面向有限元法的图神经网络形函数近似方法

doi:10.11996/JG.j.2095-302X.2025061161

图学学报 ›› 2025, Vol. 46 ›› Issue (6): 1161-1171.DOI: 10.11996/JG.j.2095-302X.2025061161

• 制造产品核心工业软件 • 上一篇下一篇

面向有限元法的图神经网络形函数近似方法

琚晨¹^,²(), 丁嘉欣¹^,², 王泽兴³, 李广钊¹^,², 管振祥⁴, 张常有¹()

¹ 中国科学院软件研究所，北京 100190
² 中国科学院大学计算机科学与技术学院，北京 100190
³ 北京国家新能源汽车技术创新中心有限公司，北京 102600
⁴ 中铁十九局集团有限公司，北京 100176

收稿日期:2025-09-18 接受日期:2025-10-29 出版日期:2025-12-30 发布日期:2025-12-27
通讯作者:张常有(1970-)，男，研究员，博士。主要研究方向为工业仿真软件和并行计算等。E-mail：changyou@iscas.ac.cn
第一作者:琚晨(2001-)，男，博士研究生。主要研究方向为系统仿真与深度学习。E-mail：juchen23@mails.ucas.ac.cn
基金资助:
国家重点研发计划(2023YFB3611303);中华人民共和国水利部重大项目(SKS-2022104)

Graph neural network-based method for approximating finite element shape functions

JU Chen¹^,²(), DING Jiaxin¹^,², WANG Zexing³, LI Guangzhao¹^,², GUAN Zhenxiang⁴, ZHANG Changyou¹()

¹ Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
² School of Computer Science and Technology, University of Chinese Academy of Sciences, Beijing 100190, China
³ Beijing National New Energy Vehicle Technology Innovation Center Co., Ltd., Beijing 102600, China
⁴ China Railway 19 Bureau Group Co., Ltd., Beijing 100176, China

Received:2025-09-18 Accepted:2025-10-29 Published:2025-12-30 Online:2025-12-27
First author：JU Chen (2001-)，PhD candidate. His main research interests cover system simulation and deep learning. E-mail：juchen23@mails.ucas.ac.cn
Supported by:
National Key Research and Development Program of China(2023YFB3611303);Major Project of the Ministry of Water Resources of the People’s Republic of China(SKS-2022104)

摘要/Abstract

摘要：

有限元(FE)计算中形函数的高效与准确评估对单元刚度组装与全局求解至关重要。本文提出了一种面向变节点数单元类型输入的图神经网络形函数近似框架。首先，通过可学习的类型嵌入将离散单元类别映射为连续向量，以支持跨类型参数共享；其次，节点编码器处理几何与目标点信息，图卷积在局部邻域内传播几何约束并结合全局池化获得单元上下文信息；最后，解码器在节点与单元级特征基础上输出形函数值并引入节点均方误差(MSE)与物理约束相结合损失函数，保证预测结果满足形函数约束条件。在模拟线性四面体单元与二次四面体单元数据集上的实验表明，模型在测试集上MSE约为0.001 8，在全局范围内能够较准确地计算形函数值，并在大规模计算环境下，神经网络推理在吞吐率上约为插值方法的3倍，为用学习方法替代或加速形函数评估提供了新的实现路径。

关键词: 图神经网络, 有限元, 形函数, CAE, 深度学习

Abstract:

Efficient and accurate evaluation of shape functions is critical for element stiffness assembly and the global solution in finite element (FE) computations. A general framework was proposed for approximating shape functions was proposed to handle multiple element types and variable node counts. First, learnable type embeddings were used to map discrete element classes to continuous vectors, enabling parameter sharing across element types. Second, geometric information and query-point features were processed by a node encoder; graph convolutions were used to propagated local geometric constraints, while global pooling was applied provided element-level context. Finally, shape function values were produced by a decoder from node- and element-level features, and the model was trained with a loss combining mean square error (MSE) and a physics-inspired sum constraint to enforce shape-function properties. Experiments on synthetic datasets of linear 4-node tetrahedra and quadratic 10-node showed demonstrated that the model achieved a test MSE of approximately 0.001 8 and that shape function values were computed accurately across the test set. Moreover, in a large-scale computing environment, the neural-network inference attained roughly three times the throughput of a conventional interpolation-based implementation. These results suggested a promising route to accelerate or replace classical shape-function evaluations using learning methods.

Key words: graph neural networks, finite element, shape function, CAE, deep learning

中图分类号:

TB115
TP183

琚晨, 丁嘉欣, 王泽兴, 李广钊, 管振祥, 张常有. 面向有限元法的图神经网络形函数近似方法[J]. 图学学报, 2025, 46(6): 1161-1171.

JU Chen, DING Jiaxin, WANG Zexing, LI Guangzhao, GUAN Zhenxiang, ZHANG Changyou. Graph neural network-based method for approximating finite element shape functions[J]. Journal of Graphics, 2025, 46(6): 1161-1171.

图/表 11

图1 基于图神经网络的形函数近似方法框架图

Fig. 1 Framework diagram of the shape function approximation method based on graph neural networks

图2 网络结构图

Fig. 2 Network structure diagram

图3 Encoder结构示意图

Fig. 3 Encoderstructure diagram

图4 消息传递示意图

Fig. 4 Message passing diagram

表1 训练参数表

Table 1 Training parameter table

训练参数	具体设置
train_ratio	0.8
batch_size	32
gnn_hidden	64
embed_dim	8
epochs	100
lr	0.001
lamda	0.1

图5 单元结构图((a) 四节点四面体单元；(b) 十节点四面体单元)

Fig. 5 Element structure diagram ((a) Four-node tetrahedral element; (b) Ten-node tetrahedral element)

表2 消融实验数据

Table 2 Ablation experiment data

模型	MSE	MSE_STD	MAE	MAE_STD	约束MSE	约束MSE_STD
full	0.174 4	0.094 6	2.425 4	0.388 3	0.064 8	0.025 0
no_cell_embed	0.182 9	0.092 1	2.472 7	0.380 2	0.085 6	0.037 4
no_norm	0.188 3	0.097 5	2.493 7	0.396 1	0.087 3	0.031 3
no_global_feat	0.255 4	0.011 8	3.025 3	0.450 8	0.145 8	0.059 5
mlp_only	0.369 0	0.012 6	3.724 0	0.519 9	0.247 4	0.111 5

表3 约束权重λ实验数据

Table 3 Experiment data of constrained weight λ

随机种子不同λ	42		101		202
随机种子不同λ	MSE	约束MSE	MSE	约束MSE	MSE	约束MSE
0	0.153 9	0.404 1	0.162 8	0.470 6	0.153 0	0.391 8
0.01	0.149 4	0.320 1	0.162 6	0.389 8	0.151 4	0.374 7
0.10	0.150 4	0.252 5	0.150 3	0.207 8	0.147 8	0.206 1
0.50	0.184 6	0.067 3	0.182 4	0.082 0	0.183 0	0.084 3
1.00	0.225 6	0.028 3	0.222 0	0.038 8	0.210 9	0.030 2
5.00	0.283 6	0.005 4	0.300 6	0.005 2	0.286 9	0.006 5

表4 模型泛化能力测试实验数据

Table 4 Experimental data for model generalization ability testing

测试类型	MSE	MSE_STD	MAE	MAE_STD	约束MSE
full	0.175 2	0.064 1	2.432 4	0.251 7	0.064 7
scale_0.5	4.785 8	0.403 7	15.588 2	0.649 9	2.410 7
scale_2.0	3.418 0	0.338 1	12.587 7	0.632 6	0.605 4
only_4node	0.151 2	0.078 8	2.332 8	0.361 5	0.032 0
only_10node	0.186 5	0.064 3	2.477 3	0.255 7	0.097 2
bad_elements	0.252 1	0.173 0	3.473 7	0.374 1	1.131 3

图6 不同λ值下的MSE与约束MSE实验结果((a) MSE结果；(b) 约束MSE结果)

Fig. 6 Experimental results of MSE and constrained MSE under different λ values ((a) MSE results; (b) Constraint MSE results)

图7 基于神经网络的和基于插值理论的计算效率对比

Fig. 7 Comparison of computational efficiency based on neural network and interpolationtheory

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面向有限元法的图神经网络形函数近似方法

Graph neural network-based method for approximating finite element shape functions

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