基于GNN/KAN的高应变速率金属材料本构关系的表征方法

袁基宸; 黄夏旭; 解国良

doi:10.11883/bzycj-2025-0103

基于GNN/KAN的高应变速率金属材料本构关系的表征方法

doi: 10.11883/bzycj-2025-0103

1.
北京科技大学机械工程学院，北京 100083
2.
北京科技大学新金属材料全国重点实验室，北京 100083

基金项目: 国家重点研发计划（2021YFB3700700）

详细信息

作者简介:
袁基宸（2002－　），男，硕士，M202320731@xs.ustb.edu.cn

通讯作者:
黄夏旭（1985－　），男，博士，副教授，huangxx@ustb.edu.cn

中图分类号: O347.3
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- 被引次数: 0
出版历程
- 收稿日期: 2025-04-02
- 修回日期: 2025-07-01
- 网络出版日期: 2025-07-01

Characterization method of material constitutive relationship at high strain rates based on GNN/KAN

1.
School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China
2.
National Key Laboratory of New Metal Materials, University of Science and Technology Beijing, Beijing 100083, China

摘要

摘要: 为准确表征金属材料在高应变速率下的应力-应变本构关系，提出了基于图神经网络（graph neural networks，GNN）和KAN（Kolmogorov-Arnold networks）的本构关系的高精度预测模型。为解决传统Johnson-Cook（JC）模型不考虑温度、应变速率与应变之间的耦合效应问题，在GNN模型中构建图结构数据以描述多维参数的非线性关联，在KAN模型中基于Kolmogorov-Arnold定理实现高维输入空间的非线性映射。基于ODS（oxide dispersion strengthened）铜合金的高应变率压缩实验，评估了GNN、KAN和JC的本构关系描述和预测精度。结果表明：GNN与KAN模型在测试集中的平均相对误差分别为8.0%与9.0%，决定系数均高于0.95，显著优于JC模型（平均相对误差为38.0%，决定系数为0.75）；将所构建的本构关系模型应用在有限元仿真中，GNN和KAN模型预测的等效塑性应变与应力分布更符合理论特征，而JC模型无法准确描述材料的软化阶段，仿真结果偏差较大。所构建的模型能有效捕捉高应变速率下材料的多场耦合特性，为极端载荷条件下的应力-应变本构关系提供了新的预测方法。
- 深度学习 /
- 高应变速率 /
- 本构关系 /
- 图神经网络 /
- KAN网络 /
- 动态力学性能预测
Abstract: To accurately characterize the stress-strain constitutive relationship of metal materials under high strain-rate conditions, a novel, high-precision constitutive-relationship-prediction model based on Graph Neural Networks (GNNs) and Kolmogorov-Arnold Networks (KANs) was developed. Traditional Johnson-Cook (JC) models often fail to account for the coupling effects among temperature, strain rate, and strain, all of which are crucial for describing the dynamic behavior of materials under extreme conditions. This limitation was addressed by constructing graph-structured data in the GNN model to capture the nonlinear correlations of multidimensional parameters and by leveraging the Kolmogorov-Arnold theorem in the KAN model to achieve precise mapping of high-dimensional input spaces. The research methodology involved several key steps. Experimental data from ODS copper subjected to high-strain-rate compression were collected using a split Hopkinson pressure bar (SHPB) system and subsequently preprocessed. The dataset included temperature, strain rate, strain, and stress. In the GNN model, when temperature and strain rate were held constant, nodes were connected sequentially based on strain values to form edges. When temperature was held constant, a reasonable threshold was established between nodes with adjacent strain rates, and nodes within this threshold were connected to form edges. The GNN employed a Message Passing Neural Network (MPNN) architecture to learn and predict material properties. Model parameters were optimized using the Adam optimizer, with the Root Mean Squared Error (RMSE) serving as the loss function. The KAN model was constructed based on the Kolmogorov-Arnold representation theorem and consisted of multiple KAN-Linear layers. Each KAN-Linear unit included base weights and spline weights. Base weights handled linear relationships through traditional linear transformations, while spline weights managed nonlinear mappings via B-spline interpolation. Both models were trained on the preprocessed dataset, and their performance was evaluated using metrics such as the Mean Relative Error (MRE), Root Mean Squared Error (RMSE), and the coefficient of determination (R²). The GNN model achieved an average MRE of 9.2% with an R² value exceeding 0.95, while the KAN model recorded an MRE of 9.1% with a similar R² value. Both models significantly outperformed the JC model, which had an MRE of 38% and an R² value of 0.75. Furthermore, the predictive capabilities of the GNN and KAN models were validated through finite element simulations. The simulation results demonstrated that the stress-strain distributions predicted by the GNN and KAN models were more consistent with theoretical expectations compared to those predicted by the JC model, particularly in capturing the material's softening phase. The findings highlight the potential of integrating advanced machine - learning techniques, such as GNNs and KANs, into the field of materials science to enhance the accuracy and efficiency of constitutive modeling. These models offer a promising alternative to traditional empirical models and hold significant implications for engineering applications in aerospace, automotive, and other industries where materials are subjected to high strain rates.
- deep learning /
- high strain rate /
- constitutive relation /
- graph neural network /
- KAN /
- prediction of dynamic mechanical properties

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图 1 研究思路

Figure 1. Research methodology

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图 2 ODS-铜合金的动态力学性能原始数据

Figure 2. Raw data of dynamic mechanical properties of ODS-copper alloys

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图 3 GNN模型结构图

Figure 3. Structure of the GNN model

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图 4 KAN模型结构图

Figure 4. Structural diagram of the KAN model

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图 5 超参数对σ_RMSE的影响

Figure 5. Effect of hyperparameters on σ_RMSE

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图 6 GNN、KAN模型与JC模型的预测精度对比

Figure 6. Comparison of the prediction accuracy of GNN and KAN models with JC models

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图 7 不同工况下JC模型、GNN模型、KAN模型的预测值与真实值对比

Figure 7. Comparison of the predicted values of the JC model, GNN model, and KAN model under different working conditions

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图 8 试样墩粗变形结果与仿真变形结果

Figure 8. The rough deformation results of the specimen pier and the simulation deformation results

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图 9 仿真应变云图

Figure 9. Comparison of the strain diagrams of the three models

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图 10 仿真应力云图

Figure 10. Comparison of stress diagrams of the three models

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表 1 模型预测结果的精度对比

Table 1. Comparison of the prediction accuracy of different models

工况	σ_MRE			R²
工况	GNN/%	KAN/%	JC/%	GNN	KAN	JC
20 ℃/4 000 s⁻¹	6.5	4.0	36.0	0.9927	0.9972	0.9073
20 ℃/4 500 s⁻¹	1.5	8.3	24.2	0.9993	0.9785	0.8978
400 ℃/6 500 s⁻¹	3.0	4.8	38.1	0.9970	0.9929	0.7975
400 ℃/7 000 s⁻¹	9.0	9.2	42.7	0.9896	0.9980	0.8590

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