Material stress-strain constitutive relation modeling method and device, program product and medium

Abstract

The invention discloses a material stress-strain constitutive relation modeling method and device, a program product and a medium, and relates to the technical field of metal material constitutive models. Uniaxial tensile experiment data of a metal material under the conditions of different temperatures and different strain rates is obtained and preprocessed, and a training data set reflecting mechanical behaviors of the material under the wide-range working condition can be constructed; constructing an initial neural network model comprising a main body network structure layer, a neuron unit, a connection parameter and a training mechanism based on the training data set, and performing iterative optimization on a hyper-parameter by adopting a Bayesian optimization algorithm to obtain a target neural network model; the nonlinear stress-strain relationship of the material under the conditions of different temperatures and strain rates can be accurately captured; the target neural network model is further converted into a user-defined material subprogram of the finite element analysis platform, and the subprogram can output a predicted material stress value according to the input strain, strain rate and temperature, so that seamless joint with a universal finite element analysis platform is realized, and the prediction precision of simulation is improved.

Description

Technical Field

[0001] This application relates to the field of constitutive model technology for metallic materials, specifically to a method, equipment, program product, and medium for modeling the constitutive relationship of material stress-strain. Background Technology

[0002] The constitutive relation of a material is a core model describing the mapping relationship between stress, strain, strain rate, and temperature during the deformation process of a material under stress, and it is the fundamental input for finite element simulation. An accurate constitutive model can improve the accuracy of simulation in predicting forming defects, thereby optimizing process parameters and reducing industrial trial-and-error costs.

[0003] Currently, the main method for describing the mechanical behavior of metallic materials is to establish stress-strain constitutive models. Traditional constitutive models typically use mathematical analytical expressions to describe the stress-strain relationship of materials, such as the Johnson-Cook model and the Zerilli-Armstrong model. These models are based on specific physical assumptions and simplification conditions, requiring extensive experiments to determine the model parameters. However, due to the complex nonlinear mechanical behavior of metallic materials under different temperatures and strain rates, traditional constitutive models often fail to accurately describe the stress-strain relationship of materials under a wide range of operating conditions, leading to significant deviations between finite element simulation results and actual engineering phenomena. Summary of the Invention

[0004] In view of this, this application provides a method, device, program product and medium for modeling material stress-strain constitutive relations.

[0005] Firstly, this application provides a method for modeling the stress-strain constitutive relationship of materials, the method comprising: Experimental data of uniaxial tension of metallic materials under different temperatures and strain rates were obtained, and the experimental data were preprocessed to obtain a training dataset. An initial neural network model is constructed based on the training dataset. The initial neural network model includes the main network structure layer, neuron units, connection parameters, and training mechanism. The hyperparameters of the initial neural network model are iteratively optimized using the Bayesian optimization algorithm to obtain the target neural network model; The target neural network model is converted into a user-defined material subroutine of the finite element analysis platform. The user-defined material subroutine is used to output the predicted material stress value based on the input strain, strain rate and temperature during the finite element simulation.

[0006] By adopting the above technical solution, and by acquiring and preprocessing uniaxial tensile test data of metallic materials under different temperatures and strain rates, a training dataset reflecting the mechanical behavior of materials under a wide range of working conditions can be constructed. Based on this training dataset, an initial neural network model containing the main network structure layer, neuron units, connection parameters, and training mechanism is constructed. The target neural network model is obtained by iteratively optimizing the hyperparameters using a Bayesian optimization algorithm, which can accurately capture the nonlinear stress-strain relationship of materials under different temperatures and strain rates. Furthermore, the target neural network model is converted into a user-defined material subroutine of the finite element analysis platform. This subroutine can output predicted material stress values ​​based on the input strain, strain rate, and temperature, thereby achieving seamless integration with the general finite element analysis platform, improving the prediction accuracy of the simulation, and providing a reliable basis for process parameter optimization and forming defect prediction.

[0007] Optionally, the preprocessing of the experimental data to obtain the training dataset includes: Calculate the mean and standard deviation of the experimental data, and retain the data whose deviation from the mean does not exceed the preset standard deviation; The nominal stress-strain data in the retained experimental data are converted into real stress-strain data, and the real stress-strain data are converted into real stress-plastic strain data based on the material's elastic modulus. The temperature, strain, strain rate, and stress in the real stress-plastic strain data are normalized to obtain the training dataset.

[0008] Optionally, the main network structure layer includes an input layer, a hidden layer, and an output layer. The input layer is used to receive temperature, strain, and strain rate parameters. The hidden layer is located between the input layer and the output layer and includes a multi-layer structure and multiple neuron units for feature extraction and nonlinear transformation of the input data. The output layer is used to output stress values. The neuron units realize inter-layer signal transmission through the connection parameters, which include weight parameters and bias parameters. The training mechanism includes forward propagation, a loss function, and an optimizer. The optimizer type includes at least one of adaptive moment estimation optimizer, stochastic gradient descent optimizer, adaptive gradient optimizer, and root mean square propagation optimizer.

[0009] Optionally, the iterative optimization of the hyperparameters of the initial neural network model using a Bayesian optimization algorithm includes: Determine the hyperparameters to be optimized, including the number of hidden layers, the number of neurons in each layer, the regularization parameter, and the activation function type; A surrogate model is established using a Gaussian process. The surrogate model is used to fit the mapping relationship between the hyperparameters and the objective function, where the objective function is the mean squared error of the training dataset. Based on the surrogate model, a new combination of hyperparameters is selected using the expected improvement function, and the initial neural network model is trained using the new combination of hyperparameters to obtain the target function value; Based on the objective function value, the surrogate model update and hyperparameter selection process is executed iteratively until the objective function convergence condition is met or the number of iterations reaches a preset threshold, thus obtaining the target hyperparameter combination.

[0010] Optionally, the step of selecting a new combination of hyperparameters based on the surrogate model using the expected improvement function includes: The predicted mean and standard deviation of the hyperparameter combination are calculated using the surrogate model. The expected improvement value of the hyperparameter combination is calculated based on the difference between the predicted mean and the current objective function value and the standard deviation. The hyperparameter combination with the maximum expected improvement value is selected as the new hyperparameter combination.

[0011] Optionally, the step of converting the target neural network model into a user-defined material subroutine for the finite element analysis platform includes: The interface parameters of the user-defined material subroutine of the finite element analysis platform are determined. The interface parameters include input parameters and output parameters, wherein the input parameters include temperature, strain and strain rate, and the output parameters include stress value and corresponding partial derivatives. Extract the network structure and connection parameters of the target neural network model; Based on the interface parameters, the network structure, the connection parameters, and the calculation process are converted into executable subroutine code. The calculation process includes input parameter preprocessing, neural network calculation, and output parameter generation. The compiler is invoked to perform syntax compilation on the subroutine code, generating a user-defined material subroutine for the finite element analysis platform.

[0012] The method further includes: The performance of the target neural network model is evaluated using performance evaluation metrics, including root mean square error, mean absolute error, comprehensive index, and coefficient of determination. The root mean square error is used to evaluate the average deviation between the predicted value and the true value; the mean absolute error is used to evaluate the relative error between the predicted value and the true value; the comprehensive index is calculated based on the root mean square error and the mean absolute error; and the coefficient of determination characterizes the goodness of fit of the predicted curve to the real data. Based on the performance evaluation indicators, analyze the degree of agreement between the stress-strain curves predicted by the target neural network model under different temperatures and strain rates and the corresponding real curves. The prediction accuracy of the target neural network model is verified based on the degree of agreement.

[0013] A second aspect of this application provides an electronic device for modeling material stress-strain constitutive relations, the electronic device comprising: one or more processors and a memory; the memory being coupled to the one or more processors, the memory being used to store computer program code including computer instructions, the one or more processors calling the computer instructions to cause the electronic device for modeling material stress-strain constitutive relations to perform the methods described in the first aspect and any possible implementation thereof.

[0014] A third aspect of this application provides a computer program product containing instructions that, when run on an electronic device for modeling material stress-strain constitutive relations, cause the electronic device to perform the methods described in the first aspect and any possible implementation thereof.

[0015] A fourth aspect of this application provides a computer-readable storage medium including instructions that, when executed on an electronic device modeling material stress-strain constitutive relations, cause the electronic device to perform the methods described in the first aspect and any possible implementation thereof.

[0016] In summary, one or more technical solutions provided in the embodiments of this application have at least the following technical effects or advantages: This application acquires and preprocesses uniaxial tensile test data of metallic materials under different temperatures and strain rates to construct a training dataset reflecting the mechanical behavior of materials under a wide range of working conditions. Based on this training dataset, an initial neural network model is constructed, including the main network structure layers, neuron units, connection parameters, and training mechanism. The hyperparameters are iteratively optimized using a Bayesian optimization algorithm to obtain a target neural network model, which can accurately capture the nonlinear stress-strain relationship of materials under different temperatures and strain rates. Furthermore, the target neural network model is converted into a user-defined material subroutine for a finite element analysis platform. This subroutine can output predicted material stress values ​​based on the input strain, strain rate, and temperature, thereby achieving seamless integration with a general finite element analysis platform, improving the prediction accuracy of simulations, and providing a reliable basis for process parameter optimization and forming defect prediction. Attached Figure Description

[0017] Figure 1 This is a flowchart illustrating a material stress-strain constitutive relation modeling method provided in an embodiment of this application; Figure 2 This is a tensile specimen diagram of an aluminum alloy provided in the embodiments of this application; Figure 3This is a schematic diagram of a graphical user interface system provided in an embodiment of this application; Figure 4 This is a schematic diagram of the main structure of an initial neural network model provided in an embodiment of this application; Figure 5 This is a comparison chart of the prediction curve and experimental curve of a target neural network model provided in an embodiment of this application; Figure 6 This is a comparison chart of the simulation curve and the actual curve provided in the embodiments of this application; Figure 7 This is a graph showing the change of a Bayesian optimization performance index as a function of the iteration process, as provided in an embodiment of this application. Figure 8 This is a schematic diagram of an exemplary hardware structure of an electronic device provided in an embodiment of this application. Detailed Implementation

[0018] To enable those skilled in the art to better understand the technical solutions in this specification, the technical solutions in the embodiments of this specification will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments.

[0019] In the description of the embodiments of this application, the words "for example" or "for instance" are used to indicate examples, illustrations, or explanations. Any embodiment or design that is described as "for example" or "for instance" in the embodiments of this application should not be construed as being more preferred or advantageous than other embodiments or design options. Rather, the use of the words "for example" or "for instance" is intended to present the relevant concepts in a specific manner.

[0020] In the description of the embodiments of this application, the term "multiple" means two or more. For example, multiple systems means two or more systems, and multiple screen terminals means two or more screen terminals. Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the indicated technical features. Thus, a feature defined with "first" or "second" may explicitly or implicitly include one or more of that feature. The terms "comprising," "including," "having," and variations thereof all mean "including but not limited to," unless otherwise specifically emphasized.

[0021] Please refer to Figure 1A flowchart illustrating a method for modeling material stress-strain constitutive relations is presented. This method can be implemented using a computer program, a microcontroller, or a device for modeling material stress-strain constitutive relations. The computer program can be integrated into the computer device or run as a standalone application. Specifically, the method includes steps 10 to 40, as follows: Step 10: Obtain experimental data of uniaxial tension of metallic materials under different temperatures and strain rates, and preprocess the experimental data to obtain a training dataset.

[0022] In this embodiment, the experimental data refers to the nominal stress-strain data obtained by performing uniaxial tensile tests on metallic materials using an MTS E45.105 high-temperature tensile testing machine. The nominal stress and nominal strain data, reflecting the mechanical response of the material, are measured by uniaxial tensile tests under different temperatures and strain rates. The nominal stress is the ratio of the force acting on the original cross-section of the specimen to the original cross-sectional area, and the nominal strain is the ratio of the increase in length of the specimen in the loading direction to its original length.

[0023] Please see Figure 2 This is a tensile specimen diagram of an aluminum alloy provided in an embodiment of this application.

[0024] Specifically, to obtain mechanical response data of the material under different working conditions, uniaxial tensile tests were conducted on 6061 aluminum alloy tensile specimens using an MTS E45.105 high-temperature tensile testing machine. The specimens were... Figure 2 The standard cylindrical tensile specimen shown has a gauge length of 30 mm and a gauge diameter of 6 mm. Before the test, the specimen is clamped in the fixture of the testing machine and heated to the target temperature by resistance heating. The temperature is measured and controlled in real time by thermocouples, with the temperature error controlled within ±2℃. To ensure uniform temperature distribution, the specimen is held at the target temperature for 10 minutes after reaching it.

[0025] For example, the experiment was conducted under five different temperatures (25℃, 200℃, 250℃, 300℃, 350℃) and three different strain rates (0.001s⁻¹, 0.01s⁻¹, 0.1s⁻¹), obtaining experimental data for a total of 15 sets of conditions. Each set of conditions was repeated three times to ensure data reliability. During the tensile process, the loading force and specimen deformation were measured using the force sensor and extensometer built into the testing machine, with a sampling frequency set to 50Hz to ensure data continuity and integrity. The system automatically recorded the raw data such as time, displacement, and load during the experiment and saved this data in real time through the testing machine's built-in data acquisition system. After obtaining the experimental data, the data was preprocessed to improve the accuracy and reliability of subsequent modeling. Preprocessing included outlier removal, true stress-strain conversion, plastic strain calculation, and normalization, ultimately resulting in a standardized training dataset.

[0026] Please see Figure 3 This diagram illustrates a graphical user interface system provided in this application embodiment. This interface system is a visualization tool developed for implementing a material stress-strain constitutive relation modeling method. Based on the PyQt5 framework, it aims to provide a complete visualized operating environment from experimental data preprocessing to neural network model training and finite element subroutine generation. The top of the system interface features six function tabs: "Data Preprocessing," "File Conversion," "Model Configuration," "Training Monitoring," "Result Analysis," and "Subroutine Generation." Each tab corresponds to a different processing step in the method of this application. The diagram shows the data preprocessing module interface, which is mainly used to process raw experimental data obtained from the MTS E45.105 high-temperature tensile testing machine. Users can upload uniaxial tensile experimental data files under different temperature and strain rate conditions through the data import area at the top of the interface. The system will automatically perform outlier removal, true stress-strain conversion, and normalization. A parameter configuration area is located in the middle of the interface for setting various parameters required for data processing, such as temperature range and strain rate. The processed standardized training dataset is automatically saved as a preset format file, providing a data foundation for subsequent neural network model training.

[0027] As an optional embodiment, the step of preprocessing the experimental data to obtain the training dataset may further include the following steps: Step 101: Calculate the mean and standard deviation of the experimental data, and retain the data whose deviation from the mean does not exceed the preset standard deviation.

[0028] Specifically, due to potential anomalies in measurement data caused by equipment fluctuations and environmental interference during the experiment, it is necessary to screen the original experimental data. First, the mean μ and standard deviation σ of all data points in the experimental dataset are calculated. The mean μ is obtained by summing the values ​​of all data points and dividing by the total number of data points n. The standard deviation σ is obtained by taking the square root of the sum of the squared differences between each data point and the mean. Then, based on the 3σ principle in statistics, a preset standard deviation of 3σ is set, and data points satisfying |x-μ|≤3σ are retained, where x is the value of a single data point. This method can eliminate significantly outliers, improving the reliability of the dataset. Step 102: Convert the nominal stress-strain data in the retained experimental data into true stress-strain data, and convert the true stress-strain data into true stress-plastic strain data based on the material's elastic modulus.

[0029] Specifically, since nominal stress and strain cannot truly reflect the mechanical response of a material during large deformation, it is necessary to convert the data into actual stress and strain data. This is achieved using conversion formulas. nominal stress Convert to real stress Using the formula nominal response Convert to real strain Next, to separate the elastic and plastic deformation of the material, it is necessary to determine the elastic modulus E. A linear segment is manually selected on the stress-strain curve as the elastic interval, and the elastic modulus E is obtained by linear fitting using the least squares method. Then, based on Hooke's law, the formula... Calculate plastic strain ,in For true plastic strain, To respond realistically, For actual stress, Let E be the nominal strain and E be the elastic modulus. The resulting true stress-plastic strain data can more accurately characterize the plastic deformation behavior of materials.

[0030] Step 103: Normalize the temperature, strain, strain rate and stress in the real stress-plastic strain data to obtain the training dataset.

[0031] Specifically, since physical quantities such as temperature, strain, strain rate, and stress have different dimensions and numerical ranges, directly using them for neural network training may lead to model convergence difficulties. Therefore, it is necessary to normalize the real stress-plastic strain data so that all features are mapped to the [0,1] interval. The min-max normalization method is used to normalize the temperature T and plastic strain data. strain rate and actual stress The normalization formula is as follows: (The process is repeated twice in the original text.) Where Xn is the normalized value, X is the original value, and Xmin and Xmax are the minimum and maximum values ​​of the physical quantity, respectively. After normalization, a standardized training dataset is obtained, in which each feature has the same numerical scale, which is beneficial for the training and convergence of the neural network model.

[0032] Step 20: Construct an initial neural network model based on the training dataset. The initial neural network model includes the main network structure layers, neuron units, connection parameters, and training mechanism.

[0033] Specifically, based on the preprocessed training dataset, an initial neural network model for predicting the stress-strain constitutive relationship of materials is constructed. This initial neural network model adopts a multi-layer feedforward neural network structure, including four core components: the main network structure layer, neuron units, connection parameters, and the training mechanism.

[0034] The main network structure consists of an input layer, hidden layers, and an output layer. The input layer receives three normalized input variables, corresponding to temperature, strain, and strain rate, respectively. Hidden layers are initially set to 1-3 layers, with the number of neurons in each layer adjustable between 8-64, determined by a Bayesian optimization algorithm. These layers are used for feature extraction and nonlinear transformation of the input data. The output layer contains one output variable, the predicted normalized stress value. Information is transferred between layers through fully connected layers. Neurons are the basic computational units of the network, each containing a weighted summation operation and a nonlinear activation function. Candidate activation functions include ReLU, tanh, sigmoid, swish, ELU, and softplus, used to introduce nonlinear transformations at the neuron output, enhancing the model's ability to express complex stress-strain relationships. Connection parameters include a weight matrix and a bias vector, used to adjust the strength and bias of signal transmission between layers. The weight matrix determines the connection strength between different neurons, while the bias vector adjusts the activation threshold of the neurons. These parameters are initialized using a random initialization method. The training mechanism defines how the model learns, including three stages: forward propagation, loss function calculation, and parameter optimization. The loss function uses mean squared error (MSE) to measure the error between the model's predictions and the actual values. The formula is: Where yi is the actual stress (the experimentally measured stress value of the hot-formed aluminum alloy), yi is the model-predicted stress, and n is the number of samples; the optimizer type is determined from Adam, SGD, Adagrad, and RMSprop using a Bayesian optimization algorithm. The optimizer is used to update the model parameters based on the gradient of the loss function. The specific configuration parameters of these components will be automatically tuned through the subsequent Bayesian optimization process to obtain the optimal model structure and training strategy, thereby improving the model's prediction accuracy of the material stress-strain relationship.

[0035] Please see Figure 4 This is a schematic diagram of the main structure of an initial neural network model provided in an embodiment of this application. The neural network model adopts a multi-layer feedforward structure, including three main parts: an input layer, a hidden layer, and an output layer. The input layer is labeled "Input" and contains three input variables. Let , represent temperature, strain, and strain rate, respectively. These three variables constitute the input vector X, serving as the input features of the model. The hidden layers, labeled "Hidden Layers," employ a two-layer structure. The first layer's neurons are denoted as al¹ᵢ (i=1,2,...,n), and the second layer's neurons are denoted as al²ᵢ (i=1,2,...,n). Each neuron receives all input signals from the previous layer through weighted connections, processes them through a nonlinear activation function, and outputs them to the next layer. The connecting lines in the diagram represent fully connected layers, with each connection corresponding to a trainable weight parameter. The output layer, labeled "Output," contains one output neuron σ, used to predict the material's stress value; its output is denoted as ŷ. The output neuron weights and combines all outputs from the last hidden layer to obtain the final prediction result. This network structure design fully considers the characteristics of material constitutive modeling, effectively capturing the complex relationship between temperature, strain, strain rate, and stress through multi-layer nonlinear transformations. The two-layer design of the hidden layer ensures that the model has sufficient feature extraction capabilities while avoiding training difficulties and overfitting problems caused by an overly complex network structure.

[0036] Step 30: Use the Bayesian optimization algorithm to iteratively optimize the hyperparameters of the initial neural network model to obtain the target neural network model.

[0037] Specifically, to improve the prediction accuracy of the neural network model, a Bayesian optimization algorithm is used to automatically optimize the hyperparameters of the initial neural network model. First, the hyperparameter optimization space is defined, including the number of hidden layers (1-3 layers), the number of neurons in each layer (8-64), the learning rate (0.0001-0.01), the activation function type (ReLU, Tanh, Sigmoid, Swish, ELU, Softplus), and the optimizer type (Adam, SGD, Adagrad, RMSprop). The optimization process uses a Gaussian process as a surrogate model to fit and predict the relationship between hyperparameters and model performance. The expected improvement criterion is selected as the sampling function to choose the most promising sampling point in the optimization space. The entire optimization process iteratively executes an optimization loop of "surrogate model update - sampling point selection - model training - performance evaluation" until the preset number of iterations or performance metric convergence is reached. In each iteration, the Gaussian process model is updated based on the model performance of the current sampling point, and then the next hyperparameter combination to be evaluated is selected using the expected improvement criterion. For example, after multiple rounds of iterative optimization, the optimal hyperparameter combination was finally obtained: two hidden layers, with 25 neurons in the first layer and 16 neurons in the second layer, a learning rate of 0.005, a sigmoid activation function, and an Adam optimizer. Using this optimal parameter set to construct the target neural network model, the root mean square error on the test set was 2.38, the mean absolute error rate was 0.78%, and the coefficient of determination reached 0.9985, indicating that the model optimized by Bayes has good prediction accuracy.

[0038] Please see Figure 5 The figure shows a comparison of the prediction curve and experimental curve of a target neural network model provided in this application embodiment. The figure shows the stress-strain response comparison at four different temperatures (200℃, 250℃, 300℃ and 350℃) under two different strain rates (0.001s⁻¹ and 0.01s⁻¹).

[0039] In the figure, the scatter plots represent the experimentally measured data points, and the solid line represents the prediction results of the neural network model. The comparison results show that under all test conditions, the model prediction curve (solid line) accurately captures the changing trends of the experimental data points (scatter plots). The prediction curve of the target neural network model has good consistency with the actual curve, demonstrating the model's high accuracy and reliability.

[0040] As an optional embodiment, the step of iteratively optimizing the hyperparameters of the initial neural network model using a Bayesian optimization algorithm to obtain the target neural network model may further include the following steps: Step 201: Determine the hyperparameters to be optimized, including the number of hidden layers, the number of neurons in each layer, the regularization parameter, and the activation function type.

[0041] Specifically, the hyperparameters to be optimized are first determined, including but not limited to two main categories: network structure parameters and training parameters. Regarding network structure parameters, a three-layer design is adopted for the hidden layers. The search range for the number of neurons in the first layer is 5-50, the second layer 3-30, and the third layer 3-20. This progressively decreasing design facilitates feature extraction layer by layer. The search range for the learning rate is set between 1e-5 and 1e-2 to ensure the stability and convergence speed of model training. To prevent overfitting, regularization parameters are introduced: the Dropout ratio is set between 0 and 0.5 to randomly deactivate some neurons during training; the L2 regularization coefficient is set between 1e-6 and 1e-3 to constrain the numerical value of model weights. Candidate activation functions include ReLU, tanh, sigmoid, swish, and ELU. Each activation function has different mathematical properties and is suitable for different types of nonlinear mappings. By selecting from these candidate activation functions, the most suitable nonlinear transformation form for modeling material constitutive relations can be found. The range of these hyperparameter values ​​is determined based on the characteristics of material constitutive relation modeling and practical engineering experience, which ensures both the sufficiency of the search space and avoids excessive consumption of computational resources.

[0042] Step 202: Use a Gaussian process to establish a surrogate model. The surrogate model is used to fit the mapping relationship between hyperparameters and the objective function, which is the mean square error of the training dataset.

[0043] Specifically, a Gaussian process (GP) is used as a surrogate model for Bayesian optimization to establish the mapping relationship between hyperparameters and model performance. A Gaussian process is a non-parametric probabilistic model whose core idea is to assume that the function values ​​at any finite number of hyperparameter points follow a multidimensional normal distribution. The kernel function of the surrogate model is a combination of a squared exponential kernel (SE) and a linear kernel. Where x1 and x2 represent any two combinations of hyperparameters, The kernel function hyperparameters are optimized using maximum likelihood estimation. The squared exponential kernel term captures local correlations in the hyperparameter space, while the linear kernel term describes the global trend. This combined kernel function reflects both the smooth impact of hyperparameter variations on model performance and adapts to the overall trend of performance changes. The mean squared error (MSE) of the training dataset is used as the objective function to evaluate the performance of the neural network model under different hyperparameter combinations. The objective function value for each hyperparameter combination is obtained through the following steps: first, a neural network model is built using this set of hyperparameters; then, it is trained on the training set; and finally, the mean squared error of the model's predictions on the validation set is calculated.

[0044] Step 203: Based on the surrogate model, select a new combination of hyperparameters using the expected improvement function, and train the initial neural network model using the new combination of hyperparameters to obtain the target function value.

[0045] Specifically, based on the constructed Gaussian process surrogate model, an expectation improvement function is used to select new hyperparameter combinations. The expectation improvement function comprehensively considers the expected value and uncertainty of prediction performance, enabling local searching near the current optimal solution while also sampling unexplored regions. For each newly selected hyperparameter combination, a neural network model is constructed using this set of parameters and trained on the training dataset. During training, a mini-batch stochastic gradient descent method with a batch size of 32 is used, with 100 training epochs, and an early stopping strategy is employed to prevent overfitting. After training, the mean squared error of the model on the validation set is calculated as the objective function value.

[0046] Step 204: Based on the objective function value, iteratively execute the surrogate model update and hyperparameter selection process until the objective function convergence condition is met or the number of iterations reaches a preset threshold, and obtain the target hyperparameter combination.

[0047] Specifically, based on the objective function value obtained in each iteration, the Gaussian process surrogate model is continuously updated, and the hyperparameter selection process is iteratively executed. The iteration termination condition for the entire optimization process is set to the objective function value improvement being less than 1e-5 after 10 consecutive iterations, or the total number of iterations reaching 100. During the iteration process, the newly evaluated hyperparameter combinations and their corresponding objective function values ​​are added to the sample set. The Gaussian process surrogate model is retrained using the expanded sample set, while the current optimal objective function value f(x⁺) is updated. Based on the updated surrogate model, the expected improvement function is calculated, and the next set of hyperparameter combinations to be evaluated is selected.

[0048] After multiple rounds of iterative optimization, the optimal hyperparameter combination was finally obtained, including: a two-layer hidden layer structure with 25 neurons in the first layer and 16 neurons in the second layer; a learning rate of 5e-3, which ensures both convergence speed and training stability; a dropout ratio of 1e-5, indicating that the model does not require strong regularization to avoid overfitting; a sigmoid activation function, which effectively captures the nonlinear characteristics in the constitutive relationship of materials; and an L2 regularization coefficient of 1e-6, providing appropriate weight constraints. This parameter configuration effectively avoids overfitting while ensuring the model's expressive power.

[0049] As an optional implementation, the step of selecting a new combination of hyperparameters based on the surrogate model and using the expected improvement function may further include the following steps: Step 301: Calculate the predicted mean and standard deviation of the hyperparameter combination using the surrogate model.

[0050] Specifically, a Gaussian process surrogate model is used to predict any point x in the hyperparameter space. Based on existing observation data, the surrogate model calculates the predicted mean μ(x) and predicted standard deviation σ(x) for that point. The predicted mean represents the expected model performance for this hyperparameter combination, while the predicted standard deviation reflects the uncertainty of the prediction result. Specifically, the correlation between the point to be predicted and all known observation points is calculated using the kernel function of the surrogate model, and the mean and variance of the predicted distribution are obtained by combining this with the observation data.

[0051] Step 302: Calculate the expected improvement value of the hyperparameter combination based on the difference between the predicted mean and the current objective function value, as well as the standard deviation.

[0052] Specifically, based on the calculated predicted mean μ(x) and standard deviation σ(x), and the current objective function value f(x⁺), the expected improvement value of the hyperparameter combination is calculated.

[0053] The expected improvement (EI) formula is: Here, μ(x) and σ(x) are the predicted mean and standard deviation of the surrogate model at point x, f(x+) is the current objective function value, and Φ(⋅) and ϕ(⋅) are the cumulative distribution function and probability density function of the standard normal distribution, respectively. The expected improvement function considers both the expected improvement in prediction performance (utilization) and the uncertainty of prediction (exploration) when selecting new hyperparameter combinations. When the predicted mean is good or the prediction uncertainty is large, the expected improvement value of the corresponding point is larger, and it is more likely to be selected as the next evaluation point.

[0054] Step 303: Select the hyperparameter combination with the maximum expected improvement value as the new hyperparameter combination.

[0055] Specifically, the hyperparameter space is sampled in a grid, the expected improvement value of each sampled point is calculated, and the hyperparameter combination with the largest expected improvement value is selected as a candidate point for the next evaluation. To improve search efficiency, the L-BFGS optimization algorithm is used to find local maxima of the expected improvement function in the continuous hyperparameter space, and the global optimum is selected from multiple local optima. This method can efficiently find the most promising hyperparameter combinations under limited computational resources. The selected new hyperparameter combinations may come from a fine search near known optima or from exploratory attempts in insufficiently explored regions, thus achieving a balance between exploration and utilization in the optimization process.

[0056] Step 40: Convert the target neural network model into a user-defined material subroutine for the finite element analysis platform. The user-defined material subroutine is used to output the predicted material stress value based on the input strain, strain rate and temperature during the finite element simulation.

[0057] Specifically, to apply the target neural network model in the finite element analysis environment, the trained neural network model needs to be converted into a user-defined material subroutine format for the finite element analysis platform. The conversion process first saves the model's network structure parameters (including the number of neurons in each layer, connection weights, and biases) and activation function types as standardized data files. Then, a user-defined material subroutine is written to implement the forward propagation calculation process of the neural network. The input parameters of this user-defined material subroutine include the temperature (T), strain (ε), and strain rate (ε') at the current time step; these physical quantities are provided by the finite element main program at each integration point. The subroutine first standardizes the input parameters, using the same normalization parameters as the training data. Subsequently, the program calculates the output values ​​of each hidden layer neuron according to the trained neural network structure. For the first hidden layer, the output of each neuron is calculated as: al¹ᵢ = f(w¹ᵢx + b¹ᵢ), where w¹ᵢ is the weight vector, b¹ᵢ is the bias term, and f is the sigmoid activation function; the calculation process for the second hidden layer is similar. Finally, the output layer neurons calculate the predicted stress value σ and perform inverse normalization to obtain the actual stress value.

[0058] As an optional embodiment, the step of converting the target neural network model into a user-defined material subroutine for a finite element analysis platform may also include the following steps: Step 401: Determine the interface parameters of the user-defined material subroutine of the finite element analysis platform. The interface parameters include input parameters and output parameters, wherein the input parameters include temperature, strain and strain rate, and the output parameters include stress value and corresponding partial derivatives.

[0059] Specifically, based on the standard interface specification of the user-defined material subroutine in the finite element analysis platform, the data structure and transmission method of the input and output parameters are determined. The input parameters include the temperature, strain tensor, and strain rate tensor of the current time step, which are passed in through the state variable array of the user-defined material subroutine interface; the output parameters include the updated stress tensor (σ) and the partial derivative matrix of stress with respect to strain, which are used to construct the tangent stiffness matrix of the material.

[0060] Step 402: Extract the network structure and connection parameters of the target neural network model.

[0061] Specifically, the complete network structure information is extracted from the trained target neural network model, including the number of neurons in each layer, the connection weight matrix W, and the bias vector b. These parameters, along with the statistics (mean and standard deviation) required for normalization, are saved as a standard format data file. The activation function type and calculation order are also recorded to ensure that the neural network's computational flow can be accurately reproduced in the subroutine.

[0062] Step 403: Based on the interface parameters, convert the network structure, connection parameters, and computation process into executable subroutine code. The computation process includes input parameter preprocessing, neural network computation, and output parameter generation.

[0063] Specifically, Python scripts are used to automatically generate standard-format subroutine code, implementing a complete computational flow from input preprocessing to final output. The computation process includes normalization of the input physical quantities, forward propagation calculations of the neural network, nonlinear activation function operations, prediction result generation, and calculation of the partial derivatives of stress with respect to strain and temperature. The entire implementation adopts a modular design approach to ensure a clear code structure, facilitating subsequent maintenance and optimization, while guaranteeing the efficiency and stability of the computational process.

[0064] Step 404: Call the compiler to perform syntax compilation on the subroutine code and generate the user-defined material subroutine for the finite element analysis platform.

[0065] Specifically, the compiler compiles the generated subroutine code, ensuring complete variable declarations, consistent array dimensions, and stable numerical calculations, while optimizing compilation options to improve runtime efficiency. The compiled dynamic link library file can be directly called by the finite element analysis platform, achieving seamless integration of the neural network model and finite element analysis. The resulting user-defined material subroutine for the finite element analysis platform not only accurately implements the ANN-based constitutive model prediction function but also possesses good numerical stability and computational efficiency, meeting the needs of practical engineering applications and providing reliable technical support for the engineering application of material constitutive models.

[0066] Please see Figure 6 This is a comparison chart of the simulated curve and the actual curve provided in the embodiments of this application, wherein the simulated curve is that of 6061 aluminum alloy at 200℃ for 0.1s. -1 Simulation curves under operating conditions.

[0067] In the figure, the red curve represents the finite element analysis (FEM) results based on the target neural network constitutive model provided by this invention, while the blue curve represents the data measured by actual experiments (EXP). Comparing the force-displacement curves, the finite element simulation results based on the neural network constitutive model (red curve) and the experimental measurement data (blue curve) show excellent overlap. Throughout the deformation process, the simulation curve accurately captures the key characteristics of the material response: from the linear response in the initial elastic stage, to the smooth transition at the yield point, to the work hardening behavior in the plastic stage, and finally to the softening stage, the simulation results are highly consistent with the experimental data. This high-precision predictive ability is not only reflected in the matching of the curve trends but also in the accuracy of the force prediction. The maximum error in load prediction is controlled within an acceptable range, demonstrating high prediction accuracy for actual tensile conditions.

[0068] As an optional embodiment, a method for modeling the constitutive relationship of material stress-strain may further include the following steps: Step 501: Use performance evaluation metrics to evaluate the performance of the target neural network model. The performance evaluation metrics include root mean square error, mean absolute error, comprehensive index, and coefficient of determination. Root mean square error is used to evaluate the average deviation between the predicted value and the true value. Mean absolute error is used to evaluate the relative error between the predicted value and the true value. The comprehensive index is calculated based on root mean square error and mean absolute error. The coefficient of determination characterizes the goodness of fit of the predicted curve to the real data.

[0069] Specifically, a multi-dimensional performance evaluation metric is used to comprehensively evaluate the model. These metrics include root mean square error (RMSE), mean absolute error (MAE), a comprehensive index, and the coefficient of determination. RMSE is calculated by taking the square root of the sum of the squares of the differences between the predicted and actual values, reflecting the absolute accuracy of the prediction. The specific calculation formula is as follows: The Mean Absolute Error (EAAR) is calculated as the percentage error between the predicted and actual values, reflecting the relative accuracy of the prediction. The specific formula is as follows: The composite metric (MIX) combines RMSE and EAAR in a weighted manner to provide a more comprehensive performance measure. The specific calculation formula is as follows: The coefficient of determination (R²) evaluates the model's fit by calculating the correlation between predicted and actual values. The specific formula is as follows: These metrics quantify the predictive performance of the model from different perspectives.

[0070] Step 502: Based on the performance evaluation indicators, analyze the degree of agreement between the stress-strain curves predicted by the target neural network model under different temperatures and strain rates and the corresponding real curves.

[0071] Specifically, the model's prediction results under multiple operating conditions were systematically analyzed. For each combination of temperature (200℃, 250℃, 300℃, and 350℃) and strain rate (0.001s⁻¹, 0.01s⁻¹), the differences between the predicted and experimental curves were compared in detail. Emphasis was placed on the linear response in the elastic stage, the accuracy of the yield inflection point, the work hardening behavior in the plastic stage, and the ability to capture stress softening characteristics. Through a combination of quantitative indicators and qualitative analysis, the model's predictive performance under different operating conditions was comprehensively evaluated.

[0072] For example, the model's root mean square error (RMSE) is 2.38 MPa, indicating a small average deviation between the predicted and actual values; the average relative error (EAAR) is only 0.78%, demonstrating high relative accuracy in the prediction results; the composite index (MIX) is 2.505, reflecting the excellent overall performance of the model; and the coefficient of determination (R²) reaches 0.9985, close to 1, indicating a very high degree of fit between the predicted curve and the experimental data. These indicators collectively confirm that the model maintains stable and accurate predictive capabilities under different temperature and strain rate conditions.

[0073] Step 503: Verify the prediction accuracy of the target neural network model based on the degree of agreement.

[0074] Specifically, the model's prediction accuracy was verified by analyzing the aforementioned quantitative indicators. The RMSE value of 2.38 MPa is far below the acceptable error range for engineering applications, indicating that the model's absolute prediction accuracy meets the requirements. The EAAR value of 0.78% is less than the empirical threshold of 1%, indicating that the relative error is controlled at a low level. The comprehensive index of 2.505 is better than the reference value of conventional methods, confirming the model's comprehensive performance. The R² value of 0.9985 is close to the ideal value of 1, indicating that the model's description of the material constitutive relationship is almost perfect. These results fully verify that the neural network-based constitutive model has excellent prediction accuracy, meets the needs of engineering applications, and can be reliably used for numerical simulation of material behavior.

[0075] Please see Figure 7This figure illustrates the performance metrics of Bayesian optimization as iteratively progresses, as provided in this embodiment of the application. The figure shows the optimization process of four metrics: fusion degree, RMSE, EAAR, and R². The green / blue / yellow / orange curves represent the search path of the Bayesian optimization, and the red curve represents the historical best value. The optimization process in the figure shows that the fusion degree fluctuates significantly in the early stages of optimization, reaching a maximum of approximately 0.3. It gradually decreases and stabilizes with iteration, eventually reaching a low level, indicating a continuous improvement in the agreement between the model's predictions and the experimental data. The RMSE metric also exhibits a clear convergence characteristic. Several large fluctuations occurred early on (reaching a maximum of approximately 100), but as the optimization process progressed, the error significantly decreased and eventually stabilized at a low level, confirming the continuous improvement in prediction accuracy. The trend of EAAR (mean relative error percentage) indicates that the optimization process effectively reduced the relative error of the prediction. Although some fluctuations occurred during iteration (reaching a maximum of approximately 60%), it eventually converged to a low level close to 1%. The change in the R² value shows the improvement in the model's goodness of fit. Through Bayesian optimization, the R² value continuously improved and eventually reached an ideal level close to 1, indicating that a model parameter combination with excellent fitting performance was obtained. These optimization results confirm the effectiveness of the Bayesian algorithm in neural network hyperparameter optimization. The final model achieved ideal levels in all performance indicators, providing a reliable guarantee for accurately predicting material constitutive relations.

[0076] This application also provides a computer storage medium that can store multiple instructions. The instructions are adapted to be loaded and executed by a processor. The method for modeling material stress-strain constitutive relations according to the above embodiments is described in detail below. The specific execution process can be found in the detailed description of the above embodiments, which will not be repeated here.

[0077] The following describes an electronic device for modeling material stress-strain constitutive relations, provided by an embodiment of this application. Figure 8 This is a schematic diagram of an exemplary hardware structure of an electronic device provided in an embodiment of this application.

[0078] In some embodiments, the electronic device for modeling the material stress-strain constitutive relationship is a computer device, or the electronic device for modeling the material stress-strain constitutive relationship includes a computer device. The computer device includes a processor, memory, and a network interface connected via a system bus. The processor of the computer device provides computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and internal memory. The non-volatile storage medium stores an operating system, computer programs, and a database. The internal memory provides an environment for the operation of the operating system and computer programs in the non-volatile storage medium. The database of the computer device stores data. The network interface of the computer device is used to communicate with other external terminals or servers via a network connection. In some embodiments, the network interface can be a wired network interface; in some embodiments, the network interface can also be a wireless network interface. When the computer program is executed by the processor, it implements the methods in the embodiments of this application.

[0079] Those skilled in the art will understand that Figure 8 The structure shown is merely a block diagram of a portion of the structure related to the present application and does not constitute a limitation on the computer device to which the present application is applied. Specific computer devices may include more or fewer components than those shown in the figure, or combine certain components, or have different component arrangements.

[0080] The above-described embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit it. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of this application.

[0081] In the above embodiments, implementation can be achieved entirely or partially through software, hardware, firmware, or any combination thereof. When implemented using software, it can be implemented entirely or partially in the form of a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, all or part of the processes or functions described in the embodiments of this application are generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (e.g., coaxial cable, fiber optic, digital subscriber line) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium that a computer can access or a data storage device such as a server or data center that integrates one or more available media. The available medium can be a magnetic medium (e.g., floppy disk, hard disk, magnetic tape), an optical medium (e.g., DVD), or a semiconductor medium (e.g., solid-state drive), etc.

[0082] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. This program can be stored in a computer-readable storage medium, and when executed, it can include the processes described in the above method embodiments. The aforementioned storage medium includes various media capable of storing program code, such as ROM or random access memory (RAM), magnetic disks, or optical disks.

Claims

1. A method for modeling the constitutive relationship of material stress-strain, characterized in that, The method includes: Experimental data of uniaxial tension of metallic materials under different temperatures and strain rates were obtained, and the experimental data were preprocessed to obtain a training dataset. An initial neural network model is constructed based on the training dataset. The initial neural network model includes the main network structure layer, neuron units, connection parameters, and training mechanism. The hyperparameters of the initial neural network model are iteratively optimized using the Bayesian optimization algorithm to obtain the target neural network model; The target neural network model is converted into a user-defined material subroutine of the finite element analysis platform. The user-defined material subroutine is used to output the predicted material stress value based on the input strain, strain rate and temperature during the finite element simulation.

2. The material stress-strain constitutive relation modeling method according to claim 1, characterized in that, The preprocessing of the experimental data to obtain the training dataset includes: Calculate the mean and standard deviation of the experimental data, and retain the data whose deviation from the mean does not exceed the preset standard deviation; The nominal stress-strain data in the retained experimental data are converted into real stress-strain data, and the real stress-strain data are converted into real stress-plastic strain data based on the material's elastic modulus. The temperature, strain, strain rate, and stress in the real stress-plastic strain data are normalized to obtain the training dataset.

3. The material stress-strain constitutive relation modeling method according to claim 1, characterized in that, The main network structure includes an input layer, a hidden layer, and an output layer. The input layer receives temperature, strain, and strain rate parameters. The hidden layer, located between the input and output layers, comprises a multi-layer structure and multiple neuron units for feature extraction and nonlinear transformation of the input data. The output layer outputs stress values. The neuron units transmit signals between layers through connection parameters, including weight parameters and bias parameters. The training mechanism includes forward propagation, a loss function, and an optimizer. The optimizer type includes at least one of adaptive moment estimation optimizer, stochastic gradient descent optimizer, adaptive gradient optimizer, and root mean square propagation optimizer.

4. The material stress-strain constitutive relation modeling method according to claim 1, characterized in that, The iterative optimization of the hyperparameters of the initial neural network model using the Bayesian optimization algorithm includes: Determine the hyperparameters to be optimized, including the number of hidden layers, the number of neurons in each layer, the regularization parameter, and the activation function type; A surrogate model is established using a Gaussian process. The surrogate model is used to fit the mapping relationship between the hyperparameters and the objective function, where the objective function is the mean squared error of the training dataset. Based on the surrogate model, a new combination of hyperparameters is selected using the expected improvement function, and the initial neural network model is trained using the new combination of hyperparameters to obtain the target function value; Based on the objective function value, the surrogate model update and hyperparameter selection process is executed iteratively until the objective function convergence condition is met or the number of iterations reaches a preset threshold, thus obtaining the target hyperparameter combination.

5. The material stress-strain constitutive relation modeling method according to claim 4, characterized in that, The step of selecting new hyperparameter combinations based on the surrogate model using the expected improvement function includes: The predicted mean and standard deviation of the hyperparameter combination are calculated using the surrogate model. The expected improvement value of the hyperparameter combination is calculated based on the difference between the predicted mean and the current objective function value and the standard deviation. The hyperparameter combination with the maximum expected improvement value is selected as the new hyperparameter combination.

6. The method for modeling material stress-strain constitutive relations according to claim 1, characterized in that, The step of converting the target neural network model into a user-defined material subroutine for a finite element analysis platform includes: The interface parameters of the user-defined material subroutine of the finite element analysis platform are determined. The interface parameters include input parameters and output parameters, wherein the input parameters include temperature, strain and strain rate, and the output parameters include stress value and corresponding partial derivatives. Extract the network structure and connection parameters of the target neural network model; Based on the interface parameters, the network structure, the connection parameters, and the calculation process are converted into executable subroutine code. The calculation process includes input parameter preprocessing, neural network calculation, and output parameter generation. The compiler is invoked to perform syntax compilation on the subroutine code, generating a user-defined material subroutine for the finite element analysis platform.

7. The method for modeling material stress-strain constitutive relations according to claim 1, characterized in that, The method further includes: The performance of the target neural network model is evaluated using performance evaluation metrics, including root mean square error, mean absolute error, comprehensive index, and coefficient of determination. The root mean square error is used to evaluate the average deviation between the predicted value and the true value; the mean absolute error is used to evaluate the relative error between the predicted value and the true value; the comprehensive index is calculated based on the root mean square error and the mean absolute error; and the coefficient of determination characterizes the goodness of fit of the predicted curve to the real data. Based on the performance evaluation indicators, analyze the degree of agreement between the stress-strain curves predicted by the target neural network model under different temperatures and strain rates and the corresponding real curves. The prediction accuracy of the target neural network model is verified based on the degree of agreement.

8. An electronic device for modeling the constitutive relationship of material stress-strain, characterized in that, The electronic device includes: one or more processors and a memory; the memory is coupled to the one or more processors, the memory is used to store computer program code, the computer program code including computer instructions, and the one or more processors call the computer instructions to cause the electronic device to perform the method as described in any one of claims 1-7.

9. A computer program product containing instructions, characterized in that, When the computer program product is run on an electronic device that models the constitutive relationship of material stress-strain, the electronic device performs the method as described in any one of claims 1-7.

10. A computer-readable storage medium comprising instructions, characterized in that, When the instruction is executed on an electronic device that models the constitutive relationship of material stress-strain, the electronic device performs the method as described in any one of claims 1-7.

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