Natural frequency detection method of thin plate in thermal environment based on PINN with optimal fitting deviation

Abstract

The present invention relates to a method for detecting the natural frequency of a thin plate in a thermal environment based on a fitting deviation optimization PINN. The method comprises: determining the control equation of the thin plate in the thermal environment; establishing a model according to the basic parameters of the thin plate to be detected, selecting the control equation data points through a fitting deviation optimization algorithm, and obtaining training data; building a forward neural network to predict the neutral plane deflection of the thin plate, and calculating the network residual; obtaining the partial differential of the predicted deflection to each parameter through an automatic differentiation module, and calculating the control equation residual and the boundary condition residual; obtaining the final loss function of the network; iteratively updating the neural network parameters to reduce the loss function to complete the training, obtain the model and calculate the natural frequency of the thin plate in the thermal environment. The technical solution of the present invention can realize the rapid and accurate detection of the natural frequency of the thin plate in the thermal environment.

Description

Technical Field

[0001] The invention relates to the field of solid mechanics calculation, and in particular to a method for detecting the natural frequency of a thin plate in a thermal environment based on a fitting deviation optimization PINN. Background Art

[0002] Thin plate structure is one of the commonly used structures in mechanical structures, such as blades, splints, wings and other structures. Such structures are in high-temperature working environment for a long time, which is prone to thermal buckling problems, destroying structural stability and leading to serious safety accidents. Therefore, the natural frequency detection problem of thin plate structure in thermal environment is one of its important research contents.

[0003] How to accurately detect the natural frequency of thin plates is an urgent problem to be solved. In the existing technology, deep learning neural networks are widely used in solving partial differential equations due to their network complexity and strong fitting ability. Compared with numerical solution methods, they have advantages such as gridless and nonlinear approximation. However, conventional deep learning networks have multiple solutions, and the network model obtained by data driving will not match the actual physical model, and the effect is poor in some engineering practices.

[0004] In summary, the existing numerical calculation methods are inefficient in detecting the natural frequency of thin plates in thermal environments, and it is difficult to quickly and accurately calculate the natural frequency of thin plates at different temperatures. Summary of the invention

[0005] In order to solve or partially solve the above problems, the present invention provides a method for detecting the natural frequency of a thin plate in a thermal environment based on a fitting deviation optimization PINN, which improves the speed and accuracy of the natural frequency detection of a thin plate in a thermal environment.

[0006] The embodiment of the present application provides a method for detecting the natural frequency of a thin plate under a thermal environment based on a fitting deviation optimization PINN, comprising the following steps:

[0007] Step 1: Determine the solid mechanics control equations and boundary conditions satisfied by the thin plate in the thermal environment;

[0008] Step 2: According to the basic parameters of the thin plate to be tested, a model is established and grid data points used for training are selected, and the data points for calculating the deviation of the control equation are determined by a fitting deviation optimization algorithm;

[0009] Step 3: Build a forward neural network to predict the neutral plane deflection of the thin plate and calculate the network residual;

[0010] Step 4: Obtain the partial differential of the predicted deflection with respect to each input parameter through the automatic differentiation module, calculate the boundary condition residual, and calculate the control equation residual by screening the obtained control equation data points;

[0011] Step 5: Obtain the final loss function of the forward network, iteratively update to obtain the optimal network parameters, complete the training and output the detection model;

[0012] Step six: testing the natural frequencies of the thin plate at different temperatures according to the testing model.

[0013] The beneficial effect of the above embodiment is that during model training, the physical information neural network (PINN) calculates the relevant deviations through the model control equations and boundary condition equations, and incorporates them into the loss function, which accelerates the network convergence and makes the neural network more consistent with the actual physical model, thereby detecting the natural frequency of the thin plate at an unknown temperature. The fitting deviation optimization PINN model can optimize the network training process through the fitting deviation optimization algorithm, quickly obtain the detection model of the thin plate in the thermal environment, and thereby calculate and obtain the natural frequency of the thin plate structure at different temperatures.

[0014] Based on the above embodiments, the present application can be further improved as follows:

[0015] In one of the embodiments of the present application, the solid mechanics control equation of the thin plate in the thermal environment in step 1 is implemented by thermal stress-strain equation and Hamilton's principle.

[0016] In one of the embodiments of the present application, in step 2, the basic parameters of the thin plate include the material elastic modulus, Poisson's ratio, density and thermal expansion coefficient, a thin plate simulation model is established by COMSOL, and the simulation results are compared with the numerical solution results to verify the correctness of the model, the midplane of the thin plate is meshed, and the grid position coordinates and their deflection values ​​at different temperatures are selected as training data.

[0017] In one of the embodiments of the present application, the fitting deviation optimization algorithm described in step 2 is specifically to select a fitting function based on the partial differential equation to be solved and the training data, calculate the fitting deviation of different grid data slices, and find the data area with the smallest deviation as the control equation data point.

[0018] Select a polynomial function to fit the grid data. The polynomial function used for fitting is expressed as:

[0019] g(x,y)=∑a ij x i y j ,0≤i,j≤m;

[0020] Among them, a ij represents the polynomial coefficient, m represents the polynomial order, x, y represent the horizontal and vertical coordinates of the training data, and i and j represent the orders of x and y respectively.

[0021] Calculate the mean square error between the fitting function and the actual deflection value, and slice it step by step from the outside of the data grid to find the data area with the smallest mean square error value as the control equation data point, which can be expressed as:

[0022]

[0023] Among them, MINMSE represents the minimum fitting error, w is the deflection value of the thin plate under the actual thermal environment, N s Indicates the number of data points after slicing.

[0024] In one of the embodiments of the present application, the forward neural network parameters described in step three include the number of hidden layers and hidden nodes, loss function, etc. The network input data is the coordinates of the plane grid nodes in the thin plate and their temperatures, and the output data is the deflection strain of the grid point under the thermal environment.

[0025] In one embodiment of the present application, the network training process in steps 4 and 5 is:

[0026] The feedforward neural network obtains the initial weights and thresholds, obtains the predicted deflection and calculates the mean square error between it and the true value; the post-automatic differentiation module obtains the partial differential of the network output with respect to the input coordinates, and obtains the control equation residual and the boundary condition equation residual through the control equation and the boundary condition equation, and calculates the final loss function; finally, the feedforward neural network parameters are optimized according to the loss function until they meet the requirements, and finally a trained detection model is obtained.

[0027] The loss function of the network is:

[0028] Loss = R I MSE I +R B MSE B +R P MSE P ;

[0029]

[0030] In the formula, MSE I , MSE B and MSE P Respectively represent the neural network residual, control equation residual and boundary condition residual; R I , R B and R P Represents the weight of each residual in the loss function; N, N P 、N B represents the number of training data points of the three residuals; w(x, y, T) is the predicted value of the neural network, T represents the temperature of the grid point, f(x, y, T) is the control equation of the thin plate model, and f Bis the boundary condition equation of the thin plate model.

[0031] One or more technical solutions provided in the embodiments of the present invention have at least the following technical effects or advantages:

[0032] 1. The present invention takes the control equation and boundary condition equation under the thermal environment of the thin plate as physical constraints, selects the control equation data points through the fitting deviation optimization algorithm, accelerates the network convergence, and constructs a detection model of the thin plate temperature and natural frequency, which can quickly and accurately detect the natural frequency of the thin plate at different temperatures.

[0033] 2. Compared with traditional methods, the present invention utilizes the strong fitting property of neural networks and the fitting deviation optimization algorithm to improve the solution efficiency and accuracy. The physical model constraints also increase the physical interpretability of the network.

[0034] Other features and advantages of the present invention will be set forth in the description which follows, and in part will be apparent from the description, or may be learned by practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following is a brief introduction to the drawings required for the specific embodiments or the description of the prior art. In all the drawings, similar elements or parts are generally identified by similar reference numerals. In the drawings, the elements or parts are not necessarily drawn according to the actual scale.

[0036] Figure 1 It is a schematic diagram of the steps of a method for detecting the natural frequency of a thin plate in a thermal environment based on a fitting deviation-optimized PINN provided in an embodiment of the present invention.

[0037] Figure 2 This is a flow chart for obtaining the natural frequency of a thin plate at different temperatures provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0038] The following embodiments of the technical solution of the present invention are described in detail in conjunction with the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and are therefore only used as examples, and cannot be used to limit the protection scope of the present invention.

[0039] It should be noted that, unless otherwise specified, the technical terms or scientific terms used in this application should have the common meanings understood by those skilled in the art to which the invention belongs.

[0040] The specific implementation of the present invention is described below with reference to the accompanying drawings.

[0041] See also Figure 1 , a method for detecting the natural frequency of a thin plate under a thermal environment based on a fitting deviation optimization PINN, comprising the following steps:

[0042] S1: Determine the solid mechanics governing equations and boundary conditions satisfied by the thin plate in thermal environment;

[0043] Among them, the control equation of the natural frequency of the thin plate in the thermal environment is realized through the thermal stress-strain equation and the Hamiltonian principle. When the temperature changes, the thin plate deforms and is affected by the stress generated by the strain and the thermal stress generated by the temperature change. From the Hamiltonian principle, it can be seen that the sum of the strain energy caused by the vibration of the thin plate in the thermal environment, the strain energy caused by the temperature field, and the kinetic energy strain energy of the thin plate is constant. Combining the two, the control equation of the thin plate in the thermal environment can be obtained.

[0044] S2: According to the basic parameters of the thin plate to be tested, a model is established and grid data points used for training are selected, and the data points for calculating the deviation of the control equation are determined by a fitting deviation optimization algorithm;

[0045] The mechanical properties of the thin plate material include elastic modulus E, Poisson's ratio v, density ρ, and thermal expansion coefficient α. At the same time, the material performance parameters are functions of the internal temperature field T, which is affected by the internal temperature field of the material. In the present invention, the temperature field is selected as a constant temperature field, and the elastic modulus of the thin plate is 200×(1-3.34×10 -4 ×(T-273.15))GPa, Poisson's ratio is 0.3×(1+1×10 -4 ×(T-273.15)), density is 7850kg / m 3 , the thermal expansion coefficient is 12.3×10 -6 1 / K, select the physical neutral surface data of the thin plate as the training data.

[0046] The model was established by the solid mechanics module in COMSOL, which is a 0.2×0.2×0.02m thin plate. The ambient temperature of the thin plate is set to T0=300K, the side of the thin plate is set to a fixed edge, and the temperature inside the plate is set to a uniform temperature field for simulation research. 41×41 grid points on the neutral surface of the thin plate in each temperature range are selected as the input data for neural network training, 20×20 grid points at the center of the neutral surface are selected as the control equation data points, and 1×1681 data on the four boundaries of the thin plate are selected as the boundary condition equation data points.

[0047] When selecting the control equation data points, you first need to determine the fitting function. The polynomial function used for fitting can be expressed as:

[0048] g(x,y)=∑a ij x i y j ,0≤i,j≤m;

[0049] Among them, a ijrepresents the polynomial coefficient, m represents the polynomial order, x, y represent the horizontal and vertical coordinates of the training data, and i and j represent the orders of x and y respectively.

[0050] Since the partial differential equation to be solved is a fourth-order partial differential equation, the fitting function used should be no less than the fourth order. The fitting polynomial coefficients are obtained by fitting with fourth-order and fifth-order polynomial functions respectively. The result is that when the fourth-order polynomial fitting is used, the quartic term x 4 、x 2 y 2 ,y 4 The coefficient is 10 -3 When using fifth-order polynomial fitting, the coefficients of the fifth-order terms are all 10 -15 The effect of the fitting effect is small, and the coefficient of the fourth-order term is very close to the coefficient of the fourth-order polynomial. Therefore, the fourth-order polynomial function is selected to fit the grid data, and slices are performed layer by layer from the boundary to the center of the thin plate to obtain the mean square error between the fitting function and the true value in different slice areas. The data area with the smallest mean square error is found as follows:

[0051]

[0052] Among them, MINMSE represents the minimum fitting error, w is the deflection value of the thin plate under the actual thermal environment, N s Indicates the number of data points after slicing. The final result is that the mean square error value is the smallest after the grid data is sliced ​​10-11 times, indicating that the fitting effect is better at this time, and the pressure of the neural network to solve the control equation is smaller, so the data points in the 20×20 area in the center of the thin plate are selected as the control equation data points.

[0053] S3: Build a forward neural network to predict the neutral plane deflection of the thin plate and calculate the network residual;

[0054] Specifically, refer to Figure 2 Construct a neural network, where the basic parameters of the forward neural network are L a ×N u , where L a is the number of hidden layers. In this invention, the hidden layers are set to 6 layers, N u is the number of hidden layer neurons, and the present invention sets the number of hidden layer neurons to 30. The neural network input is the grid coordinates and their temperature (x, y, T), the output is the corresponding deflection value w (x, y, T), the activation function is the tanh function, the number of training times is set to 25000, and the learning rate is 0.001.

[0055] S4: Obtain the partial differential of the predicted deflection with respect to each input parameter through the automatic differentiation module, calculate the boundary condition residuals, and calculate the control equation residuals by screening the obtained control equation data points.

[0056] Specifically, during each neural network training, the automatic differentiation module (AD) can obtain the high-order partial differentials of the network output with respect to the grid coordinates x and y, and calculate the relevant deviations through the control equations and boundary condition equations in the physical model. The control equation deviations and boundary condition equation deviations can generate penalty values ​​when there are differences between the forward neural network and the actual physical model, thereby imposing physical constraints on the network, making the network training results more consistent with the actual physical model.

[0057] The model control equation is as follows:

[0058]

[0059] Where ω is the circular frequency and w is the neutral plane deflection.

[0060] The boundary condition error is calculated by the boundary condition function. Common thin plate boundary conditions include fixed edge, simply supported

[0061] Edges and free edges. The fixed edge boundary condition function is:

[0062]

[0063] The boundary condition function of simply supported edge is:

[0064]

[0065] The free edge boundary condition function is:

[0066]

[0067] S5: Obtain the final loss function of the forward network, iteratively update to obtain the optimal network parameter output model, complete the training and output the detection model;

[0068] Specifically, the loss function of the network can be expressed as:

[0069] Loss = R I MSE I +R B MSE B +R P MSE P ;

[0070]

[0071] Among them, MSE I , MSE B and MSE P Respectively represent the neural network residual, control equation residual and boundary condition residual; R I , R B and R PRepresents the weight of each residual in the loss function. Since the boundary condition residual is greatly affected by the forward neural network, the forward network residual is preferentially reduced during training, and the loss function weight is set to (0.8, 0.1, 0.1); N, N P 、N B Represents the number of training data points of the three residuals; f B is the boundary condition equation of the thin plate model.

[0072] S6: According to the detection model, the natural frequency of the thin plate at different temperatures is detected.

[0073] Specifically, the training data set selects thin plate data sets at 300K, 400K, and 500K for training, and the network training effect is further optimized through multi-gradient temperature data. The test temperature range is 300K-500K, and the temperature gradient is 10K. The natural frequency of the thin plate at different temperatures is detected. After testing, the network detection performance is good, and the detection error is less than 2.4% within the training temperature range.

[0074] The advantages of the present invention are that the control equations and boundary condition equations under the thermal environment of the thin plate are used as physical constraints, and points with large residuals of the control equations at the boundary are screened and removed to accelerate network convergence, and the relationship between the temperature and the natural frequency of the thin plate is constructed, so that the natural frequency of the thin plate under different temperatures can be quickly and accurately detected.

Claims

1. A method for detecting the natural frequency of a thin plate under a thermal environment based on a PINN with a fitting deviation optimization, characterized in that: The following steps are involved: Step 1: Determine the solid mechanics control equation and boundary conditions satisfied by the thin plate in the thermal environment. The solid mechanics control equation is: Wherein, x, y represent the abscissa and ordinate, T represents temperature, E represents elastic modulus, v represents Poisson's ratio, ρ represents density, α represents thermal expansion coefficient, h represents thickness, ω represents circular frequency, and w represents deflection; The boundary conditions include: The fixed edge boundary condition function is: w＝0, The boundary condition function of simply supported edge is: w＝0, The free edge boundary condition function is: Step 2: According to the basic parameters of the thin plate to be tested, a model is established and grid data points used for training are selected, and the data points for calculating the deviation of the control equation are determined by a fitting deviation optimization algorithm. The fitting deviation optimization algorithm is specifically as follows: Select the fitting function according to the partial differential equation to be solved and the training data, calculate the fitting deviation of different grid data slices, and find the data area with the smallest deviation as the data point of the control equation; Select a polynomial function to fit the grid data. The polynomial function used for fitting is expressed as: g(x,y)=∑a ij x i and j ,0≤i, j≤m; Among them, a ij represents the polynomial coefficient, m represents the polynomial order, i and j represent the order of x and y respectively; Calculate the mean square error between the fitting function and the actual deflection value, and slice it step by step from the outside of the data grid to find the data area with the smallest mean square error value as the control equation data point, which is expressed as: Among them, MINMSE represents the minimum fitting error, N s Indicates the number of data points after slicing; Step 3: Build a forward neural network to predict the neutral plane deflection of the thin plate and calculate the network residual; Step 4: Obtain the partial differential of the predicted deflection with respect to each input parameter through the automatic differentiation module, calculate the boundary condition residual, and calculate the control equation residual by screening the obtained control equation data points; Step 5: Obtain the final loss function of the feedforward neural network, iteratively update to obtain the optimal network parameters, complete the training and output the detection model; Step six: using the detection model to detect the natural frequency of the thin plate at different temperatures.

2. The method for detecting the natural frequency of a thin plate in a thermal environment according to claim 1, characterized in that: In the step 2, the basic parameters of the thin plate include the material elastic modulus, Poisson's ratio, density and thermal expansion coefficient. A thin plate simulation model is established through COMSOL, and the simulation results are compared with the numerical solution results to verify the correctness of the model. The midplane of the thin plate is meshed, and the grid position coordinates and their deflection values ​​at different temperatures are selected as training data.

3. The method for detecting the natural frequency of a thin plate in a thermal environment according to claim 1, characterized in that: In step three, the parameters of the feedforward neural network include the number of hidden layers and nodes, and the loss function. The network input data is the coordinates of the plane grid nodes in the thin plate and their temperatures. The output data is the deflection value of the grid point under the thermal environment.

4. The method for detecting the natural frequency of a thin plate in a thermal environment according to claim 1, characterized in that: In the steps 4 and 5, the network training process is: The feedforward neural network obtains initial weights and thresholds, obtains predicted deflection and calculates the mean square error between it and the true value; the post-automatic differentiation module obtains the partial differential of the network output with respect to the input coordinates, and obtains the residual of the control equation and the residual of the boundary condition equation through the control equation and the boundary condition equation, and calculates the final loss function; the parameters of the feedforward neural network are optimized according to the final loss function until they meet the requirements, and finally the trained detection model is obtained; The loss function of the network is: Loss=R I MSE I +R B MSE B +R P MSE P ; In the formula, MSE I , MSE B and MSE P Respectively represent the neural network residual, control equation residual and boundary condition residual; R I , R B and R P Represents the weight of each residual in the loss function; N, N P 、N B represents the number of training data points of the three residuals; w(x, y, T) is the predicted value of the neural network, f(x, y, T) is the control equation of the thin plate model, and f B is the boundary condition equation of the thin plate model.

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