发表机构
Southeast University（东南大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对有限元应力预测中高频信息衰减问题，提出基于Joukowski谱提升的LiftGCN，以二阶递推实现能量保持传播，降低复杂度至O(ed)，在保持精度的同时改善应力集中重建。
AI 中文摘要
有限元应力场通常表现出强烈的局部非光滑性，其中孔洞、缺口和加载区域附近的应力集中会产生尖锐的空间梯度和高频图分量。尽管图神经网络天然适用于不规则有限元网格，但传统消息传递本质上具有平滑性，会逐渐衰减此类高频信息。酉传播通过保持谱幅度来缓解此问题，但通常依赖矩阵函数和高阶近似，传播复杂度为$O(Ked)$。我们提出LiftGCN，一种基于Joukowski谱提升的高效谱稳定图网络。LiftGCN通过Joukowski关系将归一化图算子的实谱映射到单位圆上，并将所得谱变换实现为简单的二阶递推，避免了矩阵指数、特征分解和高阶多项式截断。我们证明线性Joukowski主干具有单位模特征根，并在正定度量下允许能量保持结构，防止图频率分量随深度指数衰减。每层仅需一次稀疏邻域聚合，产生$O(ed)$传播复杂度，而轻量级局部非线性残差提供表达性特征变换。在有限元应力预测实验上，LiftGCN实现了有竞争力的整体精度，同时以大幅降低的计算成本改善了应力集中和局部高梯度结构的重建。我们的代码可在https://this URL获取。
英文摘要
Finite element stress fields often exhibit strong local non-smoothness, where stress concentrations near holes, notches, and loading regions induce sharp spatial gradients and high-frequency graph components. Although graph neural networks naturally operate on irregular finite element meshes, conventional message passing is inherently smoothing and progressively attenuates such high-frequency information. Unitary propagation alleviates this problem by preserving spectral magnitudes, but typically relies on matrix functions and high-order approximations with $O(Ked)$ propagation complexity. We propose LiftGCN, an efficient spectrally stable graph network based on Joukowski spectral lifting. LiftGCN maps the real spectrum of a normalized graph operator onto the unit circle through the Joukowski relation and realizes the resulting spectral transformation as a simple second-order recurrence, avoiding matrix exponentials, eigendecomposition, and high-order polynomial truncation. We show that the linear Joukowski backbone has unit-modulus characteristic roots and admits an energy-preserving structure under a positive-definite metric, preventing exponential attenuation of graph-frequency components with depth. Each layer requires only one sparse neighborhood aggregation, yielding $O(ed)$ propagation complexity, while lightweight local nonlinear residuals provide expressive feature transformations. Experiments on finite element stress prediction demonstrate that LiftGCN achieves competitive overall accuracy while improving reconstruction of stress concentrations and local high-gradient structures with substantially reduced computational cost. Our code is available at https://github.com/ChenZeng001/LiftGCN.
