A VUMAT Physical Information Neural Network Method for Cyclic Plastic Stress Update

Abstract

This invention discloses a VUMAT physical information neural network-based method for updating cyclic plastic stress, relating to the fields of material constitutive modeling and finite element simulation. It obtains the strain increment at the material integration point, the stress of the previous increment step, and state variables using VUMAT, calculates the test stress and test relative stress, and performs yield assessment. For the plastic response, the integration point data is input into the trained neural network to predict the plastic multiplier increment. Based on the equivalent value of the test relative stress, shear modulus, and safety factor, a radial return safety upper limit is generated to limit the plastic multiplier increment. The limited plastic multiplier increment is substituted into the Chaboche update relation to perform plastic correction on the test stress, and the back stress of each branch and the cumulative equivalent plastic strain are updated to obtain the updated stress and updated state variables. This invention reduces the repeated iterative solution of the plastic multiplier in cyclic plastic analysis and maintains consistency between stress and state variable updates.

Description

Technical Field

[0001] This invention relates to the field of material constitutive modeling and finite element simulation technology, specifically a VUMAT physical information neural network cyclic plastic stress update method. Background Technology

[0002] In finite element analysis of cyclically loaded metallic materials, existing techniques use the user material subroutine VUMAT to receive the strain increments at the material integration points, the stress from the previous increment step, and state variables, enabling elastic prediction, yield assessment, and stress and internal variable updates at the material integration points. In cyclic plastic integration methods employing implicit or semi-implicit return mappings, the plastic increment step uses Newton iteration to solve for the plastic multipliers, updating the stress, back stress, and cumulative equivalent plastic strain, before returning the results to the main finite element program. Some methods utilize neural networks to learn the mapping between loading data and material response to predict stress response.

[0003] However, under cyclic loading, material points repeatedly enter the plastic stage, and their stress state and hardening level continue to evolve. Each plastic increment step requires resolving the plastic multiplier, and the computational cost of local iteration increases with the number of material integration points and loading steps, which restricts the solution efficiency of large-scale analysis. On the other hand, even if the single-step stress prediction value is close to the reference value numerically, it cannot be guaranteed that its evolution with the back stress and cumulative equivalent plastic strain satisfies constitutive consistency and state compatibility. After the incompatible stress and state variables are introduced, they may cause deviations in the yield judgment and the calculation of plastic deformation, resulting in the local error being transmitted along the loading path, which ultimately manifests as cyclic response shift.

[0004] Therefore, the continuous incremental calculation of material integration points depends on the local solution in the plastic step, and requires that the updated stress and internal variables together constitute the effective initial state for subsequent increments. The repeated iteration of traditional return mapping brings an accumulated computational burden, while direct stress prediction is difficult to maintain this state connection relationship, making it difficult to balance reducing the solution cost of the plastic step and maintaining the constitutive consistency of continuous stress updates. Summary of the Invention

[0005] The purpose of this invention is to solve the problems of existing cyclic plastic finite element analysis, which uses back mapping to iteratively solve plastic multipliers, resulting in an accumulation of computational burden as the number of material integration points and loading steps increases, and the difficulty in maintaining constitutive consistency between stress, back stress, and accumulated equivalent plastic strain when directly predicting stress, causing state deviations to be transmitted along continuous increments and resulting in cyclic response shifts, making it difficult to balance the computational efficiency of stress updates with the consistency of state connections.

[0006] In a first aspect, this invention proposes a VUMAT physical information neural network cyclic plastic stress update method, comprising: The integration point data of the material integration point is obtained through VUMAT. The integration point data includes strain increment, stress of the previous increment step and state variables. The state variables include back stress of each branch and cumulative equivalent plastic strain. The test stress and test relative stress are calculated based on the integration point data, and yield judgment is made based on the test relative stress and the state variables. When the response is determined to be elastic, the test stress is output and the state variable is kept unchanged. When the response is determined to be plastic, the integral point data is input into the trained neural network to predict the plastic multiplier increment. Based on the equivalent value of the relative stress, shear modulus and safety factor of the test, a radial return safety upper limit is generated. The plastic multiplier increment is then subject to a non-negative limit and the radial return safety upper limit limit to obtain the limited plastic multiplier increment. Substituting the limited plastic multiplier increment into the Chaboche update relation, the test stress is plastically corrected, and the back stress of each branch and the cumulative equivalent plastic strain are updated to obtain the updated stress of the material integration point and the updated state variables. The updated stress and updated state variables are returned to the finite element main program so that the material integration point can be read as the stress and state variables of the previous increment step in the next increment step.

[0007] Preferably, the training process of the neural network includes: The strain increment, stress of the previous loading step, back stress of each branch of the previous loading step, cumulative equivalent plastic strain of the previous loading step, and reference plastic multiplier increment, reference stress and reference state variable of the material between adjacent loading steps under cyclic loading conditions are obtained. The neural network is trained by using the strain increment, the stress of the previous loading step, the back stress of each branch of the previous loading step, and the cumulative equivalent plastic strain of the previous loading step as inputs, and the reference plastic multiplier increment of the next loading step as supervision information.

[0008] Preferably, the training process of the neural network further includes: The predicted value of the plastic multiplier increment output by the neural network, along with the strain increment of the corresponding loading step, the stress of the previous loading step, and the state variables of the previous loading step, are input into a fixed Chaboche material update layer. The fixed Chaboche material update layer applies the non-negativity constraint and radial return safety upper limit constraint to the predicted value of the plastic multiplier increment, performs plastic correction on the test stress based on the limited plastic multiplier increment, and updates the back stress of each branch and the cumulative equivalent plastic strain to obtain the predicted stress and predicted state variables. A training loss is constructed based on the predicted value of the plastic multiplier increment, the predicted stress, and the predicted state variable, and the parameters of the neural network are updated according to the training loss. The fixed Chaboche material update layer does not contain trainable parameters.

[0009] Preferably, the predicted state variables include the predicted back stress of each branch and the predicted cumulative equivalent plastic strain, and the training loss includes data loss and physical loss; The predicted values ​​of the plastic multiplier increment, the predicted stress, and the predicted state variables are compared with the corresponding reference results to calculate the data loss. The updated yield residual is calculated based on the predicted stress and the predicted state variables, and a yield positive penalty is applied to the updated yield residual that is greater than zero. The back stress of each branch in the previous loading step is semi-implicitly reduced based on the plastic multiplier increment after the limit, and the relative stress equivalent value is calculated in combination with the test stress. The return mapping consistency residual is calculated based on the relative stress equivalent value, the plastic correction amount and the updated yield strength, and the return mapping consistency loss is calculated based on the return mapping consistency residual. An underprediction penalty is applied to the plastic multiplier increment after the limit is lower than the reference plastic multiplier increment; The physical loss is constructed based on the yielding positive penalty, the return mapping consistency loss, and the underprediction penalty.

[0010] Preferably, the training loss further includes a hard sample loss, the construction of which includes: The training samples are sorted from largest to smallest according to their yield positive penalty value and the square of the returned mapping consistency residual. Then, yield-difficult samples and consistency-difficult samples are selected from the current training batch according to a preset ratio. The average values ​​of the yield positive penalty value of the yield-difficult sample and the squared value of the return mapping consistency residual of the consistency-difficult sample are respectively calculated to obtain the yield-difficult sample loss and the consistency-difficult sample loss. The yielding difficulty sample loss and the consistency difficulty sample loss are weighted and included in the training loss to update the parameters of the neural network.

[0011] Preferably, the calculation of the test stress and the test relative stress, and the yield determination based on the test relative stress and the state variables, includes: The elastic stress increment is calculated by substituting the strain increment into the elastic relationship of the material, and then added to the stress of the previous increment step to obtain the test stress. Extract the deviatoric stress portion of the test stress, sum the back stresses of each branch to obtain the total back stress, and subtract the total back stress from the deviatoric stress portion to obtain the relative stress of the test stress. Calculate the equivalent value of the relative stress of the test, and calculate the yield strength based on the cumulative equivalent plastic strain. Use the difference between the equivalent value and the yield strength as the test yield function value. When the value of the trial yield function is not greater than the preset tolerance, it is determined to be an elastic response; otherwise, it is determined to be a plastic response.

[0012] Preferably, the generation of the radial return safety upper limit includes: Read the shear modulus of the material corresponding to the material integration point, and obtain a safety factor that is greater than 0 and less than 1; Divide the equivalent value of the relative stress of the test by three times the shear modulus, and multiply the resulting quotient by the safety factor to obtain the radial return safety upper limit; The plastic multiplier increment is compared with zero, and the larger value is taken as the non-negative constraint result. Then, the non-negative constraint result is compared with the radial return safety upper limit, and the smaller value is taken as the plastic multiplier increment after the limit is set, which is then substituted into the Chaboche update relation.

[0013] Preferably, substituting the limited plastic multiplier increment into the Chaboche update relation includes: The reduction coefficient is calculated based on the plastic multiplier increment after the amplitude limit and the dynamic recovery parameters corresponding to each branch. The back stress of each branch is reduced, and the plastic flow direction is calculated based on the deviatoric stress part of the back stress of each branch after reduction and the test stress. The deviatoric stress portion is plastically corrected based on the plastic flow direction and the plastic multiplier increment after the amplitude limitation, and the hydrostatic pressure portion of the test stress is added back to obtain the updated stress. According to the Chaboche update relation, the back stress of each branch is updated based on the reduction factor, the plastic flow direction, and the plastic multiplier increment after the limit, and the plastic multiplier increment after the limit is accumulated to update the cumulative equivalent plastic strain.

[0014] Preferably, the trained neural network is deployed to VUMAT in the following manner: Export the weights, biases, activation function types, input feature order, input normalization parameters, output scale factors, and material parameters of each layer of the neural network, and transcribe them into an array or include file that can be called by VUMAT; The integral point data are assembled according to the input feature order, standardized using the input standardization parameters, and forward computation is performed through the weights, biases and activation functions of each layer. The output result is processed by the softplus function and multiplied by the output scale factor to obtain the plastic multiplier increment. VUMAT performs stress and state variable updates using the same material parameters and Chaboche update relationships as the fixed Chaboche material update layer.

[0015] Preferably, before assembling the integration point data, the tensor component order of the strain increment, the stress of the previous increment step, and the back stress of each branch received by VUMAT is converted to the component order used during training, and the shear strain component is represented by tensor shear consistent with the training data. After completing the stress and state variable update, the updated stress and the updated back stress of each branch are converted back to the tensor component order required by the finite element main program. The converted updated stress is written into the stress output array, and the converted back stress of each branch and the updated cumulative equivalent plastic strain are written into the state output array. The stress output array and the state output array are returned to the finite element main program for the next incremental step to read the stress and state variables from the previous incremental step.

[0016] In a second aspect, the present invention provides a computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, it implements the VUMAT physical information neural network cyclic plastic stress update method described in any of the above claims.

[0017] Thirdly, the present invention provides an electronic device, including a processor and a memory, wherein the memory stores a computer program, and when the computer program is executed by the processor, it implements the VUMAT physical information neural network cyclic plastic stress update method described in any of the above claims.

[0018] Compared with the prior art, the present invention has the following beneficial effects: 1. By distinguishing between elastic and plastic responses through yield judgment, the neural network is used to predict the plastic multiplier increment only for the plastic response. Then, the Chaboche update relation is used to update the stress, back stress of each branch, and cumulative equivalent plastic strain. This allows the same plastic multiplier increment to be associated with stress correction and state evolution simultaneously, reducing the overhead of plastic multiplier iteration while maintaining the coordination and connection of the states required for subsequent increments.

[0019] 2. By combining underprediction penalty, return mapping consistency loss and radial return safety upper limit, the insufficient plasticity correction caused by small plasticity multiplier increment is suppressed through training, and excessive plasticity correction is restricted during invocation; combined with additional penalty for difficult samples, the constraints on samples with large yield violation and consistency deviation are strengthened, reducing the risk of local update deviation being passed on to subsequent increments and causing cyclic response shift.

[0020] 3. Training constraints are established by fixing the Chaboche material update layer, and the same material parameters, update relationships and input / output processing methods are used when deploying in VUMAT. Combined with tensor component transformation and state variable write-back, the stress and state evolution relationship constrained by the training is continued to the continuous finite element method, reducing the deviation introduced by model writing and state transfer, and improving the stability of cyclic loading analysis. Attached Figure Description

[0021] To more clearly illustrate the technical solutions of the embodiments of this application, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of this application and should not be regarded as a limitation on the scope of this application.

[0022] Figure 1 A flowchart illustrating a VUMAT physical information neural network cyclic plastic stress update method provided in an embodiment of the present invention; Figure 2 A schematic diagram of the cyclic plastic stress update process of the trained neural network in VUMAT provided in an embodiment of the present invention; Figure 3 The diagram shows a comparison of the Mises equivalent stress distribution of the traditional elastoplastic constitutive model and the plastic multiplier neural network model under a 0.9% uniaxial cyclic loading condition, provided in this embodiment of the invention. (a) represents the calculation result of the traditional elastoplastic constitutive model, and (b) represents the calculation result of the plastic multiplier neural network model.

[0023] Figure 4 A comparison of stress-strain hysteresis curves of a traditional elastoplastic constitutive model and a plastic multiplier neural network model under a 0.7% uniaxial cyclic loading condition, provided for embodiments of the present invention; Figure 5 A comparison of stress-strain hysteresis curves of a traditional elastoplastic constitutive model and a plastic multiplier neural network model under a 0.8% uniaxial cyclic loading condition, provided for embodiments of the present invention; Figure 6 A comparison of stress-strain hysteresis curves of a traditional elastoplastic constitutive model and a plastic multiplier neural network model under a 0.9% uniaxial cyclic loading condition, provided in an embodiment of the present invention. Detailed Implementation

[0024] To make the technical means, creative features, objectives, and effects of this invention easier to understand, the invention is further described below with reference to specific embodiments. However, the following embodiments are merely preferred embodiments of this invention and not all of them. Other embodiments obtained by those skilled in the art based on the embodiments described herein without creative effort are all within the protection scope of this invention.

[0025] Example 1: like Figure 1 As shown, this embodiment provides a VUMAT physical information neural network cyclic plastic stress update method, which specifically includes the following steps: S101. Obtain the integration point data of the material integration point through VUMAT. The integration point data includes strain increment, stress of the previous increment step, and state variables. The state variables include back stress of each branch and cumulative equivalent plastic strain.

[0026] Among them, the material integration point is the position where the finite element model performs material constitutive calculations, and its stress and state variables are updated step by step with the loading process; the strain increment represents the strain change of the current increment relative to the previous increment step; the back stress of each branch jointly characterizes the kinematic hardening state of the material as plastic deformation develops; the cumulative equivalent plastic strain is used to record the degree of cumulative plastic deformation generated by previous loading.

[0027] In a further implementation, VUMAT reads the strain increment and the stress of the previous increment step from the data input from the finite element master program, and reads the back stress of each branch and the cumulative equivalent plastic strain of the same material integration point from the state variable storage location, and assembles the integration point data according to the corresponding relationship.

[0028] It should be noted that the initial state set by the finite element model is used in the first call, and the state variables returned by the previous increment step are inherited in subsequent calls, so that the yield judgment and stress update are based on the existing loading state of the material integration point.

[0029] S102. Calculate the test stress and test relative stress based on the integration point data, and make a yield judgment based on the test relative stress and the state variables.

[0030] Among them, the test stress is the stress estimate obtained after applying the strain increment according to the elastic relationship, which is used to determine whether plastic correction is needed; the test relative stress represents the difference between the deviatoric stress part of the test stress and the sum of the back stresses of each branch, reflecting the stress effect that drives plastic deformation after considering the kinematic hardening state.

[0031] The strain increment is converted into an elastic stress increment based on the elastic parameters of the material, and added to the stress of the previous increment step to obtain the test stress; the deviatoric stress portion of the test stress is extracted, and the sum of the back stresses of each branch is subtracted to obtain the relative stress of the test stress.

[0032] Further, the equivalent value of the relative stress of the test is calculated, and the corresponding yield strength is calculated based on the cumulative equivalent plastic strain. The difference between the equivalent value of the relative stress of the test and the yield strength is taken as the test yield function value. When the test yield function value is not greater than the preset tolerance, it is determined to be an elastic response; otherwise, it is determined to be a plastic response.

[0033] S103. When the response is determined to be elastic, the test stress is output and the state variable is kept unchanged; when the response is determined to be plastic, the integral point data is input into the trained neural network to predict the plastic multiplier increment.

[0034] Among them, the plastic multiplier increment is a scalar that controls the amplitude of the current plastic flow, and its value is involved in stress correction, back stress update of each branch, and cumulative equivalent plastic strain update.

[0035] It should be noted that the neural network takes integral point data as input and plastic multiplier increment as prediction output. During the training phase, it uses adjacent loading step data under cyclic loading conditions and trains the neural network based on reference results and physical constraints constructed by fixing the Chaboche material update layer.

[0036] In a further implementation, for the elastic response, VUMAT directly uses the test stress as the result of this stress update, while retaining the original back stress of each branch and the cumulative equivalent plastic strain, and ends the material update process for this material integration point.

[0037] For the plastic response, the integral point data is processed according to the input organization and standardization method used during training. The forward calculation is performed through the trained neural network to obtain the plastic multiplier increment, and then the subsequent limiting and material update steps are performed.

[0038] Through the above branching process, the elastic response does not need to be predicted by the neural network, while the plastic response provides the plastic multiplier increment that originally required local iterative solution through the forward calculation of the network; the prediction results are further processed by the material update relation, so that the network prediction is clearly linked to the evolution of stress and state variables.

[0039] S104. Based on the equivalent value of the relative stress, shear modulus and safety factor of the test, a radial return safety upper limit is generated. The plastic multiplier increment is subject to non-negative limitation and the radial return safety upper limit limitation to obtain the limited plastic multiplier increment.

[0040] Among them, the radial return safety upper limit is the numerical boundary set for the plasticity correction range; the equivalent value of the test relative stress represents the relative stress level corresponding to the test loading; the shear modulus reflects the elastic response of the material to shear deformation; and the safety factor is used to retain a safety margin within the calculation boundary.

[0041] It should be noted that the radial return safety upper limit is calculated according to the current trial loading state, so that the plastic multiplier increment under different material states is constrained by the corresponding boundary.

[0042] In a further implementation, the shear modulus and safety factor of the corresponding material are read, and the radial return safety upper limit is calculated based on the equivalent value of the relative stress of the test. The plastic multiplier increment predicted by the neural network is compared with zero, and the larger value is taken. Then, the obtained value is compared with the radial return safety upper limit, and the smaller value is taken to obtain the plastic multiplier increment after the limit is obtained.

[0043] The above processing ensures that the network prediction results meet the non-negativity requirement and limits the excessive plasticity correction range, providing constrained incremental input for subsequent Chaboche updates and reducing the disturbances caused by abnormal prediction values ​​being directly passed into stress and state variable updates.

[0044] S105. Substitute the limited plastic multiplier increment into the Chaboche update relation to perform plastic correction on the test stress, and update the back stress of each branch and the cumulative equivalent plastic strain to obtain the updated stress of the material integration point and the updated state variables.

[0045] The Chaboche update relation is used to describe the evolution relationship between stress and hardening state under cyclic plastic loading.

[0046] Substitute the limited plastic multiplier increment, the test stress, and the original state variables into the Chaboche update relation to calculate the plastic flow direction and the corresponding stress correction, and complete the update of the back stress and cumulative equivalent plastic strain of each branch; during execution, the material parameters and update relation consistent with the fixed Chaboche material update layer in the training phase are used so that the material calculation relation corresponding to the training constraints is continued to the VUMAT call.

[0047] S106. Return the updated stress and updated state variables to the finite element main program so that the material integration point can be read as the stress and state variables of the previous increment step in the next increment step.

[0048] VUMAT writes the updated stress into the stress output array of the corresponding material integration point, writes the updated back stress of each branch and the cumulative equivalent plastic strain into the state output array, and returns the stress output array and the state output array to the finite element main program.

[0049] When the finite element main program enters the next increment step and calls VUMAT again, it reads the updated stress and updated state variables returned from the previous increment step at the same material integration point, and uses them as the stress and state variables of the previous increment step in this call, respectively. Together with the newly passed strain increment, they form the integration point data, and continue to perform test stress calculation, yield judgment, and corresponding material update processing.

[0050] By writing back and inheriting the updated results, the stress, back stress of each branch, and cumulative equivalent plastic strain at the same material integration point in continuous incremental calculations are kept connected, avoiding the neural network prediction results from acting only on a single increment and becoming disconnected from subsequent material states.

[0051] Based on the above technical solution, this embodiment reduces the repeated iterative solution of the plastic multiplier during cyclic loading by coordinating yield judgment, plastic multiplier increment prediction, safety limit and Chaboche material update, and makes stress correction and internal variable evolution adopt the same plastic increment basis; by inheriting and returning the state variables of the material integration point, the material state connection between adjacent increment steps is maintained, thereby taking into account both the computational efficiency of material update and the stability of continuous cyclic response.

[0052] Example 2: Based on Example 1, this example further refines the neural network training process.

[0053] The neural network training process includes training data reading and preprocessing, plastic multiplier increment prediction, fixed Chaboche material update, training loss calculation, and network parameter update. First, single-step transfer samples between adjacent loading steps are constructed based on the cyclic loading data. The strain increment, stress of the previous loading step, and state variables are used as network inputs, and the neural network outputs the predicted plastic multiplier increment. Then, the predicted plastic multiplier increment and corresponding integration point data are input into the fixed Chaboche material update layer to obtain the predicted stress and predicted state variables. A data loss is constructed based on the predicted plastic multiplier increment, predicted stress, predicted state variables, and corresponding reference results. This loss is combined with yield positive penalty, return mapping consistency loss, underprediction penalty, and hard sample loss to form the training loss, which is then updated through backpropagation to update the neural network parameters.

[0054] According to the loading path number and loading step number, the cyclic loading data obtained from the three-dimensional J2-Chaboche return mapping calculation is read, and adjacent loading step records of the same path are paired; the strain increment between adjacent loading steps, the stress of the previous loading step, the back stress of each branch of the previous loading step, and the cumulative equivalent plastic strain of the previous loading step are extracted to form the input data; the reference plastic multiplier increment of this increment transfer and the reference stress, reference back stress of each branch, and reference cumulative equivalent plastic strain of the next loading step are extracted as the corresponding supervision information; each set of input data and supervision information constitutes a single-step transfer sample, that is, the state record of the same material integration point before and after a strain increment.

[0055] The reference calculation and the fixed Chaboche material update layer use the same material parameters, yield judgment, back stress recovery, plastic flow direction and state update relationship; the reference calculation solves the reference plastic multiplier increment by returning the mapping; the fixed Chaboche material update layer uses the plastic multiplier increment predicted by the neural network and after safety constraints to perform stress and state variable updates.

[0056] This embodiment employs three back stress branches, combining six-component strain increments, six-component previous loading step stresses, eighteen back stress components, and one cumulative equivalent plastic strain into a 31-dimensional input feature. The input features are standardized based on the mean and standard deviation of each input feature from the training set, and the feature order and standardization parameters are saved. The standardized input features are then fed into a neural network for forward computation to obtain the predicted value of the plastic multiplier increment. The neural network uses a multilayer perceptron structure with two hidden layers, each containing 32 neurons.

[0057] The predicted value of the plastic multiplier increment, along with the corresponding unstandardized strain increment, stress of the previous loading step, and state variables, are fed into the fixed Chaboche material update layer. The fixed Chaboche material update layer calculates the test stress and test relative stress and performs yield judgment. For elastic response, the plastic multiplier increment used for material update is set to zero. For plastic response, non-negativity constraints and radial return safety upper limits are applied to the network predicted value.

[0058] Furthermore, by substituting the limited plastic multiplier increment into the Chaboche update relation, we obtain the predicted stress, the predicted back stress of each branch, and the predicted cumulative equivalent plastic strain.

[0059] Among them, the fixed Chaboche material update layer is a computational layer that connects the above elasticity prediction, yield judgment, amplitude limiting and material update calculation. Its material parameters remain unchanged during training and do not contain trainable parameters. The output of this layer participates in the loss calculation, and the loss gradient is passed to the neural network along the computational relationship of this layer.

[0060] For a training batch, let n represent the sample number, B represent the number of samples, and i represent the back stress branch number, with a value ranging from 1 to 3; the plastic multiplier increment actually used for material updating after the predicted plastic multiplier increment value is subject to non-negativity constraints and radial return safety upper limits is denoted as... Based on the predicted values ​​of the plastic multiplier increment and the predicted stress and predicted state variables obtained by fixing the Chaboche material update layer, the following data losses are calculated respectively: , , , , in, For reference, the plastic multiplier increment; and These represent the predicted stress and reference stress corresponding to the next loading step for the nth sample, respectively. and These represent the predicted back stress and reference back stress of the i-th branch corresponding to the n-th sample in the next loading step, respectively. and These represent the predicted cumulative equivalent plastic strain and the reference cumulative equivalent plastic strain corresponding to the next loading step for the nth sample, respectively. , , and The corresponding positive scaling factor, obtained from the training set statistics, remains unchanged during training; the norm square of the tensor bias is calculated according to the sum of squared errors of the corresponding six components.

[0061] The above losses are weighted to form data loss. : , in, , , and These are the corresponding loss weights; in an optional implementation, data loss weights are set. =1.0、 =0.5、 =0.1、 =0.5, each weight is assigned a value before training begins and remains unchanged during training, and does not participate in the parameter update of backpropagation.

[0062] The data loss simultaneously evaluates the plastic multiplier increment bias and the stress and state variable biases generated by the fixed Chaboche material renewal layer.

[0063] Extract the deviatoric stress component of the predicted stress, subtract the sum of the predicted back stresses of each branch, and calculate the updated equivalent relative stress value. The updated yield strength is calculated based on the predicted cumulative equivalent plastic strain, and the difference between the two is taken as the updated yield residual. : , in, To calculate the yield strength using given material parameters, the positive value of the updated yield residual is taken as the basis for subsequent yield positive penalty calculation.

[0064] The reduction coefficient is calculated based on the plastic multiplier increment after the amplitude limit and the dynamic recovery parameters of each branch. The back stress of each branch in the previous loading step is semi-implicitly reduced. The sum of the back stress of each branch after reduction is deducted from the deviatoric stress part of the test stress. The equivalent value of the relative stress before correction is calculated, and the mapping consistency residual is calculated and returned accordingly.

[0065] In a further implementation, the reduction factor corresponding to the i-th back stress branch is denoted as... The equivalent value of the relative stress before correction is denoted as The plasticity correction was calculated based on the shear modulus, the hardening parameters of each branch of motion, and the reduction factor, resulting in: , in, To return the mapping consistency residual, where G is the shear modulus. is the motion hardening parameter of the i-th back stress branch; the returned mapping consistency residual is used to evaluate the deviation between the pre-correction relative stress equivalent value after deducting the plastic correction and the updated yield strength.

[0066] Set a plastic sample mask based on the yield judgment result. Plastic response is set to 1, elastic response to 0; normal stress is used as the scale. The above residuals are normalized to construct the yield positive penalty loss. and return mapping consistency loss Simultaneously, the reference plastic multiplier increment is subtracted from the limited plastic multiplier increment, and the positive portion of the difference is used to construct the underprediction penalty loss. : , , , The physical loss is calculated by weighting the three losses together. , in, , and For the corresponding loss weights; in one optional implementation, the physical loss weights. =5.0、 =5.0、 =5.0.

[0067] It should be noted that the yield positive penalty evaluates whether the updated stress state exceeds the yield surface, returns the plasticity correction relationship corresponding to the plasticity multiplier increment after the mapping consistency loss constraint limit, and the underprediction penalty applies additional constraints to the part of the increment that is lower than the reference plasticity multiplier increment; the above loss is linked through the same actual plasticity multiplier increment used for material update and its update result, so that the increment evaluated by training is consistent with the increment used for material update.

[0068] Based on the yield positive penalty value and the squared value of the returned mapping consistency residual of each training sample, they are sorted from largest to smallest, and yield-difficult samples and consistency-difficult samples are selected according to a preset ratio.

[0069] Among them, the yielding difficulty sample refers to the sample with the highest yield positive penalty value in this batch, and the consistency difficulty sample refers to the sample with the highest squared residual value of the returned mapping consistency in this batch. The two types of samples are selected separately.

[0070] In a further implementation, the normalization error used for sorting and averaging is denoted as: , in, This represents the normalized yield positivity penalty value for the nth training sample. This represents the squared value of the normalized return mapping consistency residual for the nth training sample. This represents the updated yield residual of the nth training sample; the obtained and They are used for sorting and selecting two types of difficult samples and calculating the corresponding loss of difficult samples, respectively.

[0071] In this embodiment, samples with the highest ranking are selected at a ratio of 5% to form a set of indexes for samples with difficulty in yielding. Consistency-difficult sample index set Calculate the mean of the corresponding error: , , in, and These represent the number of samples in the corresponding set; Indicates the sample loss due to yielding difficulty. This represents the sample loss due to consistency difficulties.

[0072] The KKT complementary loss is calculated based on the plastic multiplier increment after amplitude limiting and the updated yield residual. These two quantities are normalized using both the plastic multiplier scaling factor and the stress scaling factor. The normalized plastic multiplier increment is then multiplied by the updated yield residual, the product is squared, and the average is calculated over the training batch to obtain the KKT complementary loss. This is used to constrain the complementary relationship between the plastic multiplier increment and the updated yield residual.

[0073] The two hard sample losses are weighted and included in the total training loss: , in, and For the corresponding loss weights, For KKT complementary loss weights; in an optional implementation, =10.0、 =5.0、 =10.0.

[0074] It should be noted that the difficult samples are selected based on the physical residuals generated by the network's current prediction, and the selection results change with training. All samples still participate in the basic loss calculation, and samples with larger residuals participate in the difficult sample loss calculation. This strengthens the constraints on a small number of samples with large yield violations or poor return mapping consistency, reducing the dilution of their errors by the batch averaging results.

[0075] Each training batch sequentially performs forward computation of the neural network, calculation of the fixed Chaboche material update layer, and calculation of the total training loss. The loss gradient is automatically differentiated and passed to the neural network along the fixed material update relationship to update the network weights and biases, while the material parameters remain unchanged. After each training round, the total training loss is calculated using the validation set; when it is lower than the previously saved result, the corresponding neural network parameters, input normalization parameters, scale factor, and material parameters are saved.

[0076] Example 3: Based on Examples 1 and 2, this example further illustrates the specific procedures for yield determination, radial return safety limit, and Chaboche stress and state variable updates.

[0077] First, read the strain increment at the material integration point, the stress of the previous increment step, and the state variables, and read the corresponding elastic parameters and Chaboche hardening parameters of the material. Based on the elastic relationship of the material, convert the strain increment into an elastic stress increment, add it to the stress of the previous increment step to obtain the test stress; extract the deviatoric stress part of the test stress, subtract the sum of the back stresses of each branch, and obtain the test relative stress; calculate the equivalent value of the test relative stress according to the J2 equivalent stress relationship, and calculate the yield strength based on the cumulative equivalent plastic strain of the previous increment step, and take the difference between the two as the test yield function value.

[0078] Yield strength is calculated based on the isotropic hardening relationship corresponding to cumulative equivalent plastic strain: , in, To include the cumulative equivalent plastic strain in this yield strength calculation, denoted as the initial yield strength, Q is the saturation parameter for isotropic hardening, and b is the parameter controlling the hardening development rate.

[0079] When performing a trial yield test, the cumulative equivalent plastic strain from the previous incremental step is used; when calculating the updated yield residual and the returned mapping consistency residual, the updated cumulative equivalent plastic strain is used.

[0080] The test yield function value is compared with a preset tolerance, which uses the same unit as the stress. The same yield judgment tolerance is used in the training phase and the VUMAT call phase. When the test yield function value is not greater than the preset tolerance, the test stress is output and the back stress of each branch and the cumulative equivalent plastic strain remain unchanged. Otherwise, the integration point data is input into the neural network according to the feature order and standardization method during training to obtain the predicted value of the plastic multiplier increment, and the amplitude limiting and material update are continued.

[0081] The radial return safety upper limit is calculated based on the equivalent value of the relative stress and the shear modulus. A safety factor is used to retain a margin within the corresponding correction scale. The predicted value of the plastic multiplier increment is then restricted to between zero and the safety upper limit. , in, This is the increment of the plastic multiplier after amplitude limiting. The predicted value of the plastic multiplier increment output by the neural network. To test the equivalent value of relative stress, G is the shear modulus. To satisfy 0 < A safety factor of less than 1; This is the upper limit for radial return safety.

[0082] The shear modulus is calculated based on the elastic modulus and Poisson's ratio of the corresponding material. The safety factor is read from the material calculation configuration, and the same value is used for training and VUMAT calls. After limiting, the obtained increment is used to participate in subsequent stress and state variable updates.

[0083] Based on the plastic multiplier increment after the amplitude limit and the dynamic recovery parameters of each branch, the reduction factor of the back stress of each branch is calculated. Specifically, the corresponding dynamic recovery parameter is multiplied by the plastic multiplier increment after the amplitude limit, and the product is multiplied by one and the reciprocal is taken to obtain the reduction factor of that branch. The back stress of each branch before the update is multiplied by the corresponding reduction factor to form the semi-implicitly reduced back stress. The sum of the back stresses of each branch after reduction is subtracted from the deviatoric stress part of the test stress to obtain the relative stress before correction, and its equivalent value is calculated.

[0084] In a further implementation, the relative stress before correction is divided by its equivalent value to form a plastic flow direction tensor that matches the subsequent Chaboche update relationship; the three times shear modulus, the plastic multiplier increment after the limit, and the plastic flow direction tensor are multiplied to obtain the deviatoric stress correction amount; the deviatoric stress correction amount is subtracted from the deviatoric stress portion of the test stress, and then added back to the hydrostatic stress portion of the test stress to obtain the updated stress at the material integration point.

[0085] The back stress of each branch is updated separately. The kinematic hardening parameter of the branch, the plastic multiplier increment after the limit is multiplied by the plastic flow direction tensor, added to the back stress of the branch before the update, and then multiplied by the corresponding reduction factor to obtain the updated branch back stress. At the same time, the plastic multiplier increment after the limit is accumulated to the cumulative equivalent plastic strain before the update to obtain the updated cumulative equivalent plastic strain.

[0086] This embodiment employs three back stress branches, with a material elastic modulus of 200204 MPa and a Poisson's ratio of 0.3. The kinematic hardening parameters of the three branches are 90500 MPa, 6000 MPa, and 350 MPa, respectively, and the corresponding dynamic recovery parameters are 480, 34, and 5, respectively. The material parameters are consistent with those used in the reference calculation and remain unchanged when the Chaboche material update layer is fixed and the VUMAT is invoked.

[0087] It should be noted that the plastic multiplier increment after the amplitude limit first changes the reduction degree of the back stress of each branch, and the reduction result is then used in the calculation of the plastic flow direction. The resulting direction is used for stress correction and back stress update at the same time. As a result, the hardening state change caused by the network prediction increment enters the current stress correction process, so that the stress and state variables are formed according to the same material update relationship, reducing the state inconsistency caused by separate prediction or separate processing.

[0088] After the calculation is completed, the updated stress, the updated back stress of each branch, and the cumulative equivalent plastic strain are used as the output of the current increment of the material integration point. The next increment step reads the above results as the stress and state variables of the previous increment step, and performs yield judgment and material update together with the newly passed strain increment. During the training phase, the above results are used as the predicted stress and predicted state variables.

[0089] Example 4: Based on Examples 1 to 3, this example further illustrates the specific steps for deploying the trained neural network to VUMAT, as well as the conversion of integral point data, return of update results, and state inheritance.

[0090] like Figure 2 As shown, the trained neural network is embedded in the VUMAT material integration point update process. After the finite element main program enters the current increment step, it calls VUMAT; it performs updates point by point according to the material integration point index k in NBLOCK; for the k-th material integration point, VUMAT reads the strain increment of the current increment step from STRAINING, the stress at the end of the previous increment step from STRESSOLD, the state variables at the end of the previous increment step from STATEOLD, and the material property parameters from PROPS, while retaining the plastic multiplier increment of the previous increment step. Yield function state quantity and predicted yield function value Where STRAINING represents the strain increment of this increment step. STRESSOLD represents the stress in the previous increment step. STATEOLD represents the state variables saved in the previous increment step, including the cumulative equivalent plastic strain. and the back stress of each branch , and PROPS represents material parameters; the read data is assembled according to the input feature order of the neural network, and the test stress and test relative stress are calculated to determine the yield.

[0091] In the process of elastic forecasting This represents the increase in elastic stress resulting from the increase in strain during this increment step. Indicates the test stress. This represents the deviatoric stress of the test stress. This represents the equivalent stress obtained based on the test deviatoric stress and back stress state. This represents the trial yield function value calculated from the trial equivalent stress and the current yield condition, and is based on... Determine whether the current material integration point enters the elastic or plastic branch; for the elastic branch, take... = Set the plastic multiplier increment to zero and maintain as well as = Where i represents the back stress branch number; for the plastic branch, the strain increment, the stress of the previous increment step, the cumulative equivalent plastic strain, and the back stress of the three branches are arranged in the order shown in the figure to form a 31-dimensional feature vector, which is then input into the plastic multiplier neural network model to obtain the plastic multiplier increment of the current increment step. After being constrained by nonnegativity and radial return safety limits, the Chaboche state update is performed using the plastic multiplier increment, wherein, Indicates the direction of plastic flow. This indicates the updated deviatoric stress. Indicates the updated stress. This represents the dynamic recovery amount in the update of the i-th back stress branch. This represents the updated back stress of the i-th branch. This represents the updated cumulative equivalent plastic strain. This represents the predicted yield function value calculated based on the updated stress and state variables.

[0092] After the update is complete, the stress is written to `STRESSNEW`, where `STATENEW` is the array of state variables updated in this increment step. Simultaneously, the updated internal energy and inelastic energy are written to `ENERINTERNNEW` and `ENERINELASNEW`, respectively. `ENERINTERNNEW` stores the updated internal energy, and `ENERINELASNEW` stores the updated inelastic energy. After writing back the data, VUMAT returns the update results to the finite element main program, proceeding to subsequent material integration points or the next increment step.

[0093] In a further implementation, the neural network parameters are first read, and the weights, biases, activation function types, input feature order, input standardization parameters, and output scale factors for each layer are derived. Simultaneously, the material parameters used in the fixed Chaboche material update layer are also derived. According to the input and output dimensions of each network layer, the weights and biases are transcribed into Fortran arrays. The input feature order, standardization parameters, output scale factors, and material parameters are written into the include file called by VUMAT and compiled together with the neural network forward computation program and the material update program. All parameters are saved using the same model, ensuring that the network parameters called by VUMAT correspond to the data processing parameters.

[0094] State variable storage locations are assigned to each material integration point to store the tensor components and cumulative equivalent plastic strain of the three back stress branches, and a consistent index correspondence is used when reading and writing. After receiving the strain increment, stress of the previous increment step and state variables from the finite element main program, VUMAT reads the data one by one according to the material integration points, performs yield judgment and corresponding processing; when it is determined to be a plastic response, it assembles the neural network input and performs forward calculation.

[0095] Before assembling the input, the tensor components of the strain increment, the stress of the previous increment step, and the back stress of each branch are converted to the component order used during training; the positions are rearranged according to the physical direction corresponding to each component so that the strain, stress, and back stress components in the same direction correspond to the training feature positions; the shear component of the strain increment adopts the tensor shear representation consistent with the training data; when the shear strain increment passed from VUMAT adopts the tensor shear representation, its value is retained, and the repeated conversion from engineering shear strain to tensor shear strain is no longer performed.

[0096] Following the order of the exported input features, the transformed strain increment, the stress of the previous increment step, the back stress of each branch, and the cumulative equivalent plastic strain are assembled into a 31-dimensional input vector. The mean of the training set is subtracted from each input feature, and then divided by the corresponding standard deviation to obtain the standardized input used for the forward computation of the network. The unstandardized integral point data is retained for subsequent Chaboche material updates.

[0097] In a further implementation, the weights and biases are read sequentially according to the network layers, a linear transformation is performed on the input vector, the derived activation function is used to calculate the output of the layer, and the output of the layer is passed to the next layer.

[0098] In this embodiment, the hidden layer uses the tanh activation function, and the scalar obtained from the output layer is processed by the softplus function and multiplied by the output scale factor to restore the plastic multiplier increment prediction value. The network structure, activation processing and output scale used in the forward computation are consistent with those in the training phase. The output scale factor uses the model-saved value and is not recalculated when VUMAT is called.

[0099] The predicted value of the plastic multiplier increment is fed into the limiting and material update process. VUMAT uses the same material parameters, safety limits and update relationships as the fixed Chaboche material update layer, and uses unstandardized integral point data to complete the stress, back stress of each branch and cumulative equivalent plastic strain update. For elastic response, the test stress and the unchanged state variables are directly used as the output.

[0100] The equivalent value of the updated relative stress is calculated based on the updated stress and the updated back stress of each branch, and the yield strength calculated based on the updated cumulative equivalent plastic strain is subtracted to obtain the updated yield residual. An independent state variable storage location is allocated to this residual, and it is written into the state output array stateNew for checking the yield consistency of the material integration points after calculation. In this embodiment, this residual is used as a monitoring quantity, and the Newton iteration is not restarted inside VUMAT based on it, nor is the global time increment of the finite element main program automatically reduced or divided inside the subroutine.

[0101] After the material update is completed, the updated stress is converted from the component order used during training back to the component order required by the finite element main program and written into the stress output array of the corresponding material integration point; the updated back stress of each branch is converted back to the component order agreed upon in the state variable storage and written into the state output array along with the updated cumulative equivalent plastic strain; the read-in conversion and write-back conversion of the back stress of each branch are inverse processes, so that the state variables read next time still conform to the predetermined storage order.

[0102] It should be noted that the input component transformation, network forward calculation, and Chaboche material update use corresponding data representations and parameters, and the storage order is restored during output, so that the stress and back stress generated in this step still correspond to the same physical direction in the next increment. This process extends the meaning of the data in the training phase to continuous material updates, reducing the propagation of state biases caused by inconsistencies in component order or shear representation.

[0103] After VUMAT completes the writing of the data for the material integration points, it returns the stress output array and state output array to the finite element main program. When the next incremental step is called, it reads the stress and state variables returned from the previous integration point of the same material, and combines them with the newly passed strain increment to form the integration point data, and continues to execute the component conversion, yield judgment and corresponding material update process.

[0104] Example 5: Based on Examples 1 to 4, this example combines... Figures 3 to 6 This illustrates the application of the method in the cyclic loading analysis of representative cubic volume elements.

[0105] A cubic finite element model was established using C3D8R solid elements. An Abaqus / Explicit analysis step was set, and the VUMAT subroutine deployed according to Example 4 was specified. The material parameters followed the settings of Example 3, and the network parameters and input / output processing parameters adopted the configuration saved after training.

[0106] Cyclic displacement boundary conditions were applied to the model to form loading amplitudes of 0.7%, 0.8%, and 0.9%, respectively. The stress distribution and stress-strain response of the corresponding loading process were extracted and compared with the traditional Chaboche return mapping calculation results.

[0107] The finite element main program receives the strain increment at the material integration point, the stress from the previous increment step, and the state variables incrementally. VUMAT then performs yield judgment and corresponding material updates. For the plastic response, the plastic multiplier increment is predicted through a neural network. After a safety limit is applied, the stress, back stress of each branch, and cumulative equivalent plastic strain are updated, and the updated results are returned to the finite element main program for use in the next increment step.

[0108] like Figure 3 As shown, under a 0.9% single-axis cyclic loading amplitude, Figure 3 (a) and Figure 3 (b) The Mises equivalent stress distributions at the same time point are given for the traditional elastoplastic constitutive model and the plastic multiplier neural network model, respectively; both sets of illustrated results show an approximately uniform stress distribution. Figure 3 (a) and Figure 3 The equivalent stresses shown in (b) are approximately 369.7 MPa and 369.0 MPa, respectively, indicating that the equivalent stress levels obtained by the two calculation methods at this loading time are similar.

[0109] Figures 4 to 6 Stress-strain hysteresis curves are presented for loading amplitudes of 0.7%, 0.8%, and 0.9%, respectively; the horizontal axis... The vertical axis represents the normal strain component in the third direction, expressed as a percentage; The values ​​represent the normal stress components in the corresponding directions, in MPa; solid lines represent the results of the traditional elastoplastic constitutive model, and dashed lines represent the results of the plastic multiplier neural network model.

[0110] like Figure 4 As shown, under a loading amplitude of 0.7%, the two sets of curves are generally similar in the initial loading segment and the main section of the hysteresis loop. After reverse loading, corresponding stress change branches are formed. By combining the curves shown in the figure, we can compare the description of the cyclic response by the material update method after the loading direction is changed, rather than being limited to the stress value at a single loading moment.

[0111] like Figure 5 As shown, after the loading amplitude increases to 0.8%, the two sets of curves still show similar hysteresis loop shapes, with similar stress levels at the tensile and compressive ends, but differences in local sections; this figure provides a comparison result at another loading amplitude to observe the stress response formed after the plastic multiplier prediction and material renewal connection.

[0112] like Figure 6As shown, at a loading amplitude of 0.9%, the two sets of curves cover a wider strain range, and remain generally similar in the loading, unloading, and reverse loading sections shown; combined with Figure 3 The stress distribution under the same amplitude can be evaluated from the stress level at a selected time and the hysteresis pattern of the loading process shown in the figure.

[0113] This embodiment uses VUMAT incremental calls to connect the plastic multiplier prediction, safety limit, and Chaboche material update, and then uses the updated state variables for subsequent loading calculations.

[0114] The present invention further provides a computer-readable storage medium storing a computer program, wherein when the computer program is executed by a processor, the processor implements the VUMAT physical information neural network cyclic plastic stress update method described in Embodiments 1 to 5.

[0115] The present invention also provides an electronic device for performing the above-described VUMAT physical information neural network cyclic plastic stress update method.

[0116] The electronic device includes a processor and a memory. The memory stores a computer program, trained neural network parameters, and parameter data required for material updates. The processor reads and executes the computer program to call VUMAT to obtain the integration point data of the material integration points, and completes the cyclic plastic stress update of the material integration points according to the yield judgment, plastic multiplier increment prediction, radial return safety limit, Chaboche stress and state variable update, and update result return process described in Examples 1 to 4.

[0117] The memory is also used to store the input feature sequence, input standardization parameters, output scale factor, and Chaboche material parameters of the neural network. When the processor executes VUMAT, it reads the above data to ensure that the data organization method, material update relationship, and parameters used in the forward calculation of the neural network correspond to those in the training stage. The processor also returns the updated stress and the updated state variables to the finite element main program for subsequent incremental calculations to continue reading.

[0118] The embodiments of the present invention described above are subject to modification and change of method by those skilled in the art without departing from the embodiments and broader aspects of the present invention. The appended claims are intended to include all such modifications and changes of method that do not depart from the present invention.

[0119] Although the embodiments of this application are described above, the content is merely an example adopted for the purpose of facilitating understanding of this application and is not intended to limit the scope and application scenarios of this application. Any person skilled in the art described in this application may make any modifications and changes in the form and details of the implementation without departing from the spirit and scope disclosed in this application, but the scope of patent protection of this application shall still be determined by the scope defined in the appended claims.

Claims

1. A VUMAT physical information neural network-based method for updating cyclic plastic stress, characterized in that, include: The integration point data of the material integration point is obtained through VUMAT. The integration point data includes strain increment, stress of the previous increment step, and state variables. The state variables include back stress of each branch and cumulative equivalent plastic strain. The test stress and test relative stress are calculated based on the integration point data, and yield judgment is made based on the test relative stress and the state variables. When the response is determined to be elastic, the test stress is output and the state variable is kept unchanged. When the response is determined to be plastic, the integral point data is input into the trained neural network to predict the plastic multiplier increment. Based on the equivalent value of the relative stress, shear modulus and safety factor of the test, a radial return safety upper limit is generated. The plastic multiplier increment is then subject to a non-negative limit and the radial return safety upper limit limit to obtain the limited plastic multiplier increment. Substituting the limited plastic multiplier increment into the Chaboche update relation, the test stress is plastically corrected, and the back stress of each branch and the cumulative equivalent plastic strain are updated to obtain the updated stress of the material integration point and the updated state variables. The updated stress and updated state variables are returned to the finite element main program so that the material integration point can be read as the stress and state variables of the previous increment step in the next increment step.

2. The method according to claim 1, characterized in that, The training process of the neural network includes: The strain increment, stress of the previous loading step, back stress of each branch of the previous loading step, cumulative equivalent plastic strain of the previous loading step, and reference plastic multiplier increment, reference stress and reference state variable of the material between adjacent loading steps under cyclic loading conditions are obtained. The neural network is trained by using the strain increment, the stress of the previous loading step, the back stress of each branch of the previous loading step, and the cumulative equivalent plastic strain of the previous loading step as inputs, and the reference plastic multiplier increment of the next loading step as supervision information.

3. The method according to claim 2, characterized in that, The training process of the neural network also includes: The predicted value of the plastic multiplier increment output by the neural network, along with the strain increment of the corresponding loading step, the stress of the previous loading step, and the state variables of the previous loading step, are input into a fixed Chaboche material update layer. The fixed Chaboche material update layer applies the non-negativity constraint and radial return safety upper limit constraint to the predicted value of the plastic multiplier increment, performs plastic correction on the test stress based on the limited plastic multiplier increment, and updates the back stress of each branch and the cumulative equivalent plastic strain to obtain the predicted stress and predicted state variables. A training loss is constructed based on the predicted value of the plastic multiplier increment, the predicted stress, and the predicted state variable, and the parameters of the neural network are updated according to the training loss. The fixed Chaboche material update layer does not contain trainable parameters.

4. The method according to claim 3, characterized in that, The predicted state variables include the predicted back stress of each branch and the predicted cumulative equivalent plastic strain; the training loss includes data loss and physical loss. The predicted values ​​of the plastic multiplier increment, the predicted stress, and the predicted state variables are compared with the corresponding reference results to calculate the data loss. The updated yield residual is calculated based on the predicted stress and the predicted state variables, and a yield positive penalty is applied to the updated yield residual that is greater than zero. The back stress of each branch in the previous loading step is semi-implicitly reduced based on the plastic multiplier increment after the limit, and the relative stress equivalent value is calculated in combination with the test stress. The return mapping consistency residual is calculated based on the relative stress equivalent value, the plastic correction amount and the updated yield strength, and the return mapping consistency loss is calculated based on the return mapping consistency residual. An underprediction penalty is applied to the plastic multiplier increment after the limit is lower than the reference plastic multiplier increment; The physical loss is constructed based on the yielding positive penalty, the return mapping consistency loss, and the underprediction penalty.

5. The method according to claim 4, characterized in that, The training loss also includes a hard sample loss, which is constructed as follows: The training samples are sorted from largest to smallest according to their yield positive penalty value and the square of the returned mapping consistency residual. Then, yield-difficult samples and consistency-difficult samples are selected from the current training batch according to a preset ratio. The average values ​​of the yield positive penalty value of the yield-difficult sample and the squared value of the return mapping consistency residual of the consistency-difficult sample are respectively calculated to obtain the yield-difficult sample loss and the consistency-difficult sample loss. The yielding difficulty sample loss and the consistency difficulty sample loss are weighted and included in the training loss to update the parameters of the neural network.

6. The method according to claim 1, characterized in that, The calculation of the test stress and the relative test stress, and the yield determination based on the relative test stress and the state variables, include: The elastic stress increment is calculated by substituting the strain increment into the elastic relationship of the material, and then added to the stress of the previous increment step to obtain the test stress. Extract the deviatoric stress portion of the test stress, sum the back stresses of each branch to obtain the total back stress, and subtract the total back stress from the deviatoric stress portion to obtain the relative stress of the test stress. Calculate the equivalent value of the relative stress of the test, and calculate the yield strength based on the cumulative equivalent plastic strain. Use the difference between the equivalent value and the yield strength as the test yield function value. When the value of the trial yield function is not greater than the preset tolerance, it is determined to be an elastic response; otherwise, it is determined to be a plastic response.

7. The method according to claim 1, characterized in that, The generation of the radial return safety limit includes: Read the shear modulus of the material corresponding to the material integration point, and obtain a safety factor that is greater than 0 and less than 1; Divide the equivalent value of the relative stress of the test by three times the shear modulus, and multiply the resulting quotient by the safety factor to obtain the radial return safety upper limit; The plastic multiplier increment is compared with zero, and the larger value is taken as the non-negative constraint result. Then, the non-negative constraint result is compared with the radial return safety upper limit, and the smaller value is taken as the plastic multiplier increment after the limit is set, which is then substituted into the Chaboche update relation.

8. The method according to claim 1, characterized in that, Substituting the limited plastic multiplier increment into the Chaboche update relation includes: The reduction coefficient is calculated based on the plastic multiplier increment after the amplitude limit and the dynamic recovery parameters corresponding to each branch. The back stress of each branch is reduced, and the plastic flow direction is calculated based on the deviatoric stress part of the back stress of each branch after reduction and the test stress. The deviatoric stress portion is plastically corrected based on the plastic flow direction and the plastic multiplier increment after the amplitude limitation, and the hydrostatic pressure portion of the test stress is added back to obtain the updated stress. According to the Chaboche update relation, the back stress of each branch is updated based on the reduction factor, the plastic flow direction, and the plastic multiplier increment after the limit, and the plastic multiplier increment after the limit is accumulated to update the cumulative equivalent plastic strain.

9. The method according to claim 3, characterized in that, The trained neural network is deployed to VUMAT in the following manner: Export the weights, biases, activation function types, input feature order, input normalization parameters, output scale factors, and material parameters of each layer of the neural network, and transcribe them into an array or include file that can be called by VUMAT; The integral point data are assembled according to the input feature order, standardized using the input standardization parameters, and forward computation is performed through the weights, biases and activation functions of each layer. The output result is processed by the softplus function and multiplied by the output scale factor to obtain the plastic multiplier increment. VUMAT performs stress and state variable updates using the same material parameters and Chaboche update relationships as the fixed Chaboche material update layer.

10. The method according to claim 9, characterized in that, Before assembling the integration point data, the tensor component order of the strain increment, the stress of the previous increment step, and the back stress of each branch received by VUMAT is converted to the component order used during training, and the shear strain component is made to adopt the tensor shear representation consistent with the training data. After completing the stress and state variable update, the updated stress and the updated back stress of each branch are converted back to the tensor component order required by the finite element main program. The converted updated stress is written into the stress output array, and the converted back stress of each branch and the updated cumulative equivalent plastic strain are written into the state output array. The stress output array and the state output array are returned to the finite element main program for the next incremental step to read the stress and state variables from the previous incremental step.

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