Inverse Design Method for Nonlinear Response Mechanical Metamaterial Structures Based on Neural Networks

Abstract

The present invention discloses a reverse design method for a non-linear response mechanical metamaterial structure based on a neural network, which relates to the technical fields of metamaterial structures and machine learning, and solves the technical problems of long time consumption and high calculation cost in the design of mechanical metamaterials. The key points of its technical solution are to change the shape parameters of the smallest periodic unit in the mechanical metamaterial, adjust the amount of internal rotation of the unit in the metamaterial structure caused by external forces, and obtain the non-linear mechanical response of the metamaterial; introduce a neural network model to economically and efficiently describe the relationship between the geometric parameters of the mechanical metamaterial and its stress-strain response; combine the neural network with an evolutionary strategy to effectively identify the structure of the mechanical metamaterial and obtain the target non-linear mechanical response, thereby constructing a reverse design model of the mechanical metamaterial structure and designing more effective energy absorption systems, soft robots, wearable structures, etc.

Description

Technical Field

[0001] The present application relates to the field of metamaterial structures and machine learning technology, and in particular to a method for inverse design of a nonlinear response mechanical metamaterial structure based on a neural network. Background Art

[0002] Metamaterials are new artificial materials that emerged in the 21st century. Their artificially designed structures present special properties that are not possessed by naturally occurring materials in nature. Among them, mechanical metamaterials have been widely used in aerospace, soft robot design and other fields in recent years due to their unique mechanical properties such as negative Poisson's ratio effect and negative stiffness effect. The properties of mechanical metamaterials do not mainly depend on their constituent materials, but on their internal artificial structures, which provides convenience for artificially adjusting the mechanical response of mechanical metamaterials. Although there are commercial software that can provide references for the structural design of mechanical metamaterials through methods such as topology optimization, their calculation cost is high and time-consuming. In addition, the structural design of mechanical metamaterials needs to consider the relationship between their geometric structure and stress-strain response, and the inverse design method of their nonlinear response mechanical metamaterial structure needs to be solved urgently. Summary of the invention

[0003] The present application provides a method for reverse design of a nonlinear response mechanical metamaterial structure based on a neural network. The technical purpose is to utilize the generalization and data fitting capabilities of neural networks, combined with algorithms such as evolutionary strategies and particle swarm optimization, to explain the relationship between the geometric parameters of mechanical metamaterials and stress-strain responses, and then realize the reverse design of mechanical metamaterial structures based on neural networks.

[0004] The above technical objectives of this application are achieved through the following technical solutions:

[0005] A method for inverse design of a nonlinear response mechanical metamaterial structure based on a neural network, comprising:

[0006] S1: changing the geometric parameters of the articulated quadrilateral mechanical metamaterial structure to obtain the internal disturbance of the unit of the mechanical metamaterial caused by the external force, thereby obtaining different stress-strain curves corresponding to different geometric parameters of the articulated quadrilateral mechanical metamaterial structure; wherein the articulated quadrilateral mechanical metamaterial structure comprises an articulated quadrilateral unit array, the articulated quadrilateral unit array comprises a plurality of minimum period units uniformly arranged in the transverse and longitudinal directions, each minimum period unit comprises four quadrilateral matrices, the vertices of adjacent quadrilateral matrices in the articulated quadrilateral unit array are connected by hinges, and there is only one vertex connection between adjacent quadrilateral matrices;

[0007] S2: Construct a neural network model, and construct a training set and a test set for stress-strain curves and geometric parameters. Train the neural network model with the training set and test the trained neural network model with the test set until the first neural network model is obtained;

[0008] S3: Construct a new training set according to the evolutionary strategy, train the first neural network model with the new training set to obtain an inverse design model for the articulated quadrilateral mechanical metamaterial structure, and design the structure of the mechanical metamaterial according to the inverse design model;

[0009] Among them, each minimum periodic unit includes eight independent vertices. The eight independent vertices include four internally connected independent vertices and four externally connected independent vertices. The four internally connected independent vertices are: there are two vertices for internal connection in each quadrilateral matrix in the minimum periodic unit with the two adjacent quadrilateral matrices in the minimum periodic unit. Regarding the two vertices at the internal connection as one vertex, then there are four internally connected independent vertices in one minimum periodic unit; the four externally connected independent vertices are: each quadrilateral matrix in the minimum periodic unit has two vertices with external connection to the quadrilateral matrices of other minimum periodic units outside. Select one vertex from the two vertices with external connection in each quadrilateral matrix as the externally connected independent vertex, then there are four externally connected independent vertices in one minimum periodic unit; the geometric structure of the minimum periodic unit is described by eight independent vertices, expressed as: where p represents the p-th minimum periodic unit; represents the displacement of the α-th independent vertex of the p-th minimum periodic unit on the x-axis, and this displacement is the displacement of the corresponding vertex of the square along the x-axis; the displacement of the α-th independent vertex of the p-th minimum periodic unit on the y-axis, and this displacement is the displacement of the corresponding vertex of the square along the y-axis; when then each quadrilateral matrix in the minimum periodic unit is a square; α = 1, 2,..., 8;

[0010] The hinge is constructed by four springs, including: a linear spring with a stiffness of k l for capturing the longitudinal response, a linear spring with a stiffness of k s for capturing the shear response, a nonlinear torsional spring with a stiffness of k θ for capturing the bending and torsion deformation, and a nonlinear rotational spring with a stiffness of k cont (β) for capturing the contact between adjacent quadrilateral matrices; where M = k θ (θ + γθ 3 ), M represents the torque applied by the spring, k θrepresents the rotational stiffness, γ represents the dimensionless material parameter, θ represents the rotational angle of the quadrilateral matrix about the z-axis; β represents the angle between two adjacent sides of the quadrilateral matrix, β0 represents the angle threshold.

[0011] Further, the step S1 includes:

[0012] S11: Change the geometric parameters of the minimum periodic unit in the hinged quadrilateral mechanical metamaterial structure to obtain a discrete model of the minimum periodic unit under different geometric parameters, and this discrete model is expressed as:

[0013]

[0014]

[0015]

[0016]

[0017] where, m [i，j] represents the mass of the [i, j] -th quadrilateral matrix, represents the acceleration of the quadrilateral matrix in the x-axis, represents the acceleration of the quadrilateral matrix in the y-axis; J [i，j] represents the moment of inertia of the [i, j] -th quadrilateral matrix, represents the angular acceleration of the quadrilateral matrix about the z-axis; represents the force in the x-axis direction generated by the hinge at the p'-th vertex of the [i, j] -th quadrilateral matrix, represents the force in the y-axis direction generated by the hinge at the p'-th vertex of the [i, j] -th quadrilateral matrix, represents the moment generated by the hinge at the p'-th vertex of the [i, j] -th quadrilateral matrix, p' = 1, 2, 3, 4; when p' is the vertex on different diagonals of the quadrilateral matrix, or represents the change in the rotational angle of the spring connected to the p'-th vertex of the [i, j] -th quadrilateral matrix; represents at the specific rotational angle θ [i，j] the position vector from the center of the [i, j] -th quadrilateral matrix to its p'-th vertex; represents the change in the lengths of the spring connected to the p'-th vertex of the [i, j] -th quadrilateral matrix along the x-axis and y-axis; represents the additional moment generated at the p'-th vertex of the [i, j] -th quadrilateral matrix, and β1 and β2 represent two angles between two adjacent sides of the quadrilateral matrix;

[0018] S12: Obtain the stress-strain curves corresponding to the articulated quadrilateral mechanical metamaterial structures with different geometric parameters according to the discrete model.

[0019] Further, in the step S2, constructing a training set and a test set regarding the stress-strain curves and geometric parameters includes: That is, on the basis that each quadrilateral matrix in the minimum periodic unit is a square with side length a, randomly perturb the vertices of the minimum periodic unit to obtain minimum periodic units with different geometric parameters. This random perturbation includes:

[0020] S21: Randomly displace the square quadrilateral matrix with side length a so that it satisfies Thus, a number of first minimum periodic units are obtained; among them, the displacement coordinates of each vertex of the quadrilateral matrix in the first minimum periodic unit are respectively and That is and

[0021] S22: Respectively, with the eight independent vertices of the first minimum periodic unit as the centers, construct eight square frames with side length 0.5a, and randomly move each independent vertex within its corresponding 0.5a square frame so that all sides of each quadrilateral matrix in the obtained second minimum periodic unit are greater than 0.2a, and all interior angles are greater than 45°;

[0022] S23: Perform rotational symmetry on the quadrilateral matrices in the second minimum periodic unit to obtain the third minimum periodic unit;

[0023] S24: Combine the second minimum periodic unit and the third minimum periodic unit to obtain a preset number of training sets and test sets.

[0024] Further, in step S2, the neural network model includes four hidden layers, and its loss function is expressed as:

[0025]

[0026] Among them, are the values of the first 10 principal components, used to represent the stress-strain curve corresponding to the pth minimum periodic unit; F s (X p ) = [F1(X p ), …, F 10 (X p )] Trepresents the corresponding neural network prediction value, β s represents the variance explained in each principal component.

[0027] Furthermore, in step S3, constructing a new training set according to the evolutionary strategy includes:

[0028] S31: On the basis that each quadrilateral matrix in the minimum periodic unit is a square with side length a, randomly move the vertices of the quadrilateral matrix to obtain the fourth minimum periodic unit;

[0029] S32: Centering on the eight independent vertices of the first minimum periodic unit respectively, construct eight square frames with side length 0.1a, and randomly move each independent vertex within its corresponding 0.1a square frame to obtain the fifth minimum periodic unit;

[0030] S33: The fourth minimum periodic unit and the fifth minimum periodic unit constitute the new training set.

[0031] The beneficial effects of this application are as follows: The reverse design method of the non-linear response mechanical metamaterial structure based on neural network described in this application changes the geometric parameters of the articulated quadrilateral mechanical metamaterial structure to obtain the internal unit perturbation of the mechanical metamaterial caused by external forces, so as to obtain different stress-strain curves corresponding to different geometric parameters of the articulated quadrilateral mechanical metamaterial structure; construct a neural network model to describe the relationship between the geometric parameters of the mechanical metamaterial and its stress-strain response through this neural network model; combine the neural network with the evolutionary strategy to effectively identify the structure of the mechanical metamaterial and obtain the target non-linear mechanical response, obtain the reverse design model of the articulated quadrilateral mechanical metamaterial structure, and design the structure of the mechanical metamaterial according to the reverse design model, and design more effective energy absorption systems, soft robots, wearable structures, etc. Description of the Drawings

[0032] Figure 1 is the design flow chart of the reverse design method of the mechanical metamaterial structure, where the solid line part is the construction process of the reverse design model, and the dotted line part is the design process of the reverse design;

[0033] Figure 2 is the cross-sectional schematic diagram of the articulated quadrilateral mechanical metamaterial structure;

[0034] Figure 3 is the schematic diagram of generating a new minimum periodic unit by randomly perturbing the positions of the vertices of the quadrilateral matrix;

[0035] Figure 4 is the schematic diagram of the discrete model using spring connections to simulate the hinges connected at the vertices of the quadrilateral matrix;

[0036] Figure 5Schematic diagram for describing the action of a non - linear rotational spring in contact with adjacent quadrilateral substrates;

[0037] Figure 6 Schematic diagram for the force analysis and displacement analysis of the [i, j] - th rigid quadrilateral substrate;

[0038] Figure 7 Schematic diagram for comparing the compression simulation results of the discrete model and the finite - element model of the articulated - quadrilateral mechanical metamaterial structure; a. Comparison of the stress - strain curves of the two; b. Comparison of the deformation results of the two;

[0039] Figure 8 Schematic diagram of the process for obtaining the geometric dimensions of a new minimum - period unit through a sampling method;

[0040] Figure 9 Schematic diagram of the model for inverse design of the articulated - quadrilateral mechanical metamaterial structure;

[0041] Figure 10 Stress - strain curve graph for testing the model;

[0042] Figure 11 Graph of error - iteration times for the evolutionary strategy algorithm. Detailed implementation manners

[0043] The technical solution of the present application will be described in detail below with reference to the accompanying drawings.

[0044] As Figure 1 shown, the inverse design method for a non - linear response mechanical metamaterial structure based on a neural network according to the present application includes:

[0045] S1: Change the geometric parameters of the articulated - quadrilateral mechanical metamaterial structure to obtain the internal disturbance of the unit of the mechanical metamaterial due to external forces, so as to obtain different stress - strain curves corresponding to different geometric parameters of the articulated - quadrilateral mechanical metamaterial structure.

[0046] Specifically, select the minimum - period unit of the articulated - quadrilateral mechanical metamaterial structure as the object, construct a non - linear deformation model (i.e., a discrete model) of the minimum - period unit under axial compression external force, and then obtain the stress - strain curve of the unit under axial compression external force when the strain ranges from 0 to - 0.1 through experimental or finite - element methods. Figure 1 The substrate unit in

[0047] is the minimum - period unit. On the premise of maintaining the unit periodicity, form a new minimum - period unit by randomly disturbing the independent vertex positions of the minimum - period unit of the mechanical metamaterial, and record the mechanical response of the new minimum - period unit. The number of repetitions of this process should be no less than 7500 (empirical value) times.

[0048] Step S1 includes:

[0049] S11: Change the geometric parameters of the minimum periodic unit in the articulated quadrilateral mechanical metamaterial structure to obtain the discrete models of the minimum periodic units under different geometric parameters.

[0050] As Figure 2 shown, the articulated quadrilateral mechanical metamaterial structure described in this application includes an array of articulated quadrilateral units. The array of articulated quadrilateral units includes several minimum periodic units that are uniformly arranged both horizontally and vertically. Each minimum periodic unit includes four quadrilateral matrices.

[0051] In the embodiment of this application, the array of articulated quadrilateral units is an array containing 10×8 quadrilateral matrices, and both sides of the array are two horizontal strip structures. All quadrilateral matrices are connected at their vertices by slender strips (i.e., hinges) with a width and length of 1 mm. The mechanical metamaterial structure maintains a certain periodicity, and the minimum periodic unit is composed of 2×2 quadrilateral matrices. The geometric shape of each periodic unit is completely described by the positions of its 12 vertices (see the points pointed by the arrows in Figure 3 ). However, due to the periodic geometric constraints between the quadrilateral matrices, only the positions of 8 vertices can be independently specified. These 8 vertices are the independent vertices of the minimum periodic unit.

[0052] Each minimum periodic unit includes eight independent vertices. The eight independent vertices include four internally connected independent vertices and four externally connected independent vertices. The four internally connected independent vertices are: in each minimum periodic unit, each quadrilateral matrix has two vertices that are internally connected to the two adjacent quadrilateral matrices in this minimum periodic unit. Regarding the two vertices at the internal connection as one vertex, a minimum periodic unit has a total of four internally connected independent vertices. The four externally connected independent vertices are: in each minimum periodic unit, each quadrilateral matrix has two vertices that are externally connected to the quadrilateral matrices of other minimum periodic units. Select one vertex from the two vertices where each quadrilateral matrix has an external connection as the externally connected independent vertex. Then, a minimum periodic unit has a total of four externally connected independent vertices. Thus, the geometric structure of the minimum periodic unit is described by eight independent vertices, expressed as: where p represents the pth minimum periodic unit; represents the displacement of the αth independent vertex of the pth minimum periodic unit along the x-axis, and this displacement is the displacement of the corresponding vertex of the square along the x-axis; the displacement of the αth independent vertex of the pth minimum periodic unit along the y-axis, and this displacement is the displacement of the corresponding vertex of the square along the y-axis; α = 1, 2, …, 8.

[0053] When When the eight independent vertices of the smallest periodic unit corresponding to the square vertices have no displacement in the x-axis and y-axis and the original square remains unchanged, each quadrilateral matrix within the smallest periodic unit is a square. In the embodiments of the present application, the diagonal length of this square is a = 10 mm.

[0054] In addition, the vertices between adjacent quadrilateral matrices in the articulated quadrilateral unit array are connected by hinges, and there is only one vertex connection between adjacent quadrilateral matrices.

[0055] In the above discrete model, it is considered that the quadrilateral matrices are rigid and three degrees of freedom are assigned to them, namely the displacement u in the X direction, the displacement v in the y direction, and the rotation θ about the z-axis. Based on this, the present application uses a combination of four springs to model the hinge, as Figure 4 shown, specifically including: a linear spring with a stiffness of k l for capturing the longitudinal response, a linear spring with a stiffness of k s for capturing the shear response, a nonlinear torsional spring with a stiffness of k θ for capturing the bending and torsion deformation, and a nonlinear rotational spring with a stiffness of k cont (β) for capturing the contact between adjacent quadrilateral matrices. Among them, M = k θ (θ + γθ 3 ), M represents the torque applied by the spring, k θ represents the rotational stiffness, γ represents the dimensionless material parameter, and θ represents the rotation angle of the quadrilateral matrix about the z-axis.

[0056] For the convenience of analysis, it is assumed that the deformation generated during compression is a small deformation, that is, the longitudinal spring and the shear spring are always parallel to the x-axis or y-axis. And it is assumed that the material is homogeneous, that is, all vertices are connected by hinges of the same shape and the connection model properties are the same everywhere. At the same time, it is assumed that the spring stiffness does not change with the different shapes of the quadrilateral matrices among various samples.

[0057] The introduction of the nonlinear rotational spring with a stiffness of k cont (β) is because during compression, the edges of adjacent quadrilateral matrices may come into contact with each other, and this contact will affect the nonlinear response of the structure. Therefore, an additional nonlinear rotational spring is needed to describe the contact between adjacent quadrilateral matrices, specifically expressed as:

[0058]

[0059] Among them, β represents the angle between two sides of adjacent quadrilateral matrices and can be determined as a function of geometric parameters, as Figure 5 shown; β0 represents the angle threshold.

[0060] The non - linear rotational spring is activated only when the angle β is less than the angle threshold β0 and an additional moment is generated at the p'-th vertex of the [i, j]-th quadrilateral substrate. For example, considering the contact between the [i, j]-th and the [i, j + 1]-th quadrilateral substrates, we can obtain:

[0061]

[0062] where β1 and β2 represent the two angles between two sides of adjacent quadrilateral substrates.

[0063] Under the above assumptions, the motion equation of the [i, j]-th quadrilateral substrate, that is, the discrete model, is expressed as:

[0064]

[0065]

[0066]

[0067] where,

[0068]

[0069]

[0070] where, m [i，j] represents the mass of the [i, j]-th quadrilateral substrate, represents the acceleration of the quadrilateral substrate in the x - axis direction, represents the acceleration of the quadrilateral substrate in the y - axis direction; J [i，j] represents the moment of inertia of the [i, j]-th quadrilateral substrate, represents the angular acceleration of the quadrilateral substrate about the z - axis; represents the force in the x - axis direction generated by the hinge at the p'-th vertex of the [i, j]-th quadrilateral substrate, represents the force in the y - axis direction generated by the hinge at the p'-th vertex of the [i, j]-th quadrilateral substrate, represents the moment generated by the hinge at the p'-th vertex of the [i, j]-th quadrilateral substrate, as Figure 6 shown, p' = 1, 2, 3, 4.

[0071] When p' is the vertex on different diagonals of the quadrilateral substrate, or represents the change in the rotational angle of the spring connected to the p'-th vertex of the [i, j]-th quadrilateral substrate; represents at a specific rotational angle θ [i，j]The position vector pointing from the center of the [i, j]-th quadrilateral matrix to its p'-th vertex; Denote the changes in the lengths of the springs connected to the p'-th vertex of the [i, j]-th quadrilateral matrix along the x-axis and y-axis. and Both represent direction vectors with a magnitude of 1.

[0072] S12: Obtain the stress-strain curves corresponding to the articulated quadrilateral mechanical metamaterial structures under different geometric parameters according to the discrete model.

[0073] To connect the discrete model with the actual physical model, we need to estimate k l , k s , k θ , γ, and β0. For this purpose, we will perform uniaxial compression simulations on a finite element model where and are both 0, that is, each quadrilateral matrix is a square with a diagonal length of 10 mm, and the vertices of each square are connected by thin beam-like hinges with a width and length of 1 mm. Finally, by combining the stress-strain curves obtained from the finite element model compression simulation with the above formulas, we determine that k l = 0.47 N / mm, k θ = 0.88 N / mm, γ = -0.2, β0 = 0.1 rad.

[0074] However, the stress-strain curve cannot determine k s . To obtain k s , assume:

[0075]

[0076] where E represents the elastic modulus during the elastic deformation process of the material, G represents the shear modulus of the material, υ represents the Poisson's ratio of the material, and takes a value of 0.5 here, and we obtain k s = 0.16 N / mm.

[0077] Figure 7 is a schematic diagram for comparing the stress-strain curves and deformation results of the discrete model and the finite element model. From Figure 7 it can be known that the discrete model of this application can better fit the compression process of the hinge quadrilateral metamaterial.

[0078] S2: Construct a neural network model, and construct a training set and a test set regarding the stress-strain curve and geometric parameters. Train the neural network model with the training set and test the trained neural network model with the test set until the final neural network model is obtained.

[0079] Specifically, a training set and a test set regarding stress-strain curves and geometric parameters are constructed, including: on the basis that each quadrilateral matrix within the i.e., each quadrilateral matrix in the minimum periodic unit is a square with side length a, the vertices of the minimum periodic unit are randomly perturbed to obtain minimum periodic units with different geometric parameters. The random perturbation includes:

[0080] S21: Randomly displace the square quadrilateral matrix with side length a so that it satisfies Thereby obtaining a number of first minimum periodic units; among them, the displacement coordinates of each vertex of the quadrilateral matrix in the first minimum periodic unit are respectively and i.e., and

[0081] S22: Respectively, with the eight independent vertices of the first minimum periodic unit as the centers, construct eight square frames with side length 0.5a, and randomly move each independent vertex within its corresponding 0.5a square frame so that all sides of each quadrilateral matrix in the obtained second minimum periodic unit are greater than 0.2a and all interior angles are greater than 45°.

[0082] S23: Rotationally symmetric the quadrilateral matrices in the second minimum periodic unit to obtain a third minimum periodic unit.

[0083] S24: Combine the second minimum periodic unit and the third minimum periodic unit to obtain a preset number of training sets and test sets.

[0084] In the embodiment of the present application, 250 highly symmetric first minimum periodic units are generated through step S21. For example, two quadrilateral matrices in the first minimum periodic unit and The displacement coordinates and indicate that the vertices of the two quadrilateral matrices have a displacement of 0 on the x-axis, and the displacement directions and magnitudes on the y-axis are the same; the displacement coordinates and as well as and indicate that the vertices of the two quadrilateral matrices have a displacement of 0 on the y-axis, and the displacement directions on the x-axis are opposite while the displacement magnitudes are the same.

[0085] In step S22, based on the design of 250 highly symmetric first minimum periodic units, respectively, with the eight independent vertices of the first minimum periodic unit as the centers, construct eight square frames with side length 0.5a, such as Figure 8As shown. Each vertex is randomly perturbed 50 times within the square frame, generating a total of 7,500 different minimum periodic cells. And in order to generate the design with deformation located at the hinge, it is restricted that all sides of each quadrilateral substrate in the minimum periodic cell are greater than 0.2a, and all interior angles are greater than 45°. At the same time, in order to expand the number of data sets, the quadrilateral substrates in the obtained minimum periodic cells are rotationally symmetric, and 30,000 training set samples are obtained based on the 7,500 geometric shapes.

[0086] To facilitate the training of the machine learning model, the method of principal component analysis (PCA) is used to reduce the dimension of the non-linear measurement. The stress-strain curve of the p-th minimum periodic cell is represented by a 100-dimensional vector:

[0087] σ p =[σ p (0), …, σ p (-0.1)] T ;

[0088] It contains the stress values at 100 equally spaced strain points.

[0089] First, the stress-strain vectors of the training set samples are assembled into a stress matrix, denoted as:

[0090] σ=[σ1, σ2, …, σ Nt T ;

[0091] And the mean value is processed for the stress matrix σ to obtain:

[0092]

[0093] where, σ p,s represents the s-th component of the p-th stress-strain vector.

[0094]

[0095] where, Nt represents the number of data points in the training set.

[0096] Then, the covariance matrix obtained is subjected to singular value decomposition and rewritten as:

[0097]

[0098] where, V is a 100×100 matrix with orthogonal columns representing the principal directions (i.e., VV T =I, where I represents the identity matrix). In addition, ∑ 2 ​= diag(β1, …, β100), which contains the eigenvalues βs that determine the variance explained in each principal component. Finally, to reduce the dimensionality of the data from 100 to n < 100, we collect the set of V columns associated with the n largest explained variances into Vn and calculate the first n principal components, denoted as:

[0099]

[0100] Here, n = 10 is selected, resulting in a training error of 0.3%. It is considered that using the first ten principal components can provide a good generalization of the stress vector.

[0101] To predict the low-dimensional representation of the stress vector for a given geometry X p of This application uses a neural network (NN) architecture with four hidden layers, and each hidden layer includes 200 neurons. The neural network model is built and trained using the pytorch package. In an extended dataset including 30,000 data points, we randomly select Nt = 0.8N data points for training, and the remaining Ntest = 0.2N for testing, and select the weights and biases of the neurons that minimize to train the neural network.

[0102] The loss function of this neural network model is denoted as:

[0103]

[0104] where are the values of the first 10 principal components, used to represent the stress-strain curve corresponding to the p-th minimum periodic unit; F s (X p ) = [F1(X p ), …, F 10 (X p )] T represents the corresponding neural network prediction value, and β s represents the variance explained in each principal component.

[0105] Specifically, the neural network model is trained using the following parameters: the activation function is the ReLU function, the Adam optimizer is used, the learning rate is 10, and the weight decay rate is 10.

[0106] S3: Construct a new training set according to the evolutionary strategy, train the first neural network model through the new training set to obtain an inverse design model of the articulated quadrilateral mechanical metamaterial structure, and design the structure of the mechanical metamaterial according to the inverse design model.

[0107] The first neural network model completed in training and testing in step S2 can effectively predict the mechanical responses of the articulated quadrilateral mechanical metamaterial structure with arbitrary matrix shapes. However, since the relationship between the geometric shape of the quadrilateral matrix and the mechanical responses of the corresponding structure cannot be ignored, an effective inverse design strategy is required to identify the mechanical metamaterials with target behaviors. To this end, as Figure 9 shown, the trained first neural network model is combined with an evolutionary strategy (ES) to obtain an effective inverse design model. To test the accuracy of this inverse design model, four stress-strain curves are selected within the range of mechanical responses that the articulated quadrilateral mechanical metamaterial structure may achieve for testing, as Figure 10 shown.

[0108] To identify the mechanical characteristics that lead to these target stress-strain curves, first, μ = 100 stress-strain curves are selected from the dataset established above to minimize err[σ t ,σ p ,-0.1]. As expected, none of the responses in the dataset match the four selected target behaviors very well, and the minimum errors between the best selection and the target behaviors are equal to: (i) 11%, (ii) 25%, (iii) 13%, and (iv) 12%.

[0109] Therefore, this application constructs a new training set according to the evolutionary strategy, including:

[0110] S31: On the basis that each quadrilateral matrix in the minimum periodic unit is a square with side length a, randomly move each vertex of the quadrilateral matrix to obtain the fourth minimum periodic unit;

[0111] S32: Centered on the eight independent vertices of the first minimum periodic unit respectively, construct eight square frames with side length 0.1a, and move each independent vertex randomly within its corresponding 0.1a square frame to obtain the fifth minimum periodic unit;

[0112] S33: The fourth minimum periodic unit and the fifth minimum periodic unit form a new training set

[0113] In the embodiments of the present application, in order to reduce these errors, we generate λ = 49 new candidate periodic cells (offspring cells) by randomly moving the eight independent vertices of each v with a minimum periodic cell (parent cell) within a frame with an edge of 0.1a. In addition, in order to better explore the design space, we generate 5000 completely arbitrary minimum periodic cells, and these arbitrary cells compete with the parent and offspring cells for consideration in the next iteration. There are a total of 10,000 geometries for the parent, offspring, and arbitrary cells, and the stress-strain curves of the above cells are determined by the first neural network model trained previously. The neural network model in the present application is a bidirectional model, which can obtain geometric parameters by inputting the stress-strain curve, and can also obtain the stress-strain curve by inputting geometric parameters.

[0114] From this set composed of the parent, offspring, and arbitrary cells, we select the parents for the next iteration by choosing μ = 100 designs that minimize and we continue this mutation / generation and selection process until is reached or the number of iterations reaches 50. Through the above processing, the errors between the finally obtained stress-strain curve and the target stress-strain curve can reach: (i) 1.9%, (ii) 4.7%, (iii) 6.6%, and (iv) 4.5%, as Figure 11 shown. Therefore, according to the inverse design model constructed by combining neural network and evolutionary strategy for the articulated quadrilateral mechanical metamaterial structure, and through the verification of the accuracy of the model, it is considered that the inverse design model can better complete the inverse design work of the articulated quadrilateral mechanical metamaterial structure.

[0115] The above is a demonstration embodiment of the present application, and the protection scope of the present application is defined by the claims and their equivalents.

Claims

1. A reverse design method for a non-linear response mechanical metamaterial structure based on a neural network, characterized in that, Including: S1: Change the geometric parameters of the articulated quadrilateral mechanical metamaterial structure to obtain the internal disturbance of the unit caused by external forces in the mechanical metamaterial, so as to obtain different stress-strain curves corresponding to different geometric parameters of the articulated quadrilateral mechanical metamaterial structure; wherein, the articulated quadrilateral mechanical metamaterial structure includes an articulated quadrilateral unit array, the articulated quadrilateral unit array includes a number of minimum period units arranged uniformly in the transverse and longitudinal directions, each minimum period unit includes four quadrilateral substrates, and the vertices between adjacent quadrilateral substrates in the articulated quadrilateral unit array are connected by hinges, and there is only one vertex connection between adjacent quadrilateral substrates; S2: Construct a neural network model, and construct a training set and a test set for the stress-strain curve and geometric parameters. Train the neural network model with the training set and test the trained neural network model with the test set until the first neural network model is obtained; S3: Construct a new training set according to the evolutionary strategy, train the first neural network model with the new training set to obtain the inverse design model of the articulated quadrilateral mechanical metamaterial structure, and design the structure of the mechanical metamaterial according to the inverse design model; Among them, each minimum periodic unit includes eight independent vertices. The eight independent vertices include four internally connected independent vertices and four externally connected independent vertices. The four internally connected independent vertices are as follows: for each quadrilateral matrix in the minimum periodic unit, there are two vertices with internal connections to the two adjacent quadrilateral matrices in the same minimum periodic unit. Considering the two vertices at the internal connection as one vertex, there are a total of four internally connected independent vertices in one minimum periodic unit. The four externally connected independent vertices are as follows: for each quadrilateral matrix in the minimum periodic unit, there are two vertices with external connections to the quadrilateral matrices of other minimum periodic units outside. Selecting one vertex from the two vertices with external connections of each quadrilateral matrix as the externally connected independent vertex, there are a total of four externally connected independent vertices in one minimum periodic unit. The geometric structure of the minimum periodic unit is described by eight independent vertices, expressed as: where p represents the p-th minimum periodic unit; represents the displacement of the α-th independent vertex of the p-th minimum periodic unit along the x-axis, and this displacement is the displacement of the corresponding vertex of the square along the x-axis; represents the displacement of the α-th independent vertex of the p-th minimum periodic unit along the y-axis, and this displacement is the displacement of the corresponding vertex of the square along the y-axis; when then each quadrilateral matrix in the minimum periodic unit is a square; α = 1, 2, ···, 8; The hinge is constructed by four springs, including: a linear spring with a stiffness of k for capturing longitudinal response, l a linear spring with a stiffness of k for capturing shear response, s a linear spring with a stiffness of k for capturing bending and torsion deformation, θ a nonlinear torsion spring with a stiffness of k for capturing the contact between adjacent quadrilateral substrates, cont (β) a nonlinear rotational spring; where M = k θ (θ + γθ 3 ), M represents the torque applied by the spring, k θ represents the rotational stiffness, γ represents a dimensionless material parameter, θ represents the rotation angle of the quadrilateral substrate about the z-axis; β represents the angle between two sides of adjacent quadrilateral substrates, β0 represents the angle threshold.

2. The reverse design method according to claim 1, characterized in that, The step S1 includes: S11: Change the geometric parameters of the minimum period unit in the articulated quadrilateral mechanical metamaterial structure to obtain the discrete model of the minimum period unit under different geometric parameters, and this discrete model is expressed as: where m [i,j] represents the mass of the [i, j] - th quadrilateral substrate, represents the acceleration of the quadrilateral substrate in the x - axis direction, represents the acceleration of the quadrilateral substrate in the y - axis direction; J [i,j] represents the moment of inertia of the [i, j] - th quadrilateral substrate, represents the angular acceleration of the quadrilateral substrate about the z - axis; represents the force in the x - axis direction generated by the hinge at the p'-th vertex of the [i, j] - th quadrilateral substrate, represents the force in the y - axis direction generated by the hinge at the p'-th vertex of the [i, j] - th quadrilateral substrate, represents the moment generated by the hinge at the p'-th vertex of the [i, j] - th quadrilateral substrate, where p' = 1, 2, 3, 4; when p' is the vertex on different diagonals of the quadrilateral substrate, or represents the change in the rotation angle of the spring connected to the p'-th vertex of the [i, j] - th quadrilateral substrate; represents at a specific rotation angle θ [i,j] the position vector from the center of the [i, j] - th quadrilateral substrate to its p'-th vertex; represents the change in the lengths of the spring connected to the p'-th vertex of the [i, j] - th quadrilateral substrate along the x - axis and y - axis; represents the additional moment generated at the p'-th vertex of the [i, j] - th quadrilateral substrate, and β1 and β2 represent two angles between two adjacent sides of adjacent quadrilateral substrates. S12: Obtain the stress-strain curve corresponding to the articulated quadrilateral mechanical metamaterial structure under different geometric parameters according to the discrete model.

3. The reverse design method according to claim 2, characterized in that, In the step S2, a training set and a test set regarding the stress-strain curve and geometric parameters are constructed, including: Based on the fact that each quadrilateral matrix in the minimum periodic unit is a square with side length a, the vertices of the minimum periodic unit are randomly perturbed to obtain minimum periodic units with different geometric parameters. The random perturbation includes: S21: Randomly displace a square quadrilateral substrate with side length a to satisfy Thereby obtaining a number of first minimum period units; among them, the displacement coordinates of each vertex of the quadrilateral substrate in the first minimum period unit are respectively and That is and S22: Respectively centered on the eight independent vertices of the first minimum period unit, construct eight square frames with a side length of 0.5a, and randomly move each independent vertex within its corresponding 0.5a square frame, so that all sides of each quadrilateral substrate in the obtained second minimum period unit are greater than 0.2a, and all interior angles are greater than 45°; S23: Perform rotational symmetry on the quadrilateral substrates in the second minimum period unit to obtain the third minimum period unit; S24: Combine the second minimum period unit and the third minimum period unit to obtain a preset number of training sets and test sets.

4. The reverse design method according to claim 3, characterized in that, In step S2, the neural network model includes four hidden layers, and its loss function is expressed as: Among them, are the values of the first 10 principal components, used to represent the stress-strain curve corresponding to the p-th smallest periodic unit; F s (X p ) = [F1(X p ), …, F 10 (X p )] T represents the corresponding neural network prediction value, and β s represents the variance explained in each principal component.

5. The reverse design method according to claim 4, characterized in that, In the step S3, constructing a new training set according to the evolutionary strategy includes: S31: On the basis that each quadrilateral substrate in the minimum period unit is a square with a side length of a, randomly move the vertices of each quadrilateral substrate to obtain the fourth minimum period unit; S32: Respectively centered on the eight independent vertices of the first minimum period unit, construct eight square frames with a side length of 0.1a, and randomly move each independent vertex within its corresponding 0.1a square frame to obtain the fifth minimum period unit; S33: The fourth minimum period unit and the fifth minimum period unit constitute the new training set.

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