often in the same order of magnitude as the internal lattice dimensions, resulting in a significant coupling effect between macroscopic deformation and microscopic structure deformation, and thus triggering the size-dependent phenomenon of the structure. In terms of numerical simulation, limited by the finite element calculation resources and time costs, there are huge challenges in performing a complete-scale modeling of metamaterials containing fully resolved microstructures. Different from the classical elasticity theory based on dimensionless strain measures, the generalized continuum mechanics can effectively characterize the size-dependent mechanical behavior of metamaterial structures by introducing a dimensional intrinsic characteristic length parameter. This means that in addition to the traditional material parameters (such as elastic modulus, density, etc.), there is also a key intrinsic characteristic length parameter with physical significance for metamaterials. At present, the precise calibration method for this parameter remains a technical problem 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 it is difficult to accurately calibrate the intrinsic characteristic parameters of metamaterials.

[0005] To solve the above problems, in a first aspect, the present invention provides a method for calibrating the intrinsic characteristic parameters of metamaterials, including: Performing homogenization analysis on the metamaterials to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of the metamaterials; Using the geometric parameter descriptors of the metamaterials as input features, and using the classical equivalent performance parameters and the size-dependent equivalent performance parameters of the metamaterials as output targets to construct a training set; Processing the training set based on the 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; Input the geometric parameter descriptors of the metamaterial to be calibrated into the Gaussian process regression model to obtain predicted values of equivalent performance parameters, and determine the intrinsic characteristic parameters of the metamaterial to be calibrated based on the predicted values of the equivalent performance parameters.

[0006] In a possible implementation, the equivalent performance parameters include: equivalent Young's modulus; Perform homogenization analysis on the metamaterial to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of the metamaterial, including: Under the condition of applying periodic boundary conditions on the boundaries of the representative volume element of the metamaterial in three different directions, and when a macroscopic tensile strain is applied to the boundary of the representative volume element in a single direction, obtain the macroscopic equivalent stress based on the local stress field of the representative volume element at the microscale, and obtain the classical equivalent performance parameters based on the macroscopic equivalent stress and the macroscopic tensile strain; Under the condition of keeping the boundary of the representative volume element of the metamaterial consistent with the macroscopic boundary conditions, obtain the macroscopic equivalent stress based on the local stress field of the representative volume element at the microscale, and obtain the size-dependent equivalent performance parameters based on the macroscopic equivalent stress and the macroscopic tensile strain.

[0007] In a possible implementation, the predicted values of the equivalent performance parameters include: predicted values of classical equivalent performance parameters and predicted values of size-dependent equivalent performance parameters; Determine the intrinsic characteristic parameters of the metamaterial to be calibrated based on the predicted values of the equivalent performance parameters, including: Determine the intrinsic characteristic parameters of the metamaterial to be calibrated based on the predicted values of the classical equivalent performance parameters, the predicted values of the size-dependent equivalent performance parameters, and the macroscopic thickness of the metamaterial structure.

[0008] In a possible implementation, the intrinsic characteristic parameters of the metamaterial to be calibrated are calculated using a non-classical homogenization model, and the non-classical homogenization model is:

[0009] where is the predicted value of the size-dependent equivalent performance parameter, is the predicted value of the classical equivalent performance parameter, is the intrinsic characteristic parameter of the metamaterial to be calibrated, that is, the non-local intrinsic characteristic length, H is the macroscopic thickness of the metamaterial structure.

[0010] In a possible implementation, process the training set based on the radial basis kernel function to construct the covariance matrix of the input features, including: Based on the radial basis kernel function, calculate the covariance between any two input features in the training set; Based on the covariance between any two input features in the training set, construct the covariance matrix of the input features.

[0011] In a possible implementation, the calculation formula of the radial basis kernel function is:

[0012] where is the covariance between any two input features, is the predicted value signal variance of the Gaussian process regression prediction model, is the length scale of the d th dimension, is the d th dimension feature of the input feature, and X is the input feature.

[0013] In a possible implementation, when optimizing the parameters in the radial basis kernel function, the following objective function is adopted:

[0014] 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 the hyperparameter combination, , T is the transpose symbol ， is the log marginal likelihood function ， I is the identity matrix.

[0015] In a possible implementation, using the geometric parameter descriptor of the metamaterial as the input feature and the classical equivalent performance parameter and the size-dependent equivalent performance parameter of the metamaterial as the output target, construct a training set, including: Using the geometric parameter descriptor of the metamaterial as the input feature and the classical equivalent performance parameter and the size-dependent equivalent performance parameter of the metamaterial as the output target, construct a basic data set; Randomly divide the training set five times, and each division obtains a group of training sets and corresponding test sets; Based on each group of test sets, evaluate the performance metrics of the Gaussian process regression model trained by the corresponding training set to obtain five groups of performance metrics; Based on the average value of the five groups of performance metrics, determine the performance metrics of the Gaussian process regression model; In the case that the performance index of the Gaussian process regression model does not meet the preset requirements, re-optimize the radial basis kernel function to update the Gaussian process regression model.

[0016] In a second aspect, the present invention provides a calibration device for the intrinsic characteristic parameters of a metamaterial, including: A homogenization processing module, configured to perform homogenization analysis on the metamaterial to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of the metamaterial; A training set construction module, configured to use the geometric parameter descriptors of the metamaterial as input features, and use the classical equivalent performance parameters and the size-dependent equivalent performance parameters of the metamaterial as output targets to construct a training set; A covariance calculation module, configured to process the training set based on the radial basis kernel function to construct a covariance matrix of the input features; A model training module, configured 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 descriptors of the metamaterial to be calibrated into the Gaussian process regression model to obtain predicted values of equivalent performance parameters, and determine the intrinsic characteristic parameters of the metamaterial to be calibrated based on the predicted values of the equivalent performance parameters.

[0017] In a third aspect, the present invention provides a non-transitory computer-readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, the steps of the calibration method for the intrinsic characteristic parameters of the metamaterial as described in any one of the above are implemented.

[0018] The beneficial effects of adopting the above implementation method are as follows: The calibration method and device for the intrinsic characteristic parameters of the metamaterial provided by the present invention obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of the metamaterial through homogenization analysis of the metamaterial; using the geometric parameter descriptors of the metamaterial as input features and the two equivalent performance parameters as output targets, the Gaussian process regression model trained by the constructed training set only needs to input the geometric parameters of the metamaterial structure into the trained Gaussian process regression model as input parameters to obtain corresponding predicted values of equivalent performance, and the obtained predicted values of equivalent performance take into account the influence of the metamaterial structure size, are more accurate than the results obtained by the classical homogenization method, and are more efficient than the non-classical homogenization method. Furthermore, the obtained intrinsic characteristic parameters of the metamaterial to be calibrated are also more accurate and efficient, thus solving the technical problem that it is difficult to accurately and efficiently calibrate the intrinsic characteristic parameters of the metamaterial. Therefore, the present invention combines data-driven (Gaussian process regression model) with physical model (homogenization theory), retains the high efficiency of machine learning, and ensures that the prediction results conform to mechanical laws. Description of the Drawings

[0019] To more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those skilled in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0020] Figure 1 It is a flowchart of an embodiment of the calibration method for the intrinsic characteristic parameters of the metamaterial provided by the present invention; Figure 2 It is a flowchart of another embodiment of the calibration method for the intrinsic characteristic parameters of the metamaterial provided by the present invention; Figure 3 It is a schematic diagram of the prediction results of the Gyroid lattice using the finite element homogenization, Gaussian process regression model, and non - classical homogenization model provided by the present invention; Figure 4 It is a principle block diagram of an embodiment of the calibration device for the intrinsic characteristic parameters of the metamaterial provided by the present invention; Figure 5 It is a schematic structural diagram of an embodiment of the electronic device provided by the present invention. Detailed implementation manners

[0021] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative efforts belong to the scope of protection of the present invention.

[0022] In the description of the embodiments of the present application, unless otherwise specified, the meaning of "a plurality of" is two or more.

[0023] In the embodiments of the present invention, the terms "include" and "have" and any variations thereof are intended to cover non - exclusive inclusion. For example, a process, method, device, product, or equipment that includes a series of steps or modules does not necessarily have to be limited to those clearly listed steps or modules, but may include other steps or modules that are not clearly listed or are inherent to these processes, methods, products, or equipment.

[0024] In the embodiments of the present invention, the naming or numbering of the steps 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 already named or numbered process steps can change the execution order according to the technical purpose to be achieved, as long as the same or similar technical effects can be achieved.

[0025] References to "embodiments" in this specification mean that the particular features, structures, or characteristics described in connection with the embodiments can be included in at least one embodiment of the invention. The phrase appears in various places in the specification and does not necessarily refer to the same embodiment, nor is it an independent or alternative embodiment mutually exclusive of other embodiments. Those skilled in the art will explicitly and implicitly understand that the embodiments described herein can be combined with other embodiments.

[0026] The present invention provides a method and device for calibrating the intrinsic characteristic parameters of a metamaterial, which will be described separately below.

[0027] As Figure 1 shown, the present invention provides a method for calibrating the intrinsic characteristic parameters of a metamaterial, including: S101. Perform homogenization analysis on the metamaterial to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of the metamaterial.

[0028] It can be understood that the homogenization process here refers to the finite element homogenization process. Size dependence can refer to thickness dependence or length dependence. The equivalent performance parameters can include the equivalent Young's modulus or the equivalent electromagnetic performance parameters (e.g., negative refractive index, equivalent dielectric constant, and permeability). Taking the equivalent Young's modulus as an example, according to the metamaterial structure, a suitable representative volume element (RVE) is selected, and by applying periodic boundary conditions to the boundaries in three directions of the representative volume element, the transfer of micro-macro mechanical quantities is realized. The equivalent Young's modulus is the modulus corresponding to the infinite array of the representative volume element of the metamaterial in three directions, which is also called the classical equivalent Young's modulus. This method is the classical homogenization technique. In some directions, the number of representative volume elements is less than five. At this time, the boundary deformation of the metamaterial structure will have a great impact on the deformation of the representative volume element, resulting in a phenomenon that the equivalent Young's modulus of the structure does not match the above prediction. In order to obtain the equivalent Young's modulus in this case, which is also called the size-dependent equivalent Young's modulus

[0029] , it is necessary to keep the boundary of the representative volume element 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, which is the so-called non-classical homogenization technique. In this embodiment, different TPMS (Triply Periodic Minimal Surface) lattices can be selected as the representative volume element. S102. Construct a training set with the geometric parameter descriptors 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.

[0030] ​

[0031] It can be understood that the geometric parameter descriptor of the metamaterial, also known as the parametric geometric description, can be the rod diameter of the metamaterial, the unit cell size, the node radius, the porosity, and the topological type encoding, etc. Using the geometric parameter descriptor of the metamaterial as the input feature and the homogenization processing result as the output target, a basic dataset can be constructed, and the basic dataset is divided into a training set and a test set according to a ratio of 8:2.

[0032] S103. Process the training set based on the radial basis kernel function to construct the covariance matrix of the input features.

[0033] It can be understood that the radial basis kernel function is also known as the Gaussian kernel function or the squared exponential kernel function. Calculate the similarity, that is, the covariance, between any two input features in the training set based on the radial basis kernel function, and construct the covariance matrix based on multiple covariances.

[0034] S104. Optimize the radial basis kernel function based on the covariance matrix to obtain the corresponding Gaussian process regression model.

[0035] It can be understood that the covariance matrix can be combined, and the Bayesian optimization method can be used to optimize the parameters in the kernel function in Gaussian process regression, that is, the radial basis kernel function. The objective function in the optimization process uses the log marginal likelihood function.

[0036] S105. Input the geometric parameter descriptor of the metamaterial to be calibrated into the Gaussian process regression model to obtain the predicted value of the equivalent performance parameter, and based on the predicted value of the equivalent performance parameter, combine with the non-classical homogenization model to determine the intrinsic characteristic parameters of the metamaterial to be calibrated.

[0037] It can be understood that the method provided by the present invention is applicable to metamaterial structures and non-classical homogenization models with any topological configuration. The Gaussian process regression model trained based on the basic dataset only needs to input the geometric parameters of the metamaterial structure as input parameters into the model to obtain the corresponding equivalent performance, and the obtained equivalent performance takes into account the influence of the structure size, and the result obtained is more accurate than that of the traditional homogenization method. Furthermore, by combining with the non-classical homogenization model, the intrinsic characteristic parameters of the metamaterial can be obtained. Therefore, by combining data-driven (GPR) with physical models (homogenization theory), both the high efficiency of machine learning is retained and the prediction result conforms to the mechanical law.

[0038] The method provided by the present invention can be implemented through an application program on a computer, and the performance and structure of the metamaterial can be simulated on the computer to obtain the corresponding equivalent performance parameters. Of course, the equivalent performance parameters can also be input into the computer after being confirmed by the user and then subsequent calculations can be performed.

[0039] 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: Under the condition of applying periodic boundary conditions to the boundaries of the representative volume element of the metamaterial in three different directions, and when a macroscopic tensile strain is applied to the boundary of the representative volume element in a single direction, based on the spatial volume occupied by the representative volume element and the local stress field at the microscale, the macroscopic equivalent stress is obtained, and based on the macroscopic equivalent stress and the macroscopic tensile strain, the classical equivalent performance parameters are obtained; Under the condition of keeping the boundary of the representative volume element of the metamaterial consistent with the macroscopic boundary conditions, based on the spatial volume occupied by the representative volume element and the local stress field at the microscale, the macroscopic equivalent stress is obtained, and based on the macroscopic equivalent stress and the macroscopic tensile strain, the size-dependent equivalent performance parameters are obtained.

[0040] It can be understood that the above single direction can be x a direction. According to the metamaterial structure, a suitable representative volume element (RVE) is selected, and the transfer of micro-macro mechanical quantities is realized by applying periodic boundary conditions to the boundaries of the representative volume element in three directions. When the macroscopic tensile strain is applied to the boundary pair of the representative volume element in the direction, the macroscopic equivalent stress can be obtained according to the following formula , and then the classical equivalent Young's modulus can be obtained.

[0041] In an actual metamaterial structure, only a finite number of representative volume elements can be filled. Even in some directions, the number of representative volume elements is less than five. At this time, the boundary deformation of the metamaterial structure will have a great impact on the deformation of the representative volume element, resulting in a phenomenon that the equivalent Young's modulus of the structure does not match 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 element 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.

[0042] In some embodiments, the predicted values of the equivalent performance parameters include: predicted values of classical equivalent performance parameters and predicted values of size-dependent equivalent performance parameters; Based on the predicted values of the equivalent performance parameters, combined with a non-classical homogenization model, the intrinsic characteristic parameters of the metamaterial to be calibrated are determined, including: Based on the predicted values of the classical equivalent performance parameters, the predicted values of the size-dependent equivalent performance parameters, and the macroscopic thickness of the metamaterial structure, the geometric non-classical homogenization model determines the intrinsic characteristic parameters of the metamaterial to be calibrated.

[0043] The non-classical homogenization model for the quantity of the intrinsic characteristic parameters of the metamaterial to be calibrated is:

[0044] Wherein, is the predicted value of the size-dependent 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.

[0045] It can be understood that the homogenization model based on the generalized continuum mechanics can describe this phenomenon by introducing the dimensional 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, and the above formula can be derived.

[0046] In some embodiments, processing the training set based on the radial basis kernel function to construct the covariance matrix of the input features includes: Calculating the covariance between any two input features in the training set based on the radial basis kernel function; Constructing the covariance matrix of the input features based on the covariance between any two input features in the training set.

[0047] Further, the calculation formula of the radial basis kernel function is:

[0048] Wherein, is the covariance between any two input features, is the predicted value signal variance of the Gaussian process regression prediction model, is the length scale of the d th dimension, is the d th dimensional feature of the input feature, and X is the input feature.

[0049] It can be understood that the signal variance controls the output amplitude of the function, that is, the signal variance controls the overall "amplitude" or the size of the change range of the output modulus of the function. The length scale of the d th dimension determines the influence range of this parameter on the output, and the d th dimensional feature can be the rod diameter, the unit cell size, etc. represents the smoothness of the control function, The larger it is, the greater the change range of the control function, and more drastic output values may occur.

[0050] In some embodiments, when optimizing the parameters in the radial basis kernel function, the following objective function is adopted:

[0051] 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 the hyperparameter combination, , T is the transpose symbol ， is the log marginal likelihood function ， I is the identity matrix.

[0052] It can be understood that by continuously adjusting the hyperparameter combination , the coefficient of determination (R²) of the model is maximized, thereby improving the prediction accuracy of the model for the equivalent Young's modulus. Also known as the observation noise variance, it represents the variance of Gaussian white noise existing in each observation value, and can also be regarded as the magnitude of measurement error or model error.

[0053] In some embodiments, taking the geometric parameter descriptor of the metamaterial as the input feature and the classical equivalent performance parameter and the size-dependent equivalent performance parameter of the metamaterial as the output target, a training set is constructed, including: Taking the geometric parameter descriptor of the metamaterial as the input feature and the classical equivalent performance parameter and the size-dependent equivalent performance parameter of the metamaterial as the output target, a basic data set is constructed; Randomly divide the training set into five parts, and each division obtains a set of training set and the corresponding test set; Based on each set of test sets, evaluate the performance metrics of the Gaussian process regression model trained by the corresponding training set to obtain five sets of performance metrics; Based on the average value of the five sets of performance metrics, determine the performance metrics of the Gaussian process regression model; In the case that the performance metrics of the Gaussian process regression model do not meet the preset requirements, re-optimize the radial basis kernel function to update the Gaussian process regression model.

[0054] It is understandable that the generalization ability of the model is evaluated using 5-fold Cross Validation to avoid overfitting and ensure prediction stability. Thus, an accurate mapping from the parametric geometry of the metamaterial to the equivalent mechanical properties is achieved, providing reliable basic data for the calibration of the characteristic length. The 5-fold cross-validation is as follows: The entire dataset is randomly divided into five subsets of equal (or nearly equal) size; Select one fold as the validation set: One of the five folds is selected as the validation set; The remaining four folds are used as the training set: The remaining four folds of data are used to train the model; Train the model: The model is trained using the training set data; Validate the model: Evaluate the performance of the model on the validation set (e.g., calculate metrics such as accuracy, F1 score, etc.); Record the results: Save the performance metrics of this validation; The results of the five validations (e.g., accuracy, error, etc.) are averaged as the final evaluation result of the model. This step can effectively reduce the accidental impact caused by data partitioning, thus more comprehensively reflecting the generalization ability of the model.

[0055] In some embodiments, as Figure 2 shown, the present invention provides a method for calibrating the intrinsic characteristic parameters of a metamaterial, specifically including the following steps: 1. Construct a basic dataset based on the finite element homogenization technique.

[0056] According to the metamaterial structure, a suitable representative volume element (RVE) is selected, and the transfer of micro-macro mechanical quantities is realized by applying periodic boundary conditions on the boundaries in three directions of the representative volume element. When the macroscopic tensile strain is applied to the boundary pair of the representative volume element in the direction, the macroscopic equivalent stress can be obtained according to the following formula :

[0057] wherein, is the spatial volume occupied by the representative volume element, 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

[0058] It should be noted that the equivalent Young's modulus at this time is that of the metamaterial representative volume element in The modulus corresponding to the infinite array in three directions, also known as the classical equivalent Young's modulus, and the corresponding method is called the classical homogenization technique. In an actual metamaterial structure, only a finite number of representative volume elements can be filled. Even in some directions, the number of representative volume elements is less than five. At this time, the boundary deformation of the metamaterial structure will have a great impact on the deformation of the representative volume element, resulting in a phenomenon that the equivalent Young's modulus of the structure does not conform to the above prediction. To obtain the equivalent Young's modulus in this case, also known as the size-dependent equivalent Young's modulus , it is necessary to make the boundary of the representative volume element 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. At this time, similarly, on the boundary pair in the direction of the representative volume element , a macroscopic tensile strain is applied , and the macroscopic equivalent stress is obtained according to formula (1) , and then the size-dependent Young's modulus is obtained according to formula (2) , and the corresponding method is called the non-classical homogenization technique.

[0059] Perform parametric geometric descriptions on a large number of representative volume elements (for example, porosity, surface area, volume, unit cell size, etc.), and then use the above classical homogenization technique and non-classical homogenization technique for homogenization to obtain the classical equivalent Young's modulus and the size-dependent equivalent Young's modulus .

[0060] 2. Train a Gaussian process regression model based on the basic dataset.

[0061] Use geometric parameter descriptors (including but not limited to rod diameter, unit cell size, node radius, porosity, and topological type encoding) as input features , and use the equivalent Young's modulus calculated by the finite element homogenization method as the output target , and construct a basic dataset. Divide the basic dataset into a training set and a test set according to a ratio of 8:2 to ensure the independence of model training and verification. Use a radial basis kernel function to process mixed-type input parameters to capture the nonlinear relationship of continuous parameters, that is, the relationship between any two input vectors is calculated for similarity through the radial basis kernel function: (3) where is the signal variance (controlling the output amplitude of the function); is the length scale of the d-th dimension (determining the influence range of this parameter on the output); is the d-th dimensional feature of the input vector (such as rod diameter, unit cell size, etc.). Based on the covariance a covariance matrix is constructed The parameters in the kernel function (3) of Gaussian process regression (GPR) are optimized using Bayesian Optimization based on the covariance matrix. The objective function for optimization is the log marginal likelihood function, i.e.: (4) where, is the observation noise variance. By continuously adjusting the hyperparameter combination , the coefficient of determination (R²) of the model is maximized, thereby improving the prediction accuracy of the model for the equivalent Young's modulus. The generalization ability of the model is evaluated using five-fold cross-validation to avoid overfitting and ensure prediction stability. Thus, an accurate mapping from the parametric geometry of the metamaterial to the equivalent mechanical properties is achieved, providing reliable basic data for the calibration of the characteristic length.

[0062] 3. Calibration of the intrinsic characteristic length.

[0063] The method provided by the present invention combines data-driven modeling (Gaussian process regression model) with a non-classical mechanics homogenization model. The mapping relationship between the geometric parameters of the metamaterial and the equivalent mechanical properties is established through a Gaussian process regression (GPR) model. The equivalent mechanical properties are dependent on the characteristic dimensions (length, width, height, etc.) of the metamaterial structure. For example, the equivalent Young's modulus of the metamaterial structure increases with the increase in the structure width. Homogenization based on generalized continuum mechanics can describe this phenomenon by introducing a dimensional intrinsic characteristic parameter. Taking the non-local homogenization model as an example, the macroscopic equivalent stress and macroscopic equivalent strain after homogenization satisfy a non-local relationship, i.e.: (5) where, is the long strain applied to the representative volume element, is the non-local intrinsic characteristic length, H is the macroscopic thickness of the metamaterial structure. The Gaussian kernel function in the formula has the following error characteristics: (6) where erf() is the Gaussian error function, and formula (5) can be transformed into: (7) Integrating the Gaussian error function over [0, H], we get: (8) 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: (9) Similarly, the relationship between the predicted value of the classical equivalent Young's modulus and the predicted value of the size-dependent equivalent Young's modulus also satisfies Equation (9). Then, the relationship between the predicted value of the classical equivalent Young's modulus and the predicted value of the size-dependent equivalent Young's modulus is as follows: (10) Wherein, is the predicted value of the size-dependent equivalent Young's modulus, is the predicted value of the classical equivalent Young's modulus, is the intrinsic characteristic parameter of the metamaterial to be calibrated, that is, the nonlocal intrinsic characteristic length, H is the macroscopic thickness of the metamaterial structure.

[0064] In some other embodiments, for the metamaterial with a Gyroid lattice, the above method is used to calibrate the intrinsic characteristic parameters, and the corresponding prediction results are as shown in Figure 3 . It can be seen from Figure 3 that there is good consistency between the results of finite element homogenization and the results predicted by the Gaussian process regression model. And the curves obtained by the non-classical homogenization model based on the finite element homogenization technology and the prediction results of the Gaussian process regression model have good consistency.

[0065] It should be noted that: for different generalized continuum mechanics, the intrinsic characteristic parameters are different. For example, the strain gradient parameter in the strain gradient theory, the torsional characteristic length and the bending characteristic length in the micropolar theory, etc. Based on the Gaussian process regression model, the prediction of the equivalent elastic properties of the metamaterial is realized and . The intrinsic characteristic parameters are obtained by fitting the parameters of the non-classical mechanics homogenization model.

[0066] The method provided by the present invention is applicable to metamaterial structures with arbitrary topological configurations and non-classical homogenization models. Based on the Gaussian process regression model trained from the basic data set, only the geometric parameters of the metamaterial structure need to be input into the model as input parameters, and the corresponding equivalent properties can be obtained. Moreover, the obtained equivalent properties take into account the influence of the structure size and are more accurate than the results obtained by the traditional homogenization method. Therefore, combining data-driven (GPR) with physical models (homogenization theory) not only retains the high efficiency of machine learning but also ensures that the prediction results conform to the mechanical laws.

[0067] It should be noted that different non-classical theories will introduce different intrinsic characteristic lengths. For example, the nonlocal theory introducing the nonlocal characteristic length, the strain gradient theory introducing the strain gradient parameter, the micropolar theory introducing the torsional characteristic length and the bending characteristic length, and so on.

[0068] The method provided by the present invention has the following beneficial effects: First, the present invention realizes the prediction of the equivalent performance of different representative volume elements through the finite element homogenization technology and constructs a basic data set; second, based on the training set of the basic data set, an optimal Gaussian process regression model is obtained; finally, the Gaussian process regression model is used to realize the online prediction of the equivalent performance of different metamaterials, and the corresponding intrinsic characteristic parameters are given based on the non-classical homogenization model with physical consistency. The present invention realizes the calibration of the intrinsic characteristic length of the metamaterial for the first time and improves the efficiency and accuracy of the prediction of the equivalent performance of the metamaterial structure.

[0069] As Figure 4 shown, the present invention also provides a calibration device 400 for the intrinsic characteristic parameters of a metamaterial, including: A homogenization processing module 401, configured to perform homogenization analysis on the metamaterial to obtain the classical equivalent performance parameters and the size-dependent equivalent performance parameters of the metamaterial; A training set construction module 402, configured to construct a training set by using the geometric parameter descriptors 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; A covariance calculation module 403, configured to process the training set based on a radial basis kernel function to construct a covariance matrix of the input features; A model training module 404, configured 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 405, configured to input the geometric parameter descriptors of the metamaterial to be calibrated into the Gaussian process regression model to obtain a predicted value of the equivalent performance parameter, and based on the predicted value of the equivalent performance parameter, combine with a non-classical homogenization model to determine the intrinsic characteristic parameters of the metamaterial to be calibrated.

[0070] In some other embodiments, the present invention also provides a calibration device for the intrinsic characteristic parameters of a metamaterial, including: An offline data set construction module: used to mobilize 3D modeling software to establish different metamaterial representative volume software, and call finite element software to calculate the equivalent mechanical properties of different metamaterial representative volume elements. And store the geometric descriptors and equivalent mechanical properties of the representative volume elements in a fixed position according to certain specifications, waiting for the Gaussian process regression model to use.

[0071] A Gaussian process regression prediction module: mainly realizes two functions, 1) based on the offline data set, with the goal of maximizing the coefficient of determination (R²), automatically realizes the hyperparameter optimization of the Gaussian process regression model; 2) inputs a specified representative volume element, and realizes the online prediction of its equivalent mechanical properties through this module.

[0072] 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.

[0073] The calibration device for the intrinsic characteristic parameters of the metamaterial provided in the above embodiments can implement the technical solutions described in the embodiments of the calibration method for the intrinsic characteristic parameters of the metamaterial. The specific implementation principles of the above modules or units can be referred to the corresponding content in the embodiments of the calibration method for the intrinsic characteristic parameters of the metamaterial, which will not be elaborated here.

[0074] As Figure 5 shown, the present invention also correspondingly provides an electronic device 500. The electronic device 500 includes a processor 501, a memory 502, and a display 503. Figure 5 Only some components of the electronic device 500 are shown, but it should be understood that it is not required to implement all the shown components, and more or fewer components can be alternatively implemented.

[0075] The memory 502 can be an internal storage unit of the electronic device 500 in some embodiments, such as the hard disk or memory of the electronic device 500. The memory 502 can also be an external storage device of the electronic device 500 in other embodiments, such as a plug-in hard disk equipped on the electronic device 500, a Smart Media Card (SMC), a Secure Digital (SD) card, a Flash Card, etc.

[0076] Furthermore, the memory 502 can also include both the internal storage unit and the external storage device of the electronic device 500. The memory 502 is used to store the application software installed on the electronic device 500 and various types of data.

[0077] The processor 501 can be a central processing unit (CPU), a microprocessor, or other data processing chips in some embodiments, and is used to run the program code stored in the memory 502 or process data, such as the calibration method for the intrinsic characteristic parameters of the metamaterial in the present invention.

[0078] The display 503 can be an LED display, a liquid crystal display, a touch liquid crystal display, and an OLED (Organic Light-Emitting Diode) toucher, etc. in some embodiments. The display 503 is used to display the information on the electronic device 500 and to display a visual user interface. The components 501 - 503 of the electronic device 500 communicate with each other through the system bus.

[0079] 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 can be achieved: Perform homogenization analysis on the metamaterial to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of the metamaterial; Construct a training set with the geometric parameter descriptors 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; Process the training set based on the radial basis kernel function to construct the covariance matrix of the input features; Optimize the radial basis kernel function based on the covariance matrix to obtain the corresponding Gaussian process regression model; Input the geometric parameter descriptors of the metamaterial to be calibrated into the Gaussian process regression model to obtain the predicted values of the equivalent performance parameters, and based on the predicted values of the equivalent performance parameters, combine with the non-classical homogenization model to determine the intrinsic characteristic parameters of the metamaterial to be calibrated.

[0080] 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, other functions can also be achieved. For specific details, reference can be made to the descriptions of the corresponding method embodiments above.

[0081] Furthermore, the type of the electronic device 500 mentioned in the embodiments of the present invention is not specifically limited. The electronic device 500 can be a portable electronic device such as a mobile phone, a tablet computer, a personal digital assistant (PDA), a wearable device, a laptop, etc. Exemplary embodiments of the portable electronic device include but are not limited to portable electronic devices equipped with IOS, android, Microsoft or other operating systems. The above portable electronic devices can also be other portable electronic devices, such as a laptop with a touch-sensitive surface (such as a touch panel). It should also be understood that in some other embodiments of the present invention, the electronic device 500 can also not be a portable electronic device, but a desktop computer with a touch-sensitive surface (such as a touch panel).

[0082] On the other hand, the present invention also provides a non-transitory computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, it is used to execute the calibration method of the intrinsic characteristic parameters of the metamaterial provided by the above methods. The method includes: Perform homogenization analysis on the metamaterial to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of the metamaterial; Taking the geometric parameter descriptors 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, a training set is constructed; Processing the training set based on the radial basis kernel function to construct the covariance matrix of the input features; Optimizing the radial basis kernel function based on the covariance matrix to obtain the corresponding Gaussian process regression model; Inputting the geometric parameter descriptors of the metamaterial to be calibrated into the Gaussian process regression model to obtain the predicted values of the equivalent performance parameters, and based on the predicted values of the equivalent performance parameters, combining with the non-classical homogenization model, determining the intrinsic characteristic parameters of the metamaterial to be calibrated.

[0083] Those skilled in the art can understand that all or part of the processes of implementing the methods in the above embodiments can be completed by instructing relevant hardware through a computer program, and the program can be stored in a computer-readable storage medium. Among them, the computer-readable storage medium is a magnetic disk, an optical disk, a read-only memory or a random access memory, etc.

[0084] The above has introduced in detail the method and device for calibrating the intrinsic characteristic parameters of the metamaterial provided by the present invention. Specific examples are used herein to elaborate on the principle and implementation manner of the present invention. The description of the above embodiments is only used to help understand the method and its core idea of the present invention; at the same time, for those skilled in the art, according to the idea of the present invention, there will be changes in the specific implementation manner and application scope. In summary, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A calibration method for the intrinsic characteristic parameters of a metamaterial, characterized in that including: performing homogenization analysis on the metamaterial to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of the metamaterial; constructing a training set with the geometric parameter descriptors 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; processing the training set based on the radial basis kernel function to construct the covariance matrix of the input features; optimizing the radial basis kernel function based on the covariance matrix to obtain the corresponding Gaussian process regression model; inputting the geometric parameter descriptors of the metamaterial to be calibrated into the Gaussian process regression model to obtain the predicted values of the equivalent performance parameters, and determining the intrinsic characteristic parameters of the metamaterial to be calibrated based on the predicted values of the equivalent performance parameters.

2. The calibration method for the intrinsic characteristic parameters of the metamaterial according to claim 1, wherein 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: under the condition of applying periodic boundary conditions on the boundaries of the representative volume element of the metamaterial in three different directions, and when the macroscopic tensile strain is applied to the boundary in a single direction of the representative volume element, based on the local stress field of the representative volume element at the microscale, obtaining the macroscopic equivalent stress, and based on the macroscopic equivalent stress and the macroscopic tensile strain, obtaining the classical equivalent performance parameters; under the condition of keeping the boundary of the representative volume element of the metamaterial consistent with the macroscopic boundary conditions, based on the local stress field of the representative volume element at the microscale, obtaining the macroscopic equivalent stress, and based on the macroscopic equivalent stress and the macroscopic tensile strain, obtaining the size-dependent equivalent performance parameters.

3. The calibration method for the intrinsic characteristic parameters of the metamaterial according to claim 1, wherein The predicted values of the equivalent performance parameters include: predicted values of the classical equivalent performance parameters and predicted values of the size-dependent equivalent performance parameters; determining the intrinsic characteristic parameters of the metamaterial to be calibrated based on the predicted values of the equivalent performance parameters, including: determining the intrinsic characteristic parameters of the metamaterial to be calibrated based on the predicted values of the classical equivalent performance parameters, the predicted values of the size-dependent equivalent performance parameters, and the macroscopic thickness of the metamaterial structure.

4. The calibration method for the intrinsic characteristic parameters of the metamaterial according to claim 3, characterized in that The intrinsic characteristic parameters of the metamaterial to be calibrated are calculated using a non-classical homogenization model, and the non-classical homogenization model is: Among them, is the predicted value of the size-dependent equivalent performance parameter, is the predicted value of the classical equivalent performance parameter, is the intrinsic characteristic parameter of the metamaterial to be calibrated, namely the nonlocal intrinsic characteristic length, H is the macroscopic thickness of the metamaterial structure.

5. The calibration method for the intrinsic characteristic parameters of the metamaterial according to claim 1, wherein processing the training set based on the radial basis kernel function to construct the 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; constructing the covariance matrix of the input features based on the covariance between any two input features in the training set.

6. The calibration method for the intrinsic characteristic parameters of the metamaterial according to claim 5, wherein The calculation formula of the radial basis kernel function is: Among them, is the covariance between any two input features, is the predicted value signal variance of the Gaussian process regression prediction model, is the d -dimensional length scale, is the d -dimensional feature of the input feature, and X is the input feature.

7. The calibration method for the intrinsic characteristic parameters of the metamaterial according to claim 6, wherein 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 the hyperparameter combination, , T is the transpose symbol ， is the log marginal likelihood function ， I is the identity matrix.

8. The calibration method for the intrinsic characteristic parameters of the metamaterial according to any one of claims 1-7, characterized in that, constructing a training set with the geometric parameter descriptors 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, including: constructing a basic data set with the geometric parameter descriptors 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. Randomly divide the training set into five parts, and each division obtains a set of training set and the corresponding test set; Based on each set of test sets, evaluate the performance metrics of the Gaussian process regression model trained by the corresponding training set, and obtain five sets of performance metrics; Based on the average value of the five sets of performance metrics, determine the performance metrics of the Gaussian process regression model; In the case that the performance metrics of the Gaussian process regression model do not meet the preset requirements, re-optimize the radial basis kernel function to update the Gaussian process regression model.

9. A calibration device for the intrinsic characteristic parameters of a metamaterial, characterized in that It includes: A homogenization processing module for performing homogenization analysis on the metamaterial to obtain the classical equivalent performance parameters and size-dependent equivalent performance parameters of the metamaterial; A training set construction module for constructing a training set with the geometric parameter descriptors 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; A covariance calculation module for processing the training set based on the radial basis kernel function to construct a covariance matrix of the input features; A model training module for optimizing the radial basis kernel function based on the covariance matrix to obtain the corresponding Gaussian process regression model; An intrinsic feature calculation module for inputting the geometric parameter descriptors of the metamaterial to be calibrated into the Gaussian process regression model to obtain predicted values of equivalent performance parameters, and determining the intrinsic feature parameters of the metamaterial to be calibrated based on the predicted values of the equivalent performance parameters.

10. 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, it implements the steps of the method for calibrating the intrinsic feature parameters of the metamaterial according to any one of claims 1 to 8.

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