A method for estimating the deflection of thin plates with unknown material parameters based on physical information network

Abstract

This invention discloses a method for estimating the deflection of thin plates with unknown material parameters based on a physical information network. This method belongs to the field of deep learning and physical neural network prediction technology. The method uses actual measured random point data as the training basis for a dual physical information network. The coordinates of points within the thin plate domain are randomly extracted and input into the dual physical information network. The method then uses a differential system of equilibrium equations and a minimum potential energy integral system in conjunction with automatic differentiation and Monte Carlo integration to respectively predict thin plate parameters and estimate deflection when material coefficients are unknown. This method replaces traditional numerical methods with gridding for interpolation and fitting, leveraging the network's anti-interference and feature extraction capabilities to convert key information, ensuring the accuracy and high reliability of the network method for thin plate deflection estimation.

Description

Technical Field

[0001] The present invention relates to the fields of deep learning and physical neural network prediction, and in particular to a method for estimating the deflection of a thin plate with unknown material parameters based on a physical information network. Background Art

[0002] Thin plates are widely used in industrial production. Their lightweight and flexible structures make them susceptible to deformation due to external interference. To rapidly optimize assembly-oriented industrial production and enable the efficient and precise assembly of complex thin-walled parts, accurate estimation of the nonlinear deformation of thin plate parts is necessary.

[0003] In traditional thin plate deformation estimation, numerical methods are often used to approximate the bending state of the plate. For some simple thin plate deformation problems, mathematical formulas are considered for solution. These methods improve computational accuracy by adding trigonometric series terms to the deflection function, supplementing and correcting the critical stress K value, or, based on elastic thin plate theory, developing governing equations and corresponding boundary conditions for the deformation of thin plates under nonuniform surface stresses. Physical analogies with thin plate thermal stresses provide solutions. In particular, a relatively detailed analysis and solution of the instability of simply supported rectangular thin plates subjected to pure shear has been presented. Due to the very limited scope of these mathematical methods, resulting in low model availability, subsequent research has attempted to perform static and vibration modal analysis of thin plates using finite element software. The Block Lanczos method is used to obtain the natural frequencies and corresponding mode shapes of the thin plates. This approach significantly reduces time complexity at the expense of a small increase in spatial complexity. Furthermore, considerations have been taken into account regarding machining technology, tool selection, fixture design, and cutting parameter selection, addressing the flatness problem of thin plates. However, the finite element method is highly dependent on initial data, making it difficult to judge and respond to data interference, and cannot meet the standards required for high-end assembly manufacturing.

[0004] Compared with traditional prediction methods, neural networks offer strong adaptability, high computational efficiency, low resource consumption, and excellent anti-interference capabilities. They transform constrained optimization problems into unconstrained optimization problems by using penalty functions to account for boundary conditions. By introducing physical models and initial condition constraints, they improve the accuracy and reliability of neural network predictions in situations with small training samples, sparse data, and noise. Currently, research in physical-informatics machine learning is still in its infancy, and the machine learning models employed are relatively simple, often used for one-way parameter estimation or model solving. New methods with high reliability and applicability are still needed for complex thin plate deformation estimation in practical engineering applications.

[0005] Therefore, it is necessary to develop a method for estimating the deflection of thin plates with unknown material parameters based on physical information networks. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to provide a method for estimating the deflection of thin plates with unknown material parameters based on a physical information network. On the basis of the original one-way training structure, the network model is completed to realize parameter prediction and deflection estimation of thin plates with unknown material parameters, which can avoid the jagged and mosaic phenomena caused by data point grid processing; a dual physical information network prediction method with greater reliability and wider applicability improves the network's ability to handle interference and extract and analyze input information, thereby reducing the degree of deviation between the network output and the actual deformation.

[0007] In order to solve the above technical problems, the technical solution adopted by the present invention is:

[0008] A method for estimating the deflection of a thin plate with unknown material parameters based on a physical information network comprises the following steps:

[0009] Step 1: Randomly measure the transverse deformation data of points within the range of the thin plate to form a measurement point set;

[0010] Step 2: Using the measurement point information obtained in step 1 as one of the training reference items of the dual physical information network, extracting a certain number of random point coordinates within the range domain as input data of the forward propagation network; the dual physical information network includes a forward propagation network module, a differential system module, an integral system module, and a feedback module;

[0011] Step 3: Select the forward propagation network as a substitute for the thin plate deformation. After the random point coordinate data is input into the forward propagation network, the forward propagation network performs linear and nonlinear recursive transformations on the input, and the preliminary training results of the forward propagation network are passed into the differential system.

[0012] Step 4: Calculate the differential system error using the initial values ​​of the material parameter variables and the preliminary output of the forward propagation network so that the prediction results meet the physical conditions imposed by the thin plate equilibrium equations and boundary conditions;

[0013] In step 5, the feedback module performs a reverse update of the weight bias of each layer of the forward propagation network based on the system error in step 4. The feedback process is as follows: Back propagation uses the chain rule for reverse calculation. If the error of the current output is large, the weights and bias values ​​of each neuron in the opposite direction of the loss function gradient are adjusted to make the new output closer to the expected value.

[0014] Step 6: Repeat the training feedback process, continuously perform data calculation and information exchange, and finally obtain the best matching value of material parameters that meets the error principle;

[0015] Step 7, introducing the material variable value obtained in step 6 to calculate the integral system penalty function, so that the deflection estimation process is carried out under the state of satisfying the minimum potential energy principle;

[0016] In step 8, the integral system penalty function is passed into the feedback module and the forward propagation network module for continuous iteration to minimize the total potential energy of the thin plate system when it meets the boundary constraints, and obtain the estimated value of the thin plate deflection.

[0017] A further improvement of the technical solution of the present invention is that in step 3, the forward propagation network module is:

[0018] Define an L-layer neural network with N neurons in the Nth layer. Define the weight matrix W and the bias vector b. Given a nonlinear activation function σ, apply the nonlinear activation function σ element-wise to finally implement the FNN recursive operation and obtain the neural network output result:

[0019] Input layer: N 0 (x) = x

[0020] Hidden layers: N l (x)=σ(W l N l-1 (x)+b l )1≤l≤L-1

[0021] Output layer: N L (x) = W L N L-1 (x)+b L .

[0022] A further improvement of the technical solution of the present invention is that: in step 4, the differential system error includes the error of the thin plate equilibrium equation f p , boundary error f bc and the measurement point error f ic The three parts are expressed as follows:

[0023] f=f p +f bc +f ic

[0024] In the thin plate small deflection bending theory, the deflection prediction of the thin plate needs to satisfy the deflection surface differential equation:

[0025]

[0026] From this we get:

[0027]

[0028] in, is the bending stiffness, P is the load function, ω is the basic field variable, E and v represent Young's modulus and Poisson's ratio respectively, and h is the thickness of the experimental elastic plate;

[0029] Deflection observations are obtained based on the forward propagation network training , and the error f of the thin plate equilibrium equation is obtained p and the measurement point error f ic :

[0030]

[0031]

[0032] Among them, D0 ′ is the corresponding bending stiffness value calculated from the elastic modulus parameter value λ obtained in training, w ic is the deformation measurement value of a random measurement point, is the corresponding point network training result;

[0033] There are three typical boundary conditions for thin plate bending problems:

[0034] (1) Geometric boundary condition Γ1: also known as fixed boundary, with a given boundary deflection ω on the boundary ′ and the boundary normal slope

[0035] (2) Mixed boundary condition Γ2: also known as simply supported boundary, the deflection ω is given at the boundary ′ and normal bending moment M n ′ ;

[0036] (3) Boundary condition Γ3: also known as free boundary, where the transverse shear force is given at the boundary and normal bending moment M n ′ ;

[0037] Where n and s correspond to the normal and tangent directions along the boundary, respectively, and the boundary error is defined as:

[0038]

[0039] f b ′ c Set parameter values ​​for the initial boundaries, The corresponding boundary parameter values ​​calculated for the network;

[0040] The error function of the differential system is calculated through the automatic difference method. Automatic difference can calculate all partial derivatives through one forward pass and one backward pass. Automatic difference is applied to repeatedly use the chain rule to obtain the derivative of the output relative to the input in the forward propagation network, and perform partial differential error function fitting.

[0041] A further improvement of the technical solution of the present invention is that: in step 7, the integral system penalty function is divided into the total potential energy of the thin plate system and the boundary constraint penalty function; the total potential energy Π of the thin plate system includes the thin plate strain energy U and the external force work W ext , the expression is as follows:

[0042] Π=U+W ext

[0043]

[0044]

[0045] After obtaining the integral formula of the total potential energy of the thin plate system, the Monte Carlo integral is performed by weighted summing the function density at the integration points to calculate the penalty function. The boundary conditions are learned by the mean square error method to minimize the total potential energy of the thin plate system under the boundary constraints:

[0046]

[0047] The strain energy of the thin plate is converted into:

[0048]

[0049] The work done by external force is converted into:

[0050]

[0051] Where A is the area of ​​the thin plate, N Ω is the number of random sampling points within the plate range, (x i ,y i ) is the coordinate of the sampling point, U(x, y) is the strain energy density function of the thin plate, and f(x, y) is the density function of the work done by the external force on the thin plate;

[0052] The boundary constraint penalty function is:

[0053]

[0054]

[0055] Γ ω =Γ1∪Γ2, are the numbers of random sampling points on the boundaries of the two groups respectively.

[0056] Due to the adoption of the above technical solution, the technical advancements achieved by the present invention are:

[0057] 1. The present invention constructs a dual physical information network including a forward propagation network module, a differential system module, an integral system module and a feedback module. The differential system and the integral system are used to realize the fitting of the forward and inverse problems of the thin plate. By enhancing the information integration and analysis capabilities of the structure, the prediction accuracy and applicability of the dual physical information network are improved. On the basis of ensuring the computational efficiency and accuracy of the dual physical information network, the entire process from sampling point to thin plate bending estimation is completed.

[0058] 2. The dual physical information network structure constructed in the present invention completes the entire process from the measurement point to the estimation of the nonlinear deformation of the thin plate. It has stronger industrial applicability and can improve the network prediction accuracy and model efficiency while consuming less resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. Those skilled in the art can derive other drawings based on these drawings without inventive effort.

[0060] Figure 1 It is a neural network diagram based on dual physical information coupling provided in an embodiment of the present invention;

[0061] Figure 2 Schematic diagram of deformation estimation of dual physical information network in an embodiment of the present invention. DETAILED DESCRIPTION

[0062] It should be noted that the terms "including" and "having" and any variations thereof in the specification and claims of the present invention and the above-mentioned drawings are intended to cover non-exclusive inclusions. For example, a process, method, system, product or apparatus comprising a series of steps or units is not necessarily limited to those steps or units clearly listed, but may include other steps or units that are not clearly listed or are inherent to these processes, methods, products or apparatuses.

[0063] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments:

[0064] like Figure 1-2 As shown, a method for estimating the deflection of a thin plate with unknown material parameters based on a physical information network includes the following steps:

[0065] Step 1: Randomly measure the transverse deformation data of points within the range of the thin plate to form a measurement point set;

[0066] Step 2: Use the measurement point information obtained in step 1 as one of the training reference items of the dual physical information network, and extract a certain number of random point coordinates within the range domain as input data of the forward propagation network;

[0067] The dual physical information network includes a forward propagation network module, a differential system module, an integral system module and a feedback module;

[0068] Step 3: Select the forward propagation network as a substitute for the thin plate deformation. After the random point coordinate data is input into the forward propagation network, the forward propagation network performs linear and nonlinear recursive transformations on the input, and the preliminary training results of the forward propagation network are passed into the differential system.

[0069] Forward propagation network module: Define an L-layer neural network with N neurons in the Nth layer. Define the weight matrix W and the bias vector b. Given a nonlinear activation function σ, apply the nonlinear activation function σ element-wise to finally implement the FNN recursive operation and obtain the neural network output result.

[0070] Input layer: N 0 (x) = x

[0071] Hidden layers: N l (x)=σ(W l N l-1 (x)+b l )1≤l≤L-1

[0072] Output layer: N L (x) = W L N L-1 (x)+b L

[0073] In step 4, the differential system error is calculated using the initial values ​​of the material parameter variables and the preliminary output of the forward propagation network so that the prediction results meet the physical conditions imposed by the thin plate equilibrium equations and boundary conditions.

[0074] The differential system error includes the error of the thin plate equilibrium equation f p , boundary error f bc and the measurement point error f ic The three parts are expressed as follows:

[0075] f=f p +f bc +f ic

[0076] In the thin plate small deflection bending theory, the deflection prediction of the thin plate needs to satisfy the deflection surface differential equation:

[0077]

[0078] From this we get:

[0079]

[0080] in, is the bending stiffness, P is the load function, ω is the basic field variable, E and v represent Young's modulus and Poisson's ratio respectively, and h is the thickness of the experimental elastic plate. The deflection observation value is obtained according to the forward propagation network training The error f of the thin plate equilibrium equation is obtained from this p and the measurement point error f ic :

[0081]

[0082]

[0083] Among them, D0 ′ is the corresponding bending stiffness value calculated from the elastic modulus parameter value λ obtained in training, w ic is the deformation measurement value of a random measurement point, is the corresponding point network training result.

[0084] There are three typical boundary conditions for thin plate bending problems:

[0085] (1) Geometric boundary condition Γ1: also known as fixed boundary, with a given boundary deflection ω on the boundary ′ and the boundary normal slope

[0086] (2) Mixed boundary condition Γ2: also known as simply supported boundary, the deflection ω is given at the boundary ′ and normal bending moment M n ′ ;

[0087] (3) Boundary condition Γ3: also known as free boundary, where the transverse shear force is given at the boundary and normal bending moment M n ′ .

[0088] Where n and s correspond to the normal and tangent directions along the boundary, respectively, and the boundary error is defined as:

[0089]

[0090] f b ′ c Set parameter values ​​for the initial boundaries, The corresponding boundary parameter values ​​calculated for the network.

[0091] The differential system error function is calculated through the automatic difference method. Automatic difference (AD) can calculate all partial derivatives through one forward pass and one backward pass. Automatic difference (AD) is applied to repeatedly use the chain rule to obtain the derivative of the output relative to the input in the forward propagation network, and perform partial differential error function fitting.

[0092] In step 5, the feedback module performs reverse propagation based on the system error in step 4 to complete the reverse update process of the weight deviation of each layer of the forward propagation network.

[0093] The feedback process is as follows: back propagation uses the chain derivative rule for reverse calculation. If the error of the current output is large, the weights and bias values ​​of each neuron in the opposite direction of the loss function gradient are adjusted to make the new output closer to the expected value.

[0094] Step 6: Repeat the training feedback process, continuously perform data calculation and information exchange, and finally obtain the best matching value of material parameters that meets the error principle.

[0095] Step 7, introducing the material variable value obtained in step 6 to calculate the integral system penalty function, so that the deflection estimation process is carried out under the state of satisfying the minimum potential energy principle;

[0096] The integral system penalty function is divided into the total potential energy of the thin plate system and the boundary constraint penalty function.

[0097] The total potential energy Π of the thin plate system includes the thin plate strain energy U and the external force work W ext

[0098] Π=U+W ext

[0099]

[0100]

[0101] After obtaining the integral formula of the total potential energy of the thin plate system, the Monte Carlo integral is performed by weighted summing the function density at the integration points to calculate the penalty function. The boundary conditions are learned by the mean square error method to minimize the total potential energy of the thin plate system under the boundary constraints:

[0102]

[0103] The strain energy of the thin plate is converted into:

[0104]

[0105] The work done by external force is converted into:

[0106]

[0107] Where A is the area of ​​the thin plate, NΩ is the number of random sampling points within the plate range, (x i ,y i ) is the coordinate of the sampling point, U(x, y) is the strain energy density function of the thin plate, and f(x, y) is the density function of the work done by the external force on the thin plate.

[0108] The boundary constraint penalty function is:

[0109]

[0110]

[0111] Γ ω =Γ1∪Γ2, are the numbers of random sampling points on the boundaries of the two groups respectively.

[0112] In step 8, the integral system penalty function is passed into the feedback module and the forward propagation network module for continuous iteration to minimize the total potential energy of the thin plate system when it meets the boundary constraints, and obtain the estimated value of the thin plate deflection.

[0113] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for estimating the deflection of a thin plate with unknown material parameters based on a physical information network, characterized in that: The following steps are involved: Step 1: Randomly measure the transverse deformation data of points within the range of the thin plate to form a measurement point set; Step 2: Using the measurement point information obtained in step 1 as one of the training reference items of the dual physical information network, extracting a certain number of random point coordinates within the range domain as input data of the forward propagation network; the dual physical information network includes a forward propagation network module, a differential system module, an integral system module, and a feedback module; Step 3: Select the forward propagation network as a substitute for the thin plate deformation. After the random point coordinate data is input into the forward propagation network, the forward propagation network performs linear and nonlinear recursive transformations on the input, and the preliminary training results of the forward propagation network are passed into the differential system. Step 4: Calculate the differential system error using the initial values ​​of the material parameter variables and the preliminary output of the forward propagation network so that the prediction results meet the physical conditions imposed by the thin plate equilibrium equations and boundary conditions; Step 5: The feedback module performs reverse propagation based on the system error in step 4 to complete the reverse update process of the weight deviation of each layer of the forward propagation network; The feedback process is as follows: Back propagation uses the chain derivative rule to perform reverse calculations. If the error of the current output is large, the weights and bias values ​​of each neuron in the opposite direction of the loss function gradient are adjusted to make the new output closer to the expected value. Step 6: Repeat the training feedback process, continuously perform data calculation and information exchange, and finally obtain the best matching value of material parameters that meets the error principle; Step 7, introducing the material variable value obtained in step 6 to calculate the integral system penalty function, so that the deflection estimation process is carried out under the state of satisfying the minimum potential energy principle; In step 8, the integral system penalty function is passed into the feedback module and the forward propagation network module for continuous iteration to minimize the total potential energy of the thin plate system when it meets the boundary constraints, and obtain the estimated value of the thin plate deflection.

2. The method for estimating the deflection of a thin plate with unknown material parameters based on a physical information network according to claim 1, characterized in that: In step 3, the forward propagation network module is: Define an L-layer neural network with N neurons in the Nth layer. Define the weight matrix W and the bias vector b. Given a nonlinear activation function σ, apply the nonlinear activation function σ element-wise to finally implement the FNN recursive operation and obtain the neural network output result: Input layer: N 0 (x) = x Hidden layers: N l (x)=σ(W l N l-1 (x)+b l )1≤l≤L-1 Output layer: N L (x) = W L N L-1 (x)+b L .

3. The method for estimating the deflection of a thin plate with unknown material parameters based on a physical information network according to claim 1, characterized in that: In step 4, the differential system error includes the error of the thin plate equilibrium equation f p , boundary error f bc and the measurement point error f ic Three parts: f=f p +f bc +f ic In the thin plate small deflection bending theory, the deflection prediction of the thin plate needs to satisfy the deflection surface differential equation: From this we get: in, is the bending stiffness, P is the load function, ω is the basic field variable, E and v represent Young's modulus and Poisson's ratio respectively, and h is the thickness of the experimental elastic plate; Deflection observations are obtained based on the forward propagation network training The error f of the thin plate equilibrium equation is obtained from this p and the measurement point error f ic : Among them, D0 ′ is the corresponding bending stiffness value calculated from the elastic modulus parameter value λ obtained in training, w ic is the deformation measurement value of a random measurement point, is the corresponding point network training result; There are three typical boundary conditions for thin plate bending problems: (1) Geometric boundary condition Γ1: also known as fixed boundary, with a given boundary deflection ω on the boundary ′ and the boundary normal slope (2) Mixed boundary condition Γ2: also known as simply supported boundary, the deflection ω is given at the boundary ′ and normal bending moment M n ′ ; (3) Boundary condition Γ3: also known as free boundary, where the transverse shear force is given at the boundary and normal bending moment M n ′ ; Where n and s correspond to the normal and tangent directions along the boundary, respectively, and the boundary error is defined as: f b ′ c Set parameter values ​​for the initial boundaries, The corresponding boundary parameter values ​​calculated for the network; The error function of the differential system is calculated through the automatic difference method. Automatic difference can calculate all partial derivatives through one forward pass and one backward pass. Automatic difference is applied to repeatedly use the chain rule to obtain the derivative of the output relative to the input in the forward propagation network, and perform partial differential error function fitting.

4. The method for estimating the deflection of a thin plate with unknown material parameters based on a physical information network according to claim 1, characterized in that: In step 7, the integral system penalty function is divided into the total potential energy of the thin plate system and the boundary constraint penalty function; the total potential energy Π of the thin plate system includes the thin plate strain energy U and the external force work W ext , the expression is as follows: P=U+W ext After obtaining the integral formula of the total potential energy of the thin plate system, the Monte Carlo integral is performed by weighted summing the function density at the integration points to calculate the penalty function. The boundary conditions are learned by the mean square error method to minimize the total potential energy of the thin plate system under the boundary constraints: The strain energy of the thin plate is converted into: The work done by external force is converted into: Where A is the area of ​​the thin plate, N Ω is the number of random sampling points within the plate range, (x i ,y i ) is the coordinate of the sampling point, U(x, y) is the strain energy density function of the thin plate, and f(x, y) is the density function of the work done by the external force on the thin plate; The boundary constraint penalty function is: Γ ω =Γ1∪Γ2, are the numbers of random sampling points on the boundaries of the two groups respectively.

Images