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。
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 0.0018 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.
作者
琚晨
丁嘉欣
王泽兴
李广钊
管振祥
张常有
JU Chen;DING Jiaxin;WANG Zexing;LI Guangzhao;GUAN Zhenxiang;ZHANG Changyou(Institute of Software,Chinese Academy of Sciences,Beijing 100190,China;School of Computer Science and Technology,University of Chinese Academy of Sciences,Beijing 100190,China;Beijing National New Energy Vehicle Technology Innovation Center Co.,Ltd.,Beijing 102600,China;China Railway 19 Bureau Group Co.,Ltd.,Beijing 100176,China)
出处
《图学学报》
北大核心
2025年第6期1161-1171,共11页
Journal of Graphics
基金
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)。
关键词
图神经网络
有限元
形函数
CAE
深度学习
graph neural networks
finite element
shape function
CAE
deep learning
