Finite element analysis (FEA) is essential for structural design but remains computationally expensive, particularly when evaluating multiple design iterations or load scenarios. Machine learning surrogate models offer a promising alternative, yet most approaches struggle with a critical limitation: generalizing across varying geometries. This work presents a mesh graph network (MGN) for predicting von Mises stress fields in 2D structural components with arbitrary hole geometries. Unlike traditional machine learning approaches that use absolute node coordinates as features, the proposed model builds on existing MGN frameworks that encode node types (e.g., fixed boundary, free surface, hole edge), relative edge features (distance between neighbors), and global features (applied load). This architecture is inherently translation- and rotation-invariant, enabling generalization to unseen geometries without retraining. The MGN was trained on 11 plate geometries under 20 load conditions and evaluated on 7 unseen geometries and 3 unseen loads. In the most favorable case, the model achieves $R^2 \geq 0.97$ on an unseen geometry and unseen load, compared to $R^2 \approx 0.01$--$0.86$ for conventional models (Random Forest, Gradient Boosting , K-Nearest Neighbors) trained on identical data. However, even in less favorable cases, the MGN model still outperforms conventional models. This work extends the mesh-based simulation framework of Pfaff et al. (arXiv:2010.03409) to structural mechanics, demonstrating that graph neural networks can serve as efficient surrogates for finite element analysis across varying geometries.

翻译：有限元分析（FEA）在结构设计中至关重要，但在评估多种设计迭代或载荷工况时仍计算成本高昂。机器学习代理模型提供了有前景的替代方案，但大多数方法面临一个关键局限：难以泛化至不同几何形状。本文提出一种网格图网络（MGN），用于预测具有任意孔洞几何形状的二维结构部件的米塞斯应力场。与采用绝对节点坐标作为特征的传统机器学习方法不同，所提模型基于现有MGN框架，该框架对节点类型（如固定边界、自由表面、孔洞边缘）、相对边特征（相邻节点间距离）及全局特征（施加载荷）进行编码。该架构具有内在的平移与旋转不变性，无需重新训练即可泛化至未见过的几何形状。MGN在20种载荷条件下的11种板状几何形状上训练，并在7种未见几何形状和3种未见载荷下进行评估。在最有利情形中，模型对未见几何形状与未见载荷的评估结果达到$R^2 \geq 0.97$，而基于相同数据训练的传统模型（随机森林、梯度提升、K近邻）的$R^2 \approx 0.01$--$0.86$。即使在较不利情形下，MGN模型仍优于传统模型。本工作将Pfaff等人（arXiv:2010.03409）的网格仿真框架拓展至结构力学领域，证明图神经网络可作为跨几何形状有限元分析的高效代理模型。