Method and system for predicting mechanical parameters of gradient nano heterogeneous material

Abstract

The invention provides a mechanical parameter prediction method and system for a gradient nano heterogeneous material, and belongs to the technical field of material mechanical property prediction and simulation. The mechanical parameter prediction method comprises the steps that test parameters of the gradient nano heterogeneous material are obtained through an experiment, and a training data set is constructed based on the test parameters; a deep residual network is trained through the training data set, and the deep residual network comprises a jump connection structure; and the trained lightweight model is converted into a format capable of being deployed in finite element software, and the finite element software can call the lightweight model. A jump connection structure is introduced into the deep residual network to execute feature matrix additive operation at the tail end of each module, the deep network gradient dissipation problem is effectively solved, meanwhile, the finite element simulation software is deeply embedded into the trained model, and the finite element simulation efficiency is improved. And the end-to-end high-precision simulation closed-loop calculation flow of the metal gradient material from the micro depth effect to the macro mechanical response is realized.

Description

Technical Field

[0001] This invention belongs to the field of material mechanical property prediction and simulation technology, and in particular relates to a method and system for predicting the mechanical parameters of gradient nanoheterogeneous materials. Background Technology

[0002] Metal gradient nanostructures, a major breakthrough in materials science, have shown revolutionary application prospects in high-end manufacturing in recent years. These materials achieve a breakthrough synergistic effect of strength and toughness by precisely controlling the grain size gradient and dislocation density distribution from the surface to the interior, and have found wide application in key areas such as aerospace engine blades, biomedical implants, and microelectromechanical systems. After surface nanostructuring, the metallic material exhibits a gradient structure from the surface inwards, consisting of nanocrystals, ultrafine grains, dislocation structures, and a coarse-grained core. This gradient structure in characteristic dimensions has a significant impact on its mechanical properties, achieving a good balance between strength and plasticity, breaking the traditional inverse relationship where increased strength comes at the cost of significant loss of plasticity. However, current accurate simulation of gradient materials faces three major technical bottlenecks: First, traditional finite element methods, based on homogenization assumptions, cannot characterize the nonlinear changes in depth-dependent mechanical responses; second, experimental characterization requires complex focused ion beam (FIB) microfabrication techniques and in-situ nanomechanical testing, which can only obtain limited data at discrete depth points, making it difficult to construct continuous gradient models; third, existing commercial simulation software lacks adaptive constitutive engines, resulting in high predicted stress errors under different gradient reinforcement layers when simulating multi-scale coupled deformation, which severely restricts the optimization design and engineering application of gradient materials in precision devices. Summary of the Invention

[0003] To address the aforementioned issues, this invention provides a method for constructing constitutive models using deep learning and deeply integrating them with finite element simulation software. This enables a closed-loop computational flow for end-to-end high-precision simulation of metallic gradient materials, from microscopic depth effects to macroscopic mechanical responses.

[0004] The first aspect of this invention provides a method for predicting the mechanical parameters of a gradient nanoheterogeneous material. The method includes: obtaining test parameters of the gradient nanoheterogeneous material through experiments; constructing a training dataset based on the test parameters; training a deep residual network using the training dataset, wherein the deep residual network includes a skip connection structure; and converting the trained model into a lightweight format that can be deployed in finite element software, enabling the finite element software to call the lightweight model.

[0005] In some implementations, training the deep residual network using the training dataset includes: inputting the training dataset into an input layer, the input layer comprising a fully connected layer and a ReLU activation layer; inputting the data output from the input layer into a residual layer to obtain predicted stress, the residual layer comprising three cascaded residual modules, each residual module comprising a skip connection structure; constructing a loss function based on the actual stress and predicted stress to obtain a loss value; backpropagating the loss value and adjusting the weight parameters of the deep residual network; repeating the above training process until the loss value continuously converges to a range less than a preset threshold.

[0006] In some implementations, the residual network also outputs predicted strain. The step of constructing a loss function based on the actual stress and predicted stress to obtain a loss value includes: calculating an estimated loss based on the difference between the actual stress and the predicted stress; forming a stress-strain graph based on the predicted strain and the predicted stress; constructing a slope loss based on the slope of the stress-strain graph; and constructing a monotonic loss based on the monotonicity of the stress-strain graph, wherein the larger the difference between the slope and a preset slope range, the larger the slope loss; and the lower the monotonicity, the larger the monotonic loss; and determining the estimated loss as a weighted sum of the slope loss and the monotonic loss.

[0007] In some implementations, converting the trained, lightweight model into a format deployable in finite element software and enabling the finite element software to call the lightweight model includes: lightweighting the trained deep residual model to obtain a lightweight model; converting the lightweight model into a platform-independent format; constructing an interface library for compiling and generating a dynamic link library, through which the lightweight model can be called across platforms; constructing a custom material file and associating the custom material file with the finite element software, wherein, in the constructed custom material file, depth and strain rate variables are passed through a material parameter array, cumulative strain is calculated in real time, and the stress response is calculated in real time by calling the lightweight model through the dynamic link library, and a Jacobian matrix is ​​constructed based on the increment of the predicted stress.

[0008] In some implementations, constructing the Jacobian matrix based on the predicted stress increment includes: using numerical differentiation to apply micro-perturbations to the corresponding variables to calculate the stress increment in order to construct the Jacobian matrix.

[0009] In some embodiments, obtaining the test parameters of the gradient nanoheterogeneous material through experiments and constructing a training dataset based on the test parameters includes: preparing and processing a preset material to obtain a focused ion beam micropillar sample; testing the ion beam micropillar sample through an in-situ compression experiment to obtain raw data, the raw data including stress and strain; performing physical constraint-based enhancement processing on the raw data, adding Gaussian noise with a standard deviation of 5% to expand the sample size during the strain hardening stage, and obtaining an amplified stress-strain curve through fitting; and sampling the amplified stress-strain curve at a preset sampling frequency to obtain a standardized training dataset.

[0010] A second aspect of this invention provides a mechanical parameter prediction system for gradient nanoheterogeneous materials. The mechanical parameter prediction system includes: a data construction module for obtaining test parameters of the gradient nanoheterogeneous material through experiments and constructing a training dataset based on the test parameters; a training module for training a deep residual network using the training dataset, wherein the deep residual network includes a skip connection structure; and a deployment module for converting the trained lightweight model into a format deployable in finite element software and enabling the finite element software to call the lightweight model.

[0011] In some implementations, the training module is further configured to input the training dataset into an input layer, the input layer comprising a fully connected layer and a ReLU activation layer; to input the data output from the input layer into a residual layer to obtain predicted stress, the residual layer comprising three cascaded residual modules, the residual modules comprising skip connection structures; and to construct a loss function based on the true stress and predicted stress to obtain a loss value, backpropagate the loss value and adjust the weight parameters of the deep residual network, repeating the above training process until the loss value continuously converges to a range less than a preset threshold.

[0012] In some implementations, the training module is further configured to calculate an estimated loss based on the difference between the actual stress and the predicted stress; to form a stress-strain graph based on the predicted strain and the predicted stress; to construct a slope loss based on the slope of the stress-strain graph; and to construct a monotonic loss based on the monotonicity of the stress-strain graph, wherein the larger the difference between the slope and a preset slope range, the larger the slope loss; and the lower the monotonicity, the larger the monotonic loss. The module is also configured to determine the estimated loss as a weighted sum of the slope loss and the monotonic loss.

[0013] In some implementations, the deployment module is further configured to: lightweight the trained deep residual model to obtain a lightweight model; convert the lightweight model into a platform-independent format; construct an interface library for compiling and generating a dynamic link library, through which the lightweight model can be called across platforms; construct a custom material file, in which depth and strain rate variables are passed through a material parameter array, cumulative strain is calculated in real time, and the stress response is calculated in real time by calling the lightweight model through the dynamic link library, and a Jacobian matrix is ​​constructed based on the increment of the predicted stress; and associate the custom material file with the finite element software.

[0014] By introducing a skip connection structure into the deep residual network to perform feature matrix addition at the end of each module, the gradient dissipation problem in deep networks is effectively solved. At the same time, by embedding the trained model deep into finite element simulation software, a closed-loop computational flow for end-to-end high-precision simulation of metallic gradient materials from microscopic depth effects to macroscopic mechanical responses is realized. Attached Figure Description

[0015] Figure 1 A flowchart illustrating a method for predicting the mechanical parameters of gradient nanoheterogeneous materials provided in an embodiment of the present invention; Figure 2 These are the original data obtained from the micropillar compression experiments by the ORI group and the AGed group, as well as the stress-strain curves obtained after calculation and fitting. The original stress-strain curves (a and b) are shown, the curves after data screening, calculation and fitting (c and d) are shown, and the final training dataset (e and f) is shown. Figure 3 This is a schematic diagram of the residual neural network model structure used for constitutive model training; Figure 4 These are parameter evolution diagrams during the training process of the deep learning models in the ORI and AGed groups, including the evolution of MSE loss (a and b), learning rate optimization (c and d), and model correctness evolution (e and f). Figure 5 These are the fitting plots of predicted and actual stress values ​​for the ORI and AGed groups, the fitting of the data validation set with the predicted values ​​(a and b), and the fitting of the model-predicted stress-strain curves with the actual data points under different conditions (c and d). Figure 6 These are feature analysis plots of the input parameters in the ORI and AGed models, showing feature importance (a and b) and the impact of different parameters on prediction error (c to h). Figure 7 These are predicted stress-strain curves of the ORI and AGed groups under different conditions. Figure 8 This is a flowchart of the process of connecting the predictive constitutive model with Abaqus umat for subroutine development; Figure 9 This is a schematic diagram of the structure of a mechanical parameter prediction system for gradient nanoheterogeneous materials provided in an embodiment of the present invention. Detailed Implementation

[0016] The specific technical features described in the various embodiments in the detailed implementation can be combined in various ways without contradiction. For example, different implementation methods can be formed by combining different specific technical features. In order to avoid unnecessary repetition, the various possible combinations of the specific technical features in this invention will not be described separately.

[0017] It should also be noted that, in order to avoid obscuring the present invention with unnecessary details, only the structures and / or processing steps closely related to the present invention are shown in the accompanying drawings, while other details that are not closely related to the present invention are omitted.

[0018] Additionally, it should be noted that the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. In the following description, the terms "first," "second," etc., are used merely to distinguish different objects and do not indicate any similarity or connection between them. It should be understood that the directional descriptions such as "above," "below," "inside," and "outside" refer to the orientation under normal use conditions.

[0019] In the following specific embodiments, the mechanical parameter prediction method for gradient nanoheterogeneous materials is used for joint simulation based on neural network models and finite element software, thereby simulating the mechanical parameters of gradient nanoheterogeneous materials more accurately. The process of this mechanical parameter prediction method is illustrated below with reference to various embodiments.

[0020] This invention provides a method for predicting the mechanical parameters of gradient nanoheterogeneous materials, such as... Figure 1 As shown, the method for predicting mechanical parameters includes: Step S101: Obtain the test parameters of the gradient nanoheterogeneous material through experiments, and construct a training dataset based on the test parameters.

[0021] This can be understood as obtaining different response data of gradient nanoheterogeneous materials under different loads through experiments. This response data includes stress and strain. By fitting, stress and strain curves can be obtained. By sampling the curves with a preset precision, multiple sets of discretized test parameters can be obtained, thus obtaining a training dataset. The testing process and data processing process are illustrated below.

[0022] The experimental and data processing procedures include: Experimental material preparation: 316L stainless steel base material ORI group with solid solution austenitic matrix was obtained and subjected to aging treatment (773.15K holding for 2h) to obtain material AG group. Then, the same parameters were used to perform surface strengthening treatment on the two groups of materials to prepare a gradient nanostructure strengthening layer of a certain thickness (200μm) on the material surface.

[0023] Sample pretreatment: Electropolishing was performed in a solution of 10 vol% perchloric acid + 90 vol% anhydrous ethanol. Then, <101> / / ND oriented grains were selected under a scanning step of 0.5 μm and an accelerating voltage of 20 kV. Pt cross marks were deposited at the target depth coordinates (20 / 50 / 80 μm) to establish a three-dimensional reference. Focused ion beam (FIB) micropillar sample preparation: First, a 5 μm diameter pre-processed area was formed by annular trenching using a 30 kV / 3 nA beam. Then, the beam current was gradually reduced to 5 kV / 50 pA to refine and shape a micropillar array with a diameter of 2.0 ± 0.05 μm and a height-to-diameter ratio of 3:1. Next, after depositing 0.2 μm wide Pt cross marks at the target depth coordinates (0.5 / 3 / 5 / 15 / 20 / 50 / 100 / 200 / 500 μm), focused ion beam (FIB) micropillar preparation was initiated to prepare a micropillar array along the depth gradient within a single <101> oriented grain. FIB layer scanning monitoring ensures depth positioning accuracy. In-situ compression experiment and data acquisition: The displacement sensor is calibrated with a laser interferometer. Initial contact is established by determining the slope change at the contact point and high frame rate SEM imaging is triggered synchronously. The instantaneous diameter di is obtained based on edge detection of real-time SEM images. Finally, a four-dimensional structured dataset of depth-strain rate-strain-stress is output. Stress-strain data processing: Converting the engineering stress-strain curves from gradient depth and multi-strain rate data obtained from micropillar compression experiments into true stress-true strain curves requires correction for the deformation characteristics of micrometer-scale specimens. This necessitates accurately characterizing instantaneous cross-sectional area changes while correcting for the interference of size effects on deformation. While macroscopic compression assumes a constant volume, micropillar compression exhibits significant size effects. The micropillar diameter-to-height ratio results in a high proportion of free surface area, leading to non-uniform lateral expansion during plastic deformation. Therefore, measured instantaneous cross-sectional area values ​​or geometric correction factors need to be introduced. True stress formula: Where Ainst is the instantaneous area.v Poisson's ratio, (Engineering strain), where A0 is the original cross-sectional area and L0 is the original length; True strain definition: And there is engineering stress =F / A0, Engineering Strain =ΔL / L0, where L is the real-time height. Finally, true stress-strain data are calculated. After obtaining the stress-strain data, the Ramberg-Osgood constitutive model is used for physical mechanism-guided curve fitting during the plastic stage. Where K(d) = Ae - Bd, n(d) = C + Dd, this model introduces a depth-dependent hardening coefficient K(d) and an exponent n(d) (A, B, C, D are fitting parameters), d = By solving LnK and n through logarithmic equations followed by linear regression, the gradual hardening behavior from surface nanocrystals to coarse grains in the core can be quantitatively characterized, overcoming the limitation of traditional single models in describing gradient effects. Then, based on dislocation density evolution theory, the fitting accuracy of the plastic segment is corrected, and parameters are optimized using nonlinear least squares method, ensuring that the coefficient of determination R² between the fitted curve and experimental data is greater than 0.99, thus establishing a four-dimensional mapping database of depth-strain rate-true stress-true strain. To address the limited sample size of the original data, a physical constraint enhancement strategy is implemented—injecting Gaussian noise with a standard deviation of 5% during the strain hardening stage (ensuring no disruption of curve convexity). This proportion originates from the statistical error of repeatability testing in micropillar compression experiments, expanding the sample size by 40% while preserving the true deformation characteristics of the material. Simultaneously, dual-source data are fused, using real experimental discrete points (and the fitted data). Dense sampling points on the curve are used to align strain coordinates via spline interpolation, constructing a hybrid training set to cover the entire strain space. Finally, standardization is performed: input features (depth, strain rate, strain variable) are standardized using StandardScaler (mean zeroed, variance 1) to eliminate the risk of gradient explosion caused by dimensional differences and normalized to the [0,1] interval, accelerating model convergence. Stratified sampling (depth-based weighting) is used to divide the training / validation / test set (8:1:1), forming a standardized training dataset that combines physical realism, statistical completeness, and numerical stability, providing high-quality input for deep learning.

[0024] in, Figure 2Figures a and b show the original stress-strain curves of 316L stainless steel with a gradient nano-reinforced layer on the surface after micropillar compression experiments. Clearly, the cross-sectional area of ​​the specimen changes during the plastic deformation stage. The instantaneous cross-sectional area during compression is larger than the original cross-sectional area. Therefore, for the same load, the true stress is always less than the engineering stress, and the engineering stress overestimates the actual stress level the material bears. In the elastic stage, the curve can almost be considered to accurately describe the deformation behavior. However, once the specimen reaches its yield strength and enters the plastic stage, the stress continues to increase due to the instantaneous increase in cross-section A (due to work hardening), causing the engineering curve to rise continuously, making it impossible to directly determine the instability point. Compression failure usually occurs after the specimen has undergone significant bulging and extremely high strain, thus failing to reveal the constitutive relationship of the material's resistance to plastic deformation at different depths. Therefore, a conversion calculation of the true stress-strain curve was performed, and unreasonable curve portions that might be due to experimental errors were screened out. The screened stress-strain curves are shown below. Figure 2 As shown in c and d; Then, the Ramberg-Osgood constitutive model is used for physics-guided curve fitting to smooth noise. Next, a necking correction algorithm is used to convert engineering values ​​into true stress and strain values, ultimately forming a continuous, smooth curve with deep parameterization characteristics. This results in a standard training dataset, as shown below. Figure 2 As shown in e and f, the core value of this processing lies in: eliminating the interference of experimental systematic errors on the characterization of hardening behavior, ensuring the comparability of curves at different depths, and supplementing the plastic segment data outside the range of the experimental equipment through extrapolation of the constitutive model; it not only preserves the authenticity of key experimental data points, but also provides dense sampling of the entire strain space through fitting curves, and enhances data diversity by adding Gaussian noise with physical constraints (simulating the distribution of experimental errors), forming a standardized dataset that combines mathematical completeness, physical authenticity and deep correlation, providing high-quality input for deep learning models that can accurately reflect the essence of gradient effects.

[0025] Step S102: Obtain the test parameters of the gradient nanoheterogeneous material through experiments, and construct a training dataset based on the test parameters.

[0026] The training dataset is input into a deep residual network for iterative training. Specifically, the loading parameters from the training data are input into the deep residual network to obtain the predicted stress output by the network. The difference between the predicted stress and the actual stress in the training dataset forms the loss value. The weight parameters of the deep residual network are adjusted through backpropagation of the loss value. The deep residual neural network includes skip connection structures to ensure that it maintains a convergence trend. The following section combines... Figure 3 The architecture and training computation process of a deep residual neural network are illustrated by way of example.

[0027] Figure 3 The demonstrated residual network architecture is designed based on deep feature reuse and cross-layer connectivity mechanisms. Its core consists of an input feature embedding layer, a third-order residual learning module, and a linear regression output layer. The input layer first maps the 3D material parameters (depth / strain rate / strain) to a 128-dimensional feature space through a fully connected layer, and then introduces nonlinear transformation capability through ReLU activation. Subsequently, it enters the residual learning backbone, where Block 1 achieves dimensionality upgrade from 128 to 256 through two fully connected layers and batch normalization, Block 2 uses identity mapping to maintain the 256-dimensional feature depth to retain the original information, and Block 3 compresses the features back through dimensionality reduction operations. The network has 128 dimensions, with each residual module embedding a Dropout layer (with a dropout rate of 0.3 / 0.3 / 0.2) to suppress overfitting. The most innovative skip connection structure performs feature matrix addition at the end of each module. When the input and output dimensions match, the identity mapping is directly passed. When the dimensions change, the number of channels is adjusted through 1×1 convolution, effectively solving the gradient vanishing problem in deep networks. Finally, the 128-dimensional feature vector generates stress prediction values ​​through a linear output layer without activation functions, avoiding distortion of physical laws caused by activation function truncation. The entire network achieves cross-scale feature extraction from microscopic dislocation evolution to macroscopic mechanical response through a 42-layer deep structure.

[0028] The training process includes establishing a high-generalization deep residual network mapping function based on the real stress-strain database: fθ: R3→R, (d, ε˙, ε)→σ, where d is the depth, ε˙ is the strain rate, ε is the strain, and σ is the predicted stress value. The hardware acceleration environment is initialized, the cuDNN acceleration library is enabled, gradient checkpointing is used, and torch.backends.cudnn.benchmark=True is set. When constructing the data pipeline, a dynamic batch size (64-256 adaptive) is selected, and the feature tensor and label tensor are transformed. The deep residual network constitutive prediction model is constructed: First, a three-layer residual module architecture is designed—the input layer (3 nodes) is mapped to a 128-dimensional feature space through a fully connected layer, and nonlinear modeling capability is introduced through the ReLU activation function; then, three residual blocks are concatenated (Block1 implements the 128→256 dimensionality upscaling transformation, Block2 maintains the 256-dimensional identity mapping, and Block3 completes the 256→128 dimensionality downscaling), whose structure can be represented as: Using the residual learning module, the structure can be expressed as: Where F is the residual function, G is the skip connection, and G is the unit mapping when dimension matching is performed. The regression output layer is 128-dimensional → linear transformation → stress prediction, which can be expressed as: Each residual block contains a linear layer, a batch normalization layer, and a Dropout regularization layer (with inactivation rates of 0.3 / 0.3 / 0.2). Its core lies in the skip connection structure. It effectively solves the gradient vanishing problem in deep networks, enabling stable convergence even at a network depth of 42 layers; the output layer uses linear regression to project 128-dimensional features onto the stress prediction value, avoiding physical distortion caused by activation function truncation.

[0029] Optionally, physical constraints are introduced when calculating the loss value to further improve training accuracy. Specifically, loss calculation and backpropagation are configured, and the loss function is dominated by the mean squared error, which can be expressed as follows: And add physical constraint enhancement The elastic modulus constraint `Lelastic` forces the predicted curve slope to be between 198 and 218 GPa, while the monotonicity constraint `Lmono` penalizes the non-monotonic behavior of the stress-strain curve. During parameter optimization and updates, the optimizer `AdamW` is configured as follows: The hyperparameters are set to .in, It is the first moment estimate after bias correction, representing the adaptive adjustment of the gradient direction. ResNet alleviates the gradient vanishing problem through skip connections, and the gradient momentum m is responsible for maintaining the consistency of the gradient propagation direction in the deep network. It is the second-order moment estimate after bias correction, representing the adaptive adjustment of the gradient magnitude. In convolutional residual blocks, shallow and deep residual blocks usually require different learning rates, while the adaptive learning rate v is responsible for implementing the layered adaptive learning rate; =0.9 controls the decay rate of the first moment m, which means retaining 90% of the historical momentum and only adding 10% of the current gradient. This setting allows the optimizer to retain more historical gradient information, thereby smoothing the update direction and reducing oscillations. =0.999 controls the decay rate of the second moment v. This means that 99.9% of the historical gradient squared information is retained, and only 0.1% of the current gradient squared is added. This setting makes the optimizer more sensitive to changes in gradient magnitude and can adaptively adjust the learning rate, which is especially helpful for stable convergence in the early stages of training.

[0030] Optionally, to accelerate the training process, the CUDA parallel computing architecture is used to initialize the hardware environment, matrix operations are optimized through tensor cores, and gradient checkpointing is implemented to reduce memory usage. During the data pipeline construction phase, dynamic batch processing (64-256 adaptive) and asynchronous loading are implemented to achieve zero-copy CPU-GPU transfer and eliminate I / O bottlenecks. The loss function is dominated by mean squared error (MSE), whose squared penalty effectively suppresses large prediction biases. Simultaneously, physical constraint enhancement terms are introduced—elastic modulus constraints (forcing the slope of the prediction curve) and monotonicity constraints (penalizing stress decreasing with strain). Adaptive weight coefficients (initial λ=0.1, decaying by 1% per cycle) balance data-driven and physical laws. Parameter optimization uses AdamW... The algorithm (learning rate η=0.001, β1=0.9, β2=0.999) combines Nesterov momentum to accelerate convergence and simultaneously implements gradient pruning (max_norm=1.0) to prevent parameter oscillations. Automatic Mixed Precision (AMP) technology is enabled during training, dynamically converting FP32 to FP16 calculations through a context manager, and using GradScaler to scale the loss value to maintain the numerical stability of small gradients, resulting in a 2.1x increase in training speed and reduced memory usage. A loss-based platform detection schedules the learning dynamic rate, setting the decay coefficient to 0.5, the allowed number of stopping epochs to 15, and the minimum learning rate threshold to 1e-6. This ensures that if the validation loss does not decrease for 15 consecutive epochs, the learning rate decays by 50%, avoiding getting trapped in local minima. An early stopping mechanism monitors the optimal validation loss in real time (training terminates if there is no improvement after 50 consecutive epochs) and saves a .pth file containing network weights, normalizers, and metadata. The training diagnostic system simultaneously deploys a gradient norm histogram (health threshold 10). -3 -10 -5 The dropout rate is automatically increased when the training / validation loss ratio is >1.2, ensuring the model's generalization ability.

[0031] Figure 4(a) and (b) show the MSE loss curves (logarithmic coordinates) for the training and validation sets. Their core value lies in revealing the model's convergence trajectory and overfitting risk—the continuous decrease in training loss while the validation loss plateaus indicates model capacity saturation. Choosing logarithmic coordinates clearly separates the early rapid decline (optimizer activation) from the later fine-tuning (feature learning) stage. The lag inflection point of the validation loss can accurately determine the timing of early stopping. (c) and (d) show the dynamic scheduling process of the learning rate using both linear and logarithmic coordinates. Their core function is to visualize the optimization strategy response mechanism. The linear coordinates show the stepwise decay of the learning rate (triggered by the stagnation of the validation loss), while the logarithmic coordinates highlight the order-of-magnitude change (0.001→1e-6). This design can correlate the learning rate adjustment with the inflection point of the loss curve, proving that dynamic scheduling effectively avoids the local minima trap. (e) and (f) show the evolution of the model's R² score on the training and validation sets during training, verifying the convergence of the model during the fitting process. The entire visualization provides direct evidence for optimizing hyperparameters such as residual network depth (42 layers) and Dropout rate (0.3), proving the effectiveness of adaptive training strategies in gradient material modeling.

[0032] Figure 5 The prediction accuracy and generalization ability of the deep learning constitutive model are verified through multi-level visualization: Figures (a) and (b) show the global prediction performance of the validation set. The stress value scatter plot is densely distributed along the diagonal of y=x, which intuitively reflects the model's generalization ability on unknown data. Selecting this figure can quantify the model bias distribution characteristics (such as the error concentration in the high stress zone) and verify the residual network's ability to capture the cross-scale mechanical response of gradient materials. Figures c / d focus on specific depth-strain rate combinations. By superimposing the model prediction curve and the experimental discrete data points, the local fitting accuracy of the model under extreme conditions is revealed. In particular, the prediction ability of the model for typical deformation characteristics of 316L stainless steel is demonstrated, including early yielding of surface nanocrystals, stress drop induced by twinning in the transition zone, continuous hardening of deep coarse grains, and adiabatic shear response under high strain rate. The positional deviation (MAE<15MPa), curve shape consistency (R²>0.99) and feature point capture ability of the predicted curve and experimental points are compared.

[0033] Figure 6 Figures (a) and (b) use horizontal bar charts generated by the importance ranking algorithm to intuitively quantify the contribution weights of the three features—depth, strain rate, and strain—to the model's prediction accuracy (calculated based on the R² decay magnitude after randomizing the features), verifying the dominant role of the depth parameter in the constitutive relation of gradient materials (physical meaning matching test). Figures (c) to (h) focus on different dimensions, discretizing the test set samples according to depth values ​​(horizontal axis) and plotting the corresponding absolute prediction error distribution (vertical axis) to detect systematic biases of the model in key gradient layers.

[0034] For specific predictions of stress-strain curves under different conditions, please refer to... Figure 7 , Figure 7 Stress-strain curves were plotted at different depths (1 / 3 / 10 / 30 / 60 / 120 / 240 μm) and different strain rates (0.1 / 1.0 s⁻¹) to verify the model's ability to capture key mechanical behaviors of gradient materials. The relationship between the horizontal axis strain and the vertical axis stress values ​​intuitively demonstrates the model's predictive performance in terms of the slope of the elastic segment, the position of the yield plateau (and the hardening trend of the plastic segment). In particular, the curve separation analysis in the strain rate abrupt change range can quantitatively evaluate the model's learning effect on strain rate sensitivity. The parallel offset characteristics of the three curves at the same depth reveal the mechanical behavior transformation from the dislocation slip-dominated region (shallow layer) to the twin-dominated region (deep layer) in the gradient nanostructure.

[0035] In some other embodiments, the trained model is manually reviewed after training to further ensure its accuracy. Specifically, after the deep learning constitutive model is trained, a four-level multi-dimensional evaluation system is implemented: First, basic quantitative indicators are calculated—the mean squared error (MSE) of the test set reflects the overall prediction bias (target value < 50 MPa²), the mean absolute error (MAE) characterizes typical prediction fluctuations, and the coefficient of determination (R²) evaluates the model's explanatory power (threshold > 0.98); second, de-standardization is performed, converting the normalized stress values ​​output by the model through a pre-stored standard scaler. Returning to the engineering unit (MPa), ensuring the evaluation indicators have physical meaning and comparability; simultaneously performing visualization analysis: generating actual-predicted stress scatter plots, overlaying a y=x reference line to verify data distribution consistency, and calculating the linear fitting slope and intercept; using the Seaborn library to add kernel density estimation cloud maps to reveal error concentration areas; key physical constraint verification steps: automatically scanning the stress-strain curves of all test samples, applying a monotonicity detection algorithm, and calculating the slope of the elastic segment to verify that the modulus value is within the reasonable range of 198-218 GPa; extreme working condition pressure testing with additional high strain rate (10s) -1 Adversarial samples in deep regions (100μm) were used to monitor model robustness (MAE increase <20%).

[0036] Manual review can be performed using the generated visual diagnostic charts. The process of visualizing training results and constructing a triple diagnostic view includes: The first view plots the training-validation loss curve, using semi-logarithmic coordinates to highlight late-stage convergence details, adding learning rate adjustment markers (marking the decay position when ReduceLROnPlateau is triggered), and simultaneously displaying the loss value and dynamic learning rate changes through a dual Y-axis design; The second view performs feature importance analysis—based on a permutation importance algorithm: the input features (depth, strain rate, and strain) are randomly shuffled 5 times each, the R² decay magnitude is calculated, and a horizontal bar chart is generated to quantify the contribution of each feature (depth parameter is usually >50%); The third view deploys gradient distribution monitoring, calculating the L2 norm of the gradient of each layer at the end of each training cycle using torch.norm, plotting a histogram (bins=50), and marking abnormal layers (gradient vanishing layers: norm <10). -6 (Explosion layer: norm > 1.0); Innovative introduction of a mechanical response comparison subplot: Selecting test samples at typical depths (5 / 15 / 50 / 200μm), and overlaying experimental curves, model prediction curves, and traditional JC model curves to highlight the ability of deep learning to capture hardening inflection points. The visualization results are saved in vector format (PDF / EPS).

[0037] In some other embodiments, the trained model is saved using a composite persistence strategy: network weights, input feature normalizers, output stress normalizers, and feature dimension metadata are encapsulated into a single .pth file using torch.save to achieve full-element freezing of the training environment; to support industrial simulation integration, the ONNX format model is further exported, specifying the input node names and output nodes, and enabling dynamic axes to support batch inference.

[0038] Step S103: Convert the lightweight model after training into a format that can be deployed in finite element software, and enable the finite element software to call the lightweight model.

[0039] The lightweighting of the trained model can be achieved in any way. For example, the original model can be used as a teacher model through knowledge distillation, and a lightweight student model can be obtained through knowledge distillation. Alternatively, the volume can be compressed by using TensorRT's FP32→INT8 quantization, and the accuracy loss can be verified to be <0.5% using the polygraphy tool. During the saving process, a configuration file is generated simultaneously to record the training hyperparameters (learning rate, batch size), evaluation metrics, and Git commit hash values ​​to ensure the reproducibility of the experiment.

[0040] To enable cross-platform model calls, lightweight models are converted into platform-independent formats. Specifically, after training and evaluating the deep learning constitutive model, the PyTorch model is converted into a C++ deployable format using TorchScript (torch.jit.trace traces the computation graph and saves it as stress_model.pt), achieving cross-platform serialization to eliminate Python environment dependencies. A C++ interface library is then built: the LibTorch framework is used to load the serialized model (torch::jit::load), the prediction function predict_stress is encapsulated (input depth / strain rate / strain variable, output stress value), the array passed from Fortran is converted to a tensor using torch::from_blob, inference is performed using model.forward and returns a double-type scalar, and a specially designed init_model global initialization function ensures thread safety. The compilation stage is optimized for different platforms: on Windows, Visual Studio is configured to link LibTorch's torch.lib to generate stress_predictor.dll; on Linux, libstress_predictor.so is compiled using g++ -shared -fPIC, with C++14 standard and ABI compatibility settings enabled.

[0041] Then, by constructing a custom material file (UMAT file) and associating the custom material file with the finite element simulation software, the trained model can be embedded into the finite element simulation software. The finite element simulation software can be any software with finite element simulation function, such as Abaqus or Hyperworks. The following uses Abaqus as the finite element simulation software as an example for explanation. Specifically, the core of the subroutine integration of the UMAT file lies in the Fortran-C mixed programming technology: declare EXTERNAL PREDICT_STRESS in umat_stress.f, pass the depth and strain rate parameters through the PROPS array (PROPS(1)=70.0 corresponds to a depth of 70μm), calculate the cumulative strain STRAIN=STRAN(1)+DSTRAN(1) in real time, and call the C++ function to obtain the stress value STRESS(1)=PREDICT_STRESS(...); to calculate the Jacobian matrix, implement a dual numerical differentiation strategy: first initialize the zero matrix DDSDDE, then apply a micro-perturbation to the corresponding variable (EPS=1.0E-6), calculate the stress increment DS and derive The tangent modulus DDSDDE(1,1) = DS / EPS ensures the convergence of the Newton iteration. In the Abaqus configuration, the material card UserMaterial,constants=2 declares the parameter dimensions, and UserSubroutine specifies the compiled UMAT target file. A three-level verification mechanism is adopted during deployment: an independent C++ test program verifies the correctness of model loading (outputs example predicted values), a Fortran-C interface test program verifies data type matching, and the UMAT debug output file (WRITE(7,*)) records the stress state at the integration points in real time. In terms of performance optimization, four acceleration technologies are introduced: model quantization, batch prediction (modifying the C++ interface to support multiple input processing of predict_stresses), OpenMP parallelization (#pragmaompparallelfor accelerates multi-integral-point calculation), and LRU caching mechanism (storing the most recent 100 sets of inputs and outputs to reduce redundant calculations). Advanced application extensions include automatic differentiation with output.backward() to accurately calculate dstress_dstrain, replacing numerical differentiation to improve the accuracy of the constitutive matrix.

[0042] Figure 8The workflow demonstrated here illustrates a method for deep integration of a predictive constitutive model with Abaqus UMAT. First, the trained PyTorch model is serialized into a platform-independent format using TorchScript. Then, a C++ interface library is built to implement model loading and stress prediction functions. This library is compiled into a dynamic link library for cross-platform use. In the UMAT subroutine, an external prediction function is declared and depth and strain rate variables are passed through a material parameter array. The cumulative strain is calculated in real time, and the stress response is obtained by calling the C++ prediction interface. Simultaneously, numerical differentiation is used to apply micro-perturbations to the corresponding variables to calculate the stress increment and construct the Jacobian matrix. The user-defined material parameter dimensions are configured in the Abaqus material definition and associated with the compiled subroutine file. This ultimately forms a closed-loop workflow from deep learning prediction to finite element solution. The entire process uses a three-level verification system to ensure integration reliability and combines model quantization and parallel computing technologies to optimize industrial simulation efficiency.

[0043] After analyzing the experimental data and model training results, the accuracy of the predictive constitutive model for gradient nanostructured 316L stainless steel under deep learning was effectively verified. Fitting micropillar compression experimental data eliminated systematic errors and constructed a deeply parameterized real stress-strain dataset. Through deep feature reuse, cross-layer identity connections, and an inactive output layer, a cross-scale mapping from microscopic dislocation evolution to macroscopic mechanical response was achieved. Prediction verification showed that the model accurately captured the yield characteristics of gradient materials at both global and local scales. Finally, through a three-level compilation and integration of TorchScript, C++, and UMAT, real-time prediction of stress response and numerical differentiation of the Jacobian matrix were achieved, constructing a closed-loop workflow from deep learning to industrial simulation. This research addresses the limitations of experimental errors and strain range in traditional gradient material constitutive characterization methods. Based on deep learning, it achieves end-to-end learning of cross-scale mechanical responses, while the seamless UMAT integration scheme translates prediction accuracy into practical industrial simulation results. This provides a high-precision numerical simulation tool for the design of gradient structural components in aerospace, nuclear power, and other fields, significantly reducing experimental costs and improving reliability prediction capabilities.

[0044] This invention also provides a system for predicting the mechanical parameters of gradient nanoheterogeneous materials, such as... Figure 9 As shown, the mechanical parameter prediction system includes a data construction module 100, a training module 200, and a deployment module 300. The data construction module 100 is used to obtain test parameters of gradient nanomaterials through experiments and construct a training dataset based on these parameters. The training module 200 is used to train a deep residual network using the training dataset, wherein the deep residual network includes a skip connection structure. The deployment module 300 is used to convert the trained, lightweight model into a format deployable in finite element software, enabling the finite element software to call the lightweight model.

[0045] In some embodiments, such as Figure 9 As shown, the training module 100 is also used to input the training dataset into the input layer, which includes a fully connected layer and a ReLU activation layer; it is also used to input the data output from the input layer into the residual layer to obtain the predicted stress, the residual layer includes three cascaded residual modules, and the residual modules include a skip connection structure; it is also used to construct a loss function based on the true stress and the predicted stress and obtain the loss value, backpropagate the loss value and adjust the weight parameters of the deep residual network, and repeat the above training process until the loss value continuously converges to a range less than a preset threshold.

[0046] In some embodiments, such as Figure 9 As shown, the training module 100 is also used to calculate the estimated loss based on the difference between the actual stress and the predicted stress; it is also used to form a stress-strain relationship diagram based on the predicted strain and the predicted stress, construct a slope loss based on the slope of the stress-strain relationship diagram, and construct a monotonic loss based on the monotonicity of the stress-strain relationship diagram, wherein the larger the difference between the slope and the preset slope range, the larger the slope loss, and the lower the monotonicity, the larger the monotonic loss; it is also used to determine the loss value by weighting the estimated loss, slope loss and monotonic loss.

[0047] In some embodiments, such as Figure 9 As shown, the deployment module 300 is also used to lightweight the trained deep residual model to obtain a lightweight model, and to convert the lightweight model into a platform-independent format; it is also used to build an interface library, which is used to compile and generate a dynamic link library, through which the lightweight model can be called across platforms; it is also used to build a custom material file, in which depth and strain rate variables are passed through a material parameter array, the cumulative strain is calculated in real time, and the stress response is calculated in real time by calling the lightweight model through the dynamic link library, and the Jacobian matrix is ​​constructed based on the increment of the predicted stress; it is also used to associate the custom material file with the finite element software.

[0048] The above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for predicting the mechanical parameters of gradient nanoheterogeneous materials, characterized in that, The method for predicting mechanical parameters includes: Test parameters of the gradient nanoheterogeneous material were obtained through experiments, and a training dataset was constructed based on the test parameters. The deep residual network is trained using the training dataset, wherein the deep residual network includes a skip connection structure; The trained and lightweight model is then converted into a format that can be deployed in finite element software, enabling the finite element software to call the lightweight model.

2. The mechanical parameter prediction method according to claim 1, characterized in that, The step of training the deep residual network using the training dataset includes: The training dataset is input into the input layer, which includes a fully connected layer and a ReLU activation layer. The data output from the input layer is input into the residual layer to obtain the predicted stress. The residual layer includes three residual modules connected in series, and each residual module includes a skip connection structure. A loss function is constructed based on the actual stress and the predicted stress to obtain the loss value. The loss value is then backpropagated and the weight parameters of the deep residual network are adjusted. The above training process is repeated until the loss value continuously converges to a range less than a preset threshold.

3. The mechanical parameter prediction method according to claim 2, characterized in that, The residual network also outputs predicted strain, and the process of constructing a loss function based on the actual stress and predicted stress to obtain the loss value includes: The estimated loss is calculated based on the difference between the actual stress and the predicted stress; A stress-strain relationship diagram is formed based on the predicted strain and the predicted stress. A slope loss is constructed based on the slope of the stress-strain relationship diagram. A monotonicity loss is constructed based on the monotonicity of the stress-strain relationship diagram. The larger the difference between the slope and the preset slope range, the larger the slope loss. The lower the monotonicity, the larger the monotonicity loss. The weighted sum of the estimated loss, the slope loss, and the monotonic loss is determined as the loss value.

4. The mechanical parameter prediction method according to claim 1, characterized in that, The step of converting the trained, lightweight model into a format deployable in finite element software, and enabling the finite element software to call the lightweight model, includes: The trained deep residual model is lightened to obtain a lightweight model, and the lightweight model is converted into a platform-independent format. An interface library is constructed, which is used to compile and generate a dynamic link library, enabling cross-platform calls to the lightweight model. A custom material file is constructed and associated with the finite element software. In the constructed custom material file, depth and strain rate variables are passed through a material parameter array. The cumulative strain is calculated in real time, and the stress response is calculated in real time by calling the lightweight model through the dynamic library. The Jacobian matrix is ​​constructed based on the increment of the predicted stress.

5. The mechanical parameter prediction method according to claim 4, characterized in that, The step of constructing the Jacobian matrix based on the predicted stress increment includes: using numerical differentiation to apply micro-perturbations to the corresponding variables to calculate the stress increment in order to construct the Jacobian matrix.

6. The mechanical parameter prediction method according to claim 1, characterized in that, The step of obtaining the test parameters of the gradient nanoheterogeneous material through experiments and constructing a training dataset based on the test parameters includes: A focused ion beam micropillar sample was obtained by preparing and processing the preset material. The original data, including stress and strain, were obtained by testing the ion beam micropillar sample through in-situ compression experiments. The original data is augmented based on physical constraints. Gaussian noise with a standard deviation of 5% is added during the strain hardening stage to expand the sample size. The amplified stress-strain curve is obtained by fitting. A standardized training dataset is obtained by sampling the amplified stress-strain curves at a preset sampling frequency.

7. A mechanical parameter prediction system for gradient nanoheterogeneous materials, characterized in that, The mechanical parameter prediction system includes: The data construction module is used to obtain the test parameters of the gradient nanoheterogeneous material through experiments and to construct a training dataset based on the test parameters. A training module is used to train a deep residual network using the training dataset, wherein the deep residual network includes a skip connection structure; The deployment module is used to convert the lightweight model after training into a format that can be deployed in finite element software, and to enable the finite element software to call the lightweight model.

8. The mechanical parameter prediction system according to claim 7, characterized in that, The training module is further configured to input the training dataset into the input layer, which includes a fully connected layer and a ReLU activation layer; to input the data output from the input layer into the residual layer to obtain the predicted stress, the residual layer including three cascaded residual modules, each including a skip connection structure; and to construct a loss function based on the true stress and the predicted stress to obtain a loss value, backpropagate the loss value and adjust the weight parameters of the deep residual network, repeating the above training process until the loss value continuously converges to a range less than a preset threshold.

9. The mechanical parameter prediction system according to claim 8, characterized in that, The training module is further configured to calculate an estimated loss based on the difference between the actual stress and the predicted stress; to form a stress-strain graph based on the predicted strain and the predicted stress; to construct a slope loss based on the slope of the stress-strain graph; and to construct a monotonic loss based on the monotonicity of the stress-strain graph, wherein the larger the difference between the slope and a preset slope range, the larger the slope loss; and the lower the monotonicity, the larger the monotonic loss. The module is also configured to determine the estimated loss as a weighted sum of the slope loss and the monotonic loss.

10. The mechanical parameter prediction system according to claim 7, characterized in that, The deployment module is also used to lightweight the trained deep residual model to obtain a lightweight model, and convert the lightweight model into a platform-independent format; it is also used to build an interface library, which is used to compile and generate a dynamic link library, through which the lightweight model can be called across platforms; it is also used to build a custom material file, in which depth and strain rate variables are passed through a material parameter array, the cumulative strain is calculated in real time, and the stress response is calculated in real time by calling the lightweight model through the dynamic link library, and a Jacobian matrix is ​​constructed based on the increment of the predicted stress; It is also used to associate the material custom file with finite element software.

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