Physical information neural network method for porous functional gradient plate analysis and parameter identification

Abstract

The invention discloses a physical information neural network method for porous functional gradient plate analysis and parameter identification. The physical information neural network method for porous functional gradient plate analysis comprises the steps that a physical information neural network with a space coordinate point as input and a displacement field component as output is built according to a known elastic modulus distribution expression of an FGM, and the physical information neural network is used for predicting the mechanical response of a porous functional gradient plate; the physical information neural network method for hole-containing functional gradient plate parameter identification comprises the steps that on the basis of a physical information neural network for hole-containing functional gradient plate analysis, a new network for independent parameter identification is additionally arranged, and a full-field displacement data set obtained based on a digital speckle test and space coordinates form a mixed input mode; and finally, the inversion calculation of the functional gradient parameter distribution is realized through joint optimization. According to the method, the mechanical response of the porous functional gradient plate and the elastic modulus distribution of the functional gradient plate are predicted.

Description

Technical Field

[0001] The present invention relates to the technical field of functional gradient material analysis and design in the field of solid mechanics, and in particular to a physical information neural network method for analyzing and parameter identifying a functional gradient plate containing holes. Background Art

[0002] Functionally graded materials (FGMs) are innovative composite materials whose microscopic composition and properties vary continuously across spatial locations. Based on design requirements, FGMs can non-uniformly and continuously combine materials with varying properties within a structure to form composite materials with specific functions. Therefore, FGMs are suitable for meeting the diverse performance requirements of various complex engineering structures and hold broad application prospects in fields such as aerospace, energy, and biomedicine.

[0003] Since the concept of FGM was proposed, the analysis of its mechanical behavior has attracted widespread attention from the international academic and industrial communities. The main methods for analyzing the mechanical properties of FGM include analytical solutions based on specific gradient assumptions and numerical solutions considering general gradient distributions. There are many limitations in analytically solving the mechanical behavior of FGM. The scope of application of analytical solutions is relatively small. Especially for FGMs with complex gradient distributions, analytical solutions often cannot effectively reflect the gradual characteristics of the material at different positions, affecting the accuracy of the results. Numerical solutions also have limitations in describing the behavior of complex gradient materials. The accuracy of numerical solutions depends on the setting of mesh division and boundary conditions. If the mesh division is too coarse or the boundary conditions are not properly selected, numerical errors or unstable results may occur. Although numerical solution methods are suitable for accurate prediction of responses such as stress and deformation under complex loads, they are highly complex and computationally expensive when applied, especially for problems such as parameter identification.

[0004] In recent years, physical-information neural networks (PINNs), a new numerical computational method, have successfully implemented artificial neural network solutions for complex physical fields by directly embedding physical laws into loss functions. PINNs offer advantages such as low data dependency, no meshing requirements, and the ability to efficiently solve problems with complex boundary conditions and multiple physical fields. However, designing physical-information neural networks for the analysis and parameter identification of functionally graded plates (FGMs) with holes remains a challenging issue in the research of FGM machine learning algorithms. Summary of the Invention

[0005] To overcome the shortcomings of the aforementioned prior art, the present invention provides a physical information neural network method for the analysis and parameter identification of functionally gradient plates (FGPs) containing holes. This method uses a physical information neural network to predict the mechanical response and elastic modulus distribution of FGPs containing holes. For FGP analysis, displacement and stress fields can be predicted with high accuracy without relying on measured data. For parameter identification of FGPs containing holes, elastic modulus distribution prediction is achieved using a differentiable PINN framework based on a small amount of displacement field data.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is:

[0007] A physical information neural network method for analyzing and identifying parameters of a functionally gradient plate containing holes comprises the following steps:

[0008] The physical information neural network method is divided into analysis of functionally graded plates with holes and parameter identification of functionally graded plates with holes;

[0009] The physical information neural network method for analyzing functionally graded plates with holes is to build a physical information neural network with spatial coordinate points as input and displacement field components as output based on the known elastic modulus distribution expression of the FGM to predict the mechanical response of the functionally graded plates with holes.

[0010] The physical information neural network method for parameter identification of functionally gradient plates with holes is based on the physical information neural network for analysis of functionally gradient plates with holes. By adding a new network for independent parameter identification, the full-field displacement data set obtained based on digital speckle testing and the spatial coordinates are combined into a hybrid input mode to construct a representation network for the elastic modulus distribution. Finally, the inverse calculation of the functional gradient parameter distribution is achieved through joint optimization.

[0011] The specific scheme is:

[0012] For the analysis of functionally gradient plates with holes (i.e. solving the displacement field of functionally gradient plates with holes), a physical information network NET is built. uv , the input is the two-dimensional coordinate position point (x, y) of the functional gradient plate with holes, and the output is the displacement field distribution u pred (x, y), v pred (x, y);

[0013] U pred = NET uv (X;θ) (1)

[0014] Among them, NET uv Represents a fully connected neural network, the coordinate position point X=(x,y) of the functional gradient plate with holes, U pred =(u pred (x,y),vpred (x,y)) are the predicted values ​​of the displacement field components in the x and y directions respectively, and θ represents the network NET uv The trainable parameters here refer to the weights w and bias b of the fully connected neural network.

[0015] Using the physical information neural network NET uv The output of the physical information neural network is combined with the automatic differentiation technology of the physical information neural network. The functional gradient plate with holes satisfies the control equations derived from the basic equations of classical elastic mechanics, the load restrictions imposed by the boundary, as well as the displacement boundary conditions and stress boundary conditions, thereby obtaining the relevant physical constraints satisfied by the physical information network.

[0016] The physical constraints include three constraints, which act together on the training process of the physical information neural network;

[0017] Constraint 1 control equations ensure that the model satisfies the basic equations of the functionally gradient plate in terms of physical laws;

[0018] Constraint 2 displacement boundary conditions ensure that the predicted displacement field satisfies the Dirichlet boundary conditions of the functionally graded plate with holes;

[0019] Constraint 3 stress boundary conditions ensure that the predicted stress field meets the requirements of the external load at the boundary of the functionally gradient plate with holes. Through the combined effect of these three physical constraints, an accurate prediction of the displacement field of the functionally gradient plate with holes under tensile load is achieved.

[0020] The constraint 1 is specifically:

[0021] According to the classical elastic theory, the geometric equation satisfied by the functionally graded plate with holes is:

[0022]

[0023] Among them, ε x , ε y and γ xy are normal strain and shear strain respectively;

[0024] The constitutive equation is

[0025]

[0026] Among them, σ x ,σ y and t xy are the normal stress and shear stress, respectively, E(x,y) is the general form of the elastic modulus of the functionally gradient material, and μ is the Poisson's ratio;

[0027] The equilibrium equation is

[0028]

[0029] By substituting the geometric equation (2) and the constitutive equation (3) into the equilibrium equation (4), the control equation based on the displacement form can be further obtained as follows:

[0030]

[0031] The constraint 2 is specifically:

[0032]

[0033] That is, in the boundary region Γ D On the top, the displacement boundary conditions in the x-direction and y-direction are given;

[0034] The constraint 3 is specifically:

[0035] σ i (x,y)=p, i=x,y (7)

[0036] Where p represents the tensile load of the boundary;

[0037] Stress boundary conditions at the hole edge

[0038]

[0039] Where l and m are the sine and direction cosine of the outer normal direction of the arc at the hole, respectively, and (x, y) is the position coordinate of the hole boundary point;

[0040] Based on the above physical information neural network, NET uv By embedding the control equation (5), displacement boundary conditions (6) and stress boundary conditions (7) and (8) into the loss function, a framework with strong physical constraints is constructed. By optimizing the total loss function value, the neural network solution strictly satisfies the multi-physics laws of gradient material inhomogeneity, hole boundary coupling effect and load balance, thereby achieving accurate prediction of the mechanical response of functionally gradient plates with holes.

[0041] The total loss function Loss for predicting the physical field of functionally graded plates with holes under tensile load based on the physical information neural network framework total ; Defined as the mean square error Loss of the control equation PDE ; Mean square error Loss of displacement boundary conditions b1 ; Mean square error Loss of the stress boundary condition on the upper boundary b2 ; Mean square error Loss of the stress boundary condition on the right boundary b3 ; Mean square error Loss of the stress boundary condition at the hole edge b4

[0042] Loss total =LossPDE +Loss b1 +Loss b2 +Loss b3 +Loss b4 (9)

[0043] in

[0044]

[0045] Among them, N f is the number of collocation points randomly sampled in the domain, N b1 is the number of sampling points for displacement boundary conditions, N b2 is the number of sampling points for the upper boundary condition, N b3 is the number of sampling points for the right boundary condition, N b4 is the number of sampling points along the hole edge.

[0046] For parameter identification of functionally graded plates with holes, two independent fully connected networks Net are built uv and NET E ;

[0047] The digital speckle analysis system was used to test the functionally gradient plate with holes under tensile load to obtain the full-field displacement data set u * (x,y),v * (x,y), the input is the two-dimensional coordinate position point (x,y) of the functional gradient plate with holes, two fully connected networks Net uv and NET E The outputs are the displacement field u pred (x,y),v pred (x,y) distribution and distribution of elastic modulus E(x,y);

[0048] U pred = Net uv (X; θ uv ) (16)

[0049] E pred = NET E (X; θ E ) (17)

[0050] Among them, U pred =(u pred (x,y),v pred (x,y)) is the network Net uv Output displacement field u pred (x,y),v pred (x,y) distribution, E pred =Epred (x,y), is the network NET E Output elastic modulus E pred (x, y) distribution, X = (x, y) is the input two-dimensional coordinate position point of the functional gradient plate with holes, θ uv and θ E Represents the network Net uv and NET E The trainable parameters here refer to the weights w and bias b of the fully connected neural network.

[0051] Based on the above physical information neural network, a physical information neural network with spatial coordinate points as input and displacement field components and elastic modulus as output is built to identify the elastic modulus distribution of functionally graded plates with holes. By combining physical constraints including control equations, displacement boundary conditions, stress boundary conditions and other physical equation constraints as well as real displacement field data u * (x,y),v * (x,y) and the predicted displacement field data u pred (x,y),v pred The mean square error between (x, y) is used as the loss function term, by reducing the total loss function Loss total The trained model is trained so that it satisfies the requirements of the functional gradient plate containing holes;

[0052] The total loss function Loss for predicting the physical field of functionally graded plates with holes under tensile load based on the physical information neural network framework total , defined as the mean square error Loss of the control equation PDE ; Mean square error Loss of displacement boundary conditions b1 ; Mean square error Loss of the stress boundary condition on the upper boundary b2 ; Mean square error Loss of the stress boundary condition on the right boundary b3 ; Mean square error Loss of the stress boundary condition at the hole edge b4 ; Mean square error Loss of displacement data data

[0053] Loss total =Loss PDE +Loss b1 +Loss b2 +Loss b3 +Loss b4 +Loss data (18)

[0054] The partial differential equation loss term, displacement boundary condition loss term and stress boundary condition loss term in the above formula are the same as the loss function of the displacement field solution of the functionally gradient plate with holes (see formula (9)).data is the displacement data loss component, as follows

[0055]

[0056] Among them, N a is the number of input real displacement field data points, u * (x,y),v * (x,y) is the full-field real displacement data set constructed by the digital speckle analysis system based on the test of the functional gradient plate with holes under tensile load. pred (x,y),v pred (x,y) is a physical information neural network Net uv The predicted value of the displacement field distribution is obtained.

[0057] Based on the above physical information neural network, Net uv and NET E By embedding the control equation (5), displacement boundary conditions (6), and stress boundary conditions (7), (8), and (19) into the loss function, a framework with strong physical constraints is constructed. By optimizing the total loss function value, the neural network solution strictly satisfies the multi-physical laws of gradient material inhomogeneity, hole boundary coupling effect, and load balance, thereby realizing the prediction of the elastic modulus distribution of functional gradient plates with holes.

[0058] Beneficial effects of the present invention:

[0059] This paper uses a physical information neural network framework for analyzing functionally gradient plates with holes to predict the distribution of their mechanical response under tensile loads based on physical information constraints. This framework also uses a physical information neural network framework for parameter identification of functionally gradient plates with holes to identify the elastic modulus distribution of these plates under tensile loads based on a full-field displacement dataset obtained through digital speckle testing.

[0060] The machine method of physical information neural network proposed in the present invention realizes the prediction of the displacement field distribution of the functionally gradient plate with holes under tensile load and the distribution of the elastic modulus of the functionally gradient plate with holes.

[0061] The proposed machine learning method for a physical information neural network, based on the known elastic modulus distribution expression for functionally gradient materials, constructs a physical information neural network for the mechanical analysis of functionally gradient plates with holes, using spatial coordinate points as input and displacement field components as output. This approach is based on the constraints of the fundamental equations of elasticity and the physical information constraints of boundary conditions. By optimizing the total loss function, the neural network solution is made to strictly satisfy the multi-physics laws of gradient material inhomogeneity, hole-boundary coupling effects, and load balance. This allows for the precise prediction of the displacement and stress field distribution of functionally gradient plates with holes under applied tensile loads. Because it is independent of experimental data, the model has greater applicability and can be widely applied to loading problems in thin plates with holes of various shapes or multiple holes.

[0062] In addition, the method proposed in the present invention realizes the full-field displacement data set obtained by digital speckle testing. Through this deep learning method based on physical information, the true distribution of the elastic modulus of the functional gradient plate containing holes can be predicted and analyzed. This method only requires prior physical information and part of the real displacement field data. The present invention provides a physical information neural network method for the analysis and parameter identification of functional gradient plates containing holes. Unlike the traditional elastic modulus function form that relies on prior knowledge, this method can use real displacement field data to accurately predict the actual distribution of the elastic modulus. Therefore, the use of physical information neural network methods provides a new approach to the stress optimization problem of thin plates containing holes. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 Schematic diagram of the physical information neural network for the analysis of functionally graded plates with holes.

[0064] Figure 2 Schematic diagram of the physical information neural network for parameter identification of functionally graded plates with holes.

[0065] Figure 3 The graphs show the mechanical response and elastic modulus distribution of the functionally gradient plate with holes under load. DETAILED DESCRIPTION

[0066] The present invention will be described in further detail below with reference to the accompanying drawings.

[0067] The present invention realizes the prediction of mechanical response and elastic modulus distribution of functionally gradient plates with holes under load based on a physical information neural network method for analyzing and identifying parameters of functionally gradient plates with holes.

[0068] The following is a detailed description of the present invention and the specific implementation process is as follows: Figure 1 and Figure 2 As shown:

[0069] Physical information neural network NET for the analysis of functionally graded plates with holesuv By building a neural network NET uv , performs mapping from space-time coordinate input to displacement field output, and can automatically learn and predict the distribution of displacement field of complex functional gradient plates with holes under load conditions without relying on grid division, and through automatic differentiation technology, it can further realize the distribution prediction of stress field of functional gradient plates with holes under load conditions.

[0070] Building a fully connected physical information network NET uv (X; θ) is used to predict the displacement field distribution U of the functionally gradient plate with holes under load conditions pred , where the fully connected neural network input is the two-dimensional coordinate position point X = (x, y) of the functional gradient plate with holes, and the output is the displacement field distribution U pred =(u pred (x,y),v pred (x,y)), as shown in the attached Figure 1 shown.

[0071] U pred = NET uv (X;θ) (1)

[0072] Among them, NET uv represents a fully connected neural network, θ represents a fully connected neural network NET uv The trainable parameters here refer to the weights w and bias b of the fully connected neural network.

[0073] Using the physical information neural network NET uv Output U pred =(u pred (x,y),v pred (x,y)), combined with the automatic differentiation technology of physical information neural network, the stress field distribution of functional gradient plate with holes can be further solved. Figure 1 As shown in the figure, the experimental values ​​adopted by the study are that the length of the functionally gradient square plate is d, the hole radius is r, the right end is stretched to a uniform tensile load p, and the elastic modulus expression is given as E(x,y)=E0sin(x 2 +y 2 According to the load on the functionally gradient plate with holes and the simply supported constraints imposed on the left and lower boundaries of the functionally gradient plate, the relevant physical constraints satisfied by the physical information network are obtained.

[0074] Constraint 1: Control equation constraints

[0075] According to the classical elasticity theory, the geometric equation for the two-dimensional problem is

[0076]

[0077] Among them, ε x , ε y , γ xy are the normal strain component and shear strain component along the x and y directions respectively.

[0078] The constitutive equation for the plane stress problem is

[0079]

[0080] Among them, σ x ,σ y , τ xy They are the principal stress and shear stress components along the x and y directions respectively. E(x,y) is the general form of the elastic modulus. Here we study E(x,y)=E0sin(x 2 +y 2 ), E0 = 120 GPa, μ is Poisson's ratio.

[0081] The equilibrium equation is

[0082]

[0083] By substituting the geometric equation (2) and the constitutive equation (3) into the equilibrium equation (4), the control equation constraint based on the displacement form can be further obtained as follows:

[0084]

[0085] Constraint 2: Displacement field boundary condition:

[0086] In this paper, the left and lower boundaries of the functionally graded plate with holes are simply supported (the displacement of the left boundary is zero, and the displacement of the lower boundary is zero).

[0087] u(x,0)=0; v(0,y)=0 (6)

[0088] Constraint 3: Stress boundary condition: (stress boundary condition at the hole edge, tensile load on the right boundary, free upper boundary)

[0089] Stress boundary condition on the upper boundary

[0090] σ y (x,d)=0;σ xy =(x,d)=0(7a)

[0091] Stress boundary condition on the right boundary

[0092] σ y (d,y)=0;σ xy =(d,y)=0(7b)

[0093] Where d is the length of the functionally graded square plate, and p is the uniform tensile load applied at the right boundary.

[0094] Stress boundary conditions at the hole edge

[0095]

[0096] Among them, l and m are the sine and direction cosine of the outer normal direction of the arc at the hole, respectively, and (x,y) is the position coordinate of the hole boundary point.

[0097] Based on the above physical information neural network, NET uv By embedding the control equation (5), displacement boundary conditions (6) and stress boundary conditions (7), (8) into the loss function, training is performed to reduce the total loss function Loss total , so that the trained model satisfies the requirements of functionally gradient plates with holes; the total loss function Loss for predicting the physical field of functionally gradient plates with holes under tensile load based on the physical information neural network framework total , defined as the mean square error Loss of the control equation PDE ; Mean square error Loss of displacement boundary conditions b1 ; Mean square error Loss of the stress boundary condition on the upper boundary b2 ; Mean square error Loss of the stress boundary condition on the right boundary b3 ; Mean square error Loss of the stress boundary condition at the hole edge b4

[0098] Loss total =Loss PDE +Loss b1 +Loss b2 +Loss b3 +Loss b4 (9)

[0099] in

[0100]

[0101]

[0102] Among them, N f is the number of collocation points randomly sampled in the domain, N b1 is the number of sampling points for displacement boundary conditions, N b2 is the number of sampling points for the upper boundary condition, N b3 is the number of sampling points for the right boundary condition, N b4 is the number of sampling points along the hole edge.

[0103] The physical information neural network training process adopts the strategy of decreasing learning rate, and the learning rate for the first n times is 1×10 -4 After several trainings, the learning rate is changed to 0.1 times of the original learning rate for network training. uv Loss function total When the number of training times reaches the maximum, the network training is stopped; the final output of the neural network is the predicted result of the displacement field distribution of the functionally gradient plate with holes under the action of simply supported constraints on the left and bottom ends and a uniform tensile load on the right end. Combined with the automatic differentiation technology, the stress field distribution can be further predicted.

[0104] Physical information neural network NET for elastic modulus distribution of functionally graded plates with holes under load conditions E By building a neural network NET E , executing the spatial-temporal coordinate input to the output of the elastic modulus distribution, it is possible to learn and predict the distribution of the elastic modulus of complex functionally gradient plates with holes under load conditions without relying on mesh division.

[0105] A digital speckle analysis system was used to conduct tests on functionally gradient plates with holes under tensile loads to construct a full-field real displacement dataset u * (x,y),v * (x,y). Build a physical information neural network Net uv The input is the two-dimensional coordinate point position X=(x,y) of the functional gradient plate with holes, and the output is the displacement data U of the functional gradient plate with holes under load conditions. pred =(u pred (x,y),v pred (x,y)). Physical information neural network NET for the elastic modulus distribution of functionally gradient plates with holes under load conditions E Build a physical information network NET E The input is the two-dimensional coordinate point position X=(x,y) of the functional gradient plate with holes, and the output is the elastic modulus distribution E pred =E pred (x,y), as attached Figure 2 shown.

[0106] U pred = Net uv (X;θ uv ) (16)

[0107] E pred = NET E (X;θ E ) (17)

[0108] Among them, Netuv and NET E Represented as two fully connected neural networks with common input, θ uv ,θ E Represents the network Net uv and NET E The trainable parameters here refer to the weights w and bias b of the fully connected neural network;

[0109] Based on the above physical information neural network, a physical information neural network for parameter identification of functional gradient plates with holes is built with spatial coordinate points as input and displacement field components and elastic modulus as output, which is used to identify the elastic modulus distribution of functional gradient plates with holes. By combining physical constraints including control equations, displacement boundary conditions and stress boundary conditions and real displacement field data u * (x,y),v * (x, y) is embedded into the total loss function, and the total loss function Loss is reduced through training. The total loss function Loss of the physical field under tensile load of functional gradient plate with holes is predicted based on the physical information neural network framework. total , defined as the mean square error Loss of the control equation PDE ; Mean square error Loss of displacement boundary conditions b1 ; Mean square error Loss of the stress boundary condition on the upper boundary b2 ; Mean square error Loss of the stress boundary condition on the right boundary b3 ; Mean square error Loss of the stress boundary condition at the hole edge b4 ; Mean square error Loss of displacement field data data

[0110] Loss total =Loss PDE +Loss b1 +Loss b2 +Loss b3 +Loss b4 +Loss data (28)

[0111] The partial differential equation loss term, displacement boundary condition and stress boundary condition loss term in the above formula are the same as the loss function of the above scheme 1. data is the displacement data loss component, as follows

[0112]

[0113] Among them, N a is the number of input real displacement field data points, u * (x,y),v *(x,y) is the full-field real displacement data set constructed by the digital speckle analysis system based on the test of the functional gradient plate with holes under tensile load. pred (x,y),v pred (x,y) is a neural network framework based on physical information Net uv The predicted value of the full-field displacement distribution is obtained.

[0114] When the physical information neural network NET uv Loss function total When the number of training times reaches the maximum, the network training is stopped; the final output of the neural network is the predicted value of the mechanical response of the functionally gradient plate with holes under the action of constraints on the left and lower ends and a uniform tensile load on the right end.

[0115] Through the above two schemes, the physical information neural network framework for the analysis of functional gradient plates with holes can finally predict the distribution of mechanical responses of functional gradient plates with holes under tensile load based on physical information constraints. The physical information neural network framework for parameter identification of functional gradient plates with holes can finally identify the elastic modulus distribution of functional gradient plates with holes under tensile load based on the full-field displacement data set obtained by digital speckle testing.

[0116] In summary, the machine method of physical information neural network proposed in the present invention realizes the prediction of displacement field distribution and stress field distribution of functional gradient plates with holes under tensile load, as well as the distribution prediction of elastic modulus of functional gradient plates with holes.

[0117] Example:

[0118] The invention discloses an application of a physical information neural network method for identifying the displacement field distribution and elastic modulus distribution of a functionally gradient plate containing holes under a tensile load to a functionally gradient plate containing holes.

[0119] Through the prediction of the mechanical response of the functionally graded plate with a central hole under uniaxial tensile load and the identification of the elastic modulus distribution of the functionally graded plate under the framework of the physical information neural network of the present invention, as shown in the attached Figure 2 As shown in the figure, a square functionally graded plate with holes of size 20mm×20mm is constrained at the left and bottom boundaries, and the right end is subjected to a uniform tensile load p. In the analysis and solution of the functionally graded plate with holes, the elastic modulus of the functionally graded plate is assumed to be The mechanical parameters are E0 = 120 GPa, μ = 0.3, the radius of the central hole is 2 mm, and the uniaxial tensile load is p = 1 MPa. Based on the framework of the new physical neural network, the activation function is determined to be Tanh, and a network with 10 hidden layers, 20 neurons per layer, and a learning rate of 1×10 -4The physical information neural network framework is used to train the physical information neural network for the analysis and parameter identification of functional gradient plate with holes under load, aiming to find the optimal network parameters. The present invention adopts a decreasing learning rate strategy, with a learning rate of 1×10 -4 When the learning rate of the next 10,000 times is changed to 0.1 times the original learning rate, the network training is stopped when the total number of physical information network training times exceeds the predetermined 40,000 times.

[0120] The hidden layer of the fully connected neural network is iteratively optimized to finally obtain the displacement field distribution prediction of the mechanical response as follows: Figure 3 As shown in (a)-(b), the stress field distribution is predicted as Figure 3 (c)-(e) and the distribution of elastic modulus is shown in Figure 3 (f) shown.

[0121] The solution proposed in the present invention is based on the constraints of the control equations, displacement boundary conditions, and stress boundary conditions, and realizes the accurate prediction of the displacement field of the functional gradient plate containing holes under uniaxial tensile load through the training of physical information neural network. Further based on the constraints of physical information, by obtaining a small amount of real displacement field data and the mean square error between the predicted displacement field as a constraint embedded into the physical information neural network, the identification of the elastic modulus distribution of the functional gradient plate containing holes under uniaxial tensile load is realized through the training of the physical information neural network. The method provided by the present invention applies the physical information neural network to the prediction of the mechanical response and elastic modulus distribution of the functional gradient plate, and also provides a new analytical means for the future optimization design of functional gradient structures.

Claims

1. A physical information neural network method for analysis and parameter identification of functionally graded plates containing holes, characterized by: The following steps are included: The physical information neural network method is divided into analysis of functionally graded plates with holes and parameter identification of functionally graded plates with holes; The physical information neural network method for analyzing functionally graded plates with holes is to build a physical information neural network with spatial coordinate points as input and displacement field components as output based on the known elastic modulus distribution expression of the FGM to predict the mechanical response of the functionally graded plates with holes. The physical information neural network method for parameter identification of functionally gradient plates with holes is based on the physical information neural network for analysis of functionally gradient plates with holes. By adding a new network for independent parameter identification, the full-field displacement data set obtained based on digital speckle testing and the spatial coordinates are combined into a hybrid input mode to construct a representation network for the elastic modulus distribution. Finally, the inverse calculation of the functional gradient parameter distribution is achieved through joint optimization.

2. The physical information neural network method for analyzing and identifying parameters of functionally gradient plates containing holes according to claim 1, characterized in that: Specifically: Build a physical information network NET for the analysis of functionally graded plates with holes uv , the input is the two-dimensional coordinate position point (x, y) of the functional gradient plate with holes, and the output is the displacement field distribution u pred (x,y),v pred (x,y); U pred =NOT uv (X;θ) (1) Among them, NET uv Represents a fully connected neural network, the coordinate position point X=(x,y) of the functional gradient plate with holes, U pred =(u pred (x,y),v pred (x,y)) are the predicted values ​​of the displacement field components in the x and y directions respectively, and θ represents the network NET uv The trainable parameters here refer to the weights w and bias b of the fully connected neural network; Using the physical information neural network NET uv The output of the physical information neural network is combined with the automatic differentiation technology of the physical information neural network. The functional gradient plate with holes satisfies the control equations derived from the basic equations of classical elastic mechanics, the load restrictions imposed by the boundary, as well as the displacement boundary conditions and stress boundary conditions, thereby obtaining the relevant physical constraints satisfied by the physical information network.

3. The physical information neural network method for analyzing and identifying parameters of functionally gradient plates containing holes according to claim 2, characterized in that: The physical constraints include three constraints, which act together on the training process of the physical information neural network; Constraint 1 control equations ensure that the model satisfies the basic equations of the functionally gradient plate in terms of physical laws; Constraint 2 displacement boundary conditions ensure that the predicted displacement field satisfies the Dirichlet boundary conditions of the functionally graded plate with holes; Constraint 3 stress boundary conditions ensure that the predicted stress field meets the requirements of the external load at the boundary of the functionally gradient plate with holes. Through the combined effect of these three physical constraints, an accurate prediction of the displacement field of the functionally gradient plate with holes under tensile load is achieved.

4. The physical information neural network method for analyzing and identifying parameters of functionally gradient plates containing holes according to claim 3, characterized in that: The constraint 1 is specifically: According to the classical elastic theory, the geometric equation satisfied by the functionally graded plate with holes is: Among them, ε x , ε y and γ xy are normal strain and shear strain respectively; The constitutive equation is Among them, σ x ,σ y and τ xy are the normal stress and shear stress, respectively, E(x,y) is the general form of the elastic modulus of the functionally gradient material, and μ is the Poisson's ratio; The equilibrium equation is By substituting the geometric equation (2) and the constitutive equation (3) into the equilibrium equation (4), the control equation based on the displacement form is further obtained as follows:

5. The physical information neural network method for analyzing and identifying parameters of functionally gradient plates containing holes according to claim 3, characterized in that: The constraint 2 is specifically: That is, in the boundary region Γ D On the surface, displacement boundary conditions in the x-direction and y-direction are given.

6. The physical information neural network method for analyzing and identifying parameters of functionally gradient plates containing holes according to claim 3, characterized in that: The constraint 3 is specifically: σ i (x,y)=p,i=x,y (7) Where p represents the tensile load of the boundary; Stress boundary conditions at the hole edge Where l and m are the sine and direction cosine of the outer normal direction of the arc at the hole, respectively, and (x, y) is the position coordinate of the hole boundary point; Based on the above physical information neural network, NET uv By embedding the control equation (5), displacement boundary conditions (6) and stress boundary conditions (7) and (8) into the loss function, a framework with strong physical constraints is constructed. By optimizing the total loss function value, the neural network solution strictly satisfies the multi-physical laws of gradient material inhomogeneity, hole boundary coupling effect and load balance, thereby achieving accurate prediction of the mechanical response of functional gradient plates with holes.

7. The physical information neural network method for analyzing and identifying parameters of functionally gradient plates containing holes according to claim 3, characterized in that: The total loss function Loss for predicting the physical field of functionally graded plates with holes under tensile load based on the physical information neural network framework total ; Defined as the mean square error Loss of the control equation PDE ; Mean square error Loss of displacement boundary conditions b1 ; Mean square error Loss of the stress boundary condition on the upper boundary b2 ; Mean square error Loss of the stress boundary condition on the right boundary b3 ; Mean square error Loss of the stress boundary condition at the hole edge b4 Loss total =Loss PDE +Loss b1 +Loss b2 +Loss b3 +Loss b4 (9) in Among them, N f is the number of collocation points randomly sampled in the domain, N b1 is the number of sampling points for displacement boundary conditions, N b2 is the number of sampling points for the upper boundary condition, N b3 is the number of sampling points for the right boundary condition, N b4 is the number of sampling points along the hole edge.

8. The physical information neural network method for analyzing and identifying parameters of functionally gradient plates containing holes according to claim 1, characterized in that: For parameter identification of functionally graded plates with holes, two independent fully connected networks Net are built uv and NET E ; The digital speckle analysis system was used to test the functionally gradient plate with holes under tensile load to obtain the full-field displacement data set u * (x,y),v * (x,y), the input is the two-dimensional coordinate position point (x,y) of the functional gradient plate with holes, two fully connected networks Net uv and NET E The outputs are the displacement field u pred (x,y),v pred (x,y) distribution and distribution of elastic modulus E(x,y); U pred =No uv (X;θ uv ) (16) E pred =NOT E (X;θ E ) (17) Among them, U pred =(u pred (x,y),v pred (x,y)) is the network Net uv Output displacement field u pred (x,y),v pred (x,y) distribution, E pred =E pred (x,y), is the network NET E Output elastic modulus E pred (x, y) distribution, X = (x, y) is the input two-dimensional coordinate position point of the functional gradient plate with holes, θ uv and θ E Represents the network Net uv and NET E The trainable parameters here refer to the weights w and bias b of the fully connected neural network; Based on the above physical information neural network, a physical information neural network with spatial coordinate points as input and displacement field components and elastic modulus as output is built to identify the elastic modulus distribution of functionally graded plates with holes. By combining physical constraints including control equations, displacement boundary conditions, stress boundary conditions and other physical equation constraints as well as real displacement field data u * (x,y),v * (x,y) and the predicted displacement field data u pred (x,y),v pred The mean square error between (x, y) is used as the loss function term, by reducing the total loss function Loss total The trained model is trained so that it satisfies the requirements of functional gradient plates with holes.

9. The physical information neural network method for analyzing and identifying parameters of functionally gradient plates containing holes according to claim 8, characterized in that: The total loss function Loss for predicting the physical field of functionally graded plates with holes under tensile load based on the physical information neural network framework total , defined as the mean square error Loss of the control equation PDE ; Mean square error Loss of displacement boundary conditions b1 ; Mean square error Loss of the stress boundary condition on the upper boundary b2 ; Mean square error Loss of the stress boundary condition on the right boundary b3 ; Mean square error Loss of the stress boundary condition at the hole edge b4 ; Mean square error Loss of displacement data data Loss total =Loss PDE +Loss b1 +Loss b2 +Loss b3 +Loss b4 +Loss data (18) The partial differential equation loss term, displacement boundary condition loss term and stress boundary condition loss term in the above formula are the same as the loss function of the displacement field solution of the functionally gradient plate with holes (see formula (9)). data is the displacement data loss component, as follows Among them, N a is the number of input real displacement field data points, u * (x,y),v * (x,y) is the full-field real displacement data set constructed by the digital speckle analysis system based on the test of the functional gradient plate with holes under tensile load. pred (x,y),v pred (x,y) is a physical information neural network Net uv The predicted value of the displacement field distribution is obtained.

10. The physical information neural network method for analyzing and identifying parameters of functionally gradient plates containing holes according to claim 9, characterized in that: Based on the above physical information neural network, Net uv and NET E By embedding the control equation (5), displacement boundary conditions (6), and stress boundary conditions (7), (8), and (19) into the loss function, a framework with strong physical constraints is constructed. By optimizing the total loss function value, the neural network solution strictly satisfies the multi-physical laws of gradient material inhomogeneity, hole boundary coupling effect, and load balance, thereby realizing the prediction of the elastic modulus distribution of functional gradient plates with holes.

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