用图神经网络预测微尺度应力分布，兼顾物理规律与周期边界条件。
Physics-Informed Graph Neural Networks to Reconstruct Local Fields Considering Finite Strain Hyperelasticity
- 将周期性微观结构建模为图，结合消息传递GNN进行应力场重建。
- 非线性超弹性下计算速度远超有限元模拟，提升效率。
- 适合需要局部应力分析的断裂与疲劳评估场景。
我们提出一种称为P-DivGNN的物理信息机器学习框架，用于在多尺度模拟中，基于周期性微观结构网格和宏观平均应力值，重构微尺度局部应力场。该方法将周期性微观结构表示为图，并结合消息传递图神经网络进行建模。能够恢复出由均场降维模型（ROM）或有限元（FE）模拟在宏观尺度上产生的平均应力值。局部应力场的准确预测对断裂分析或局部疲劳准则定义至关重要。模型在训练中引入物理约束，确保局部应力场处于平衡状态，并采用周期图表示以满足周期边界条件。所提物理信息图神经网络在不同几何形态下的线性和非线性超弹性响应中均表现优异。在非线性超弹性情况下，相比有限元模拟实现显著计算加速，特别适用于大规模应用。
原文摘要 · Abstract (English)
We propose a physics-informed machine learning framework called P-DivGNN to reconstruct local stress fields at the micro-scale, in the context of multi-scale simulation given a periodic micro-structure mesh and mean, macro-scale, stress values. This method is based in representing a periodic micro-structure as a graph, combined with a message passing graph neural network. We are able to retrieve local stress field distributions, providing average stress values produced by a mean field reduced order model (ROM) or Finite Element (FE) simulation at the macro-scale. The prediction of local stress fields are of utmost importance considering fracture analysis or the definition of local fatigue criteria. Our model incorporates physical constraints during training to constraint local stress field equilibrium state and employs a periodic graph representation to enforce periodic boundary conditions. The benefits of the proposed physics-informed GNN are evaluated considering linear and non linear hyperelastic responses applied to varying geometries. In the non-linear hyperelastic case, the proposed method achieves significant computational speed-ups compared to FE simulation, making it particularly attractive for large-scale applications.
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