一种用于非均质固体力学的变分物理信息图神经网络
A variational physics-informed graph neural network for heterogeneous solid mechanics
- Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出变分物理信息图神经网络，通过离散化承载非均质性，最小化离散势能，实现无惩罚界面表示，在刚度对比扫描中精度优于强形式PINN，适用于参数化代理和逆识别。
AI中文摘要：
非均质固体中的应力局部化由双材料界面控制，其中位移场保持$C^0$连续性，而面内应力由于刚度不匹配而跳跃。基于坐标的物理信息神经网络（PINNs）通过预设的正则化宽度或加权界面惩罚来表示这种跳跃，使其精度对相衬变化的处理方式敏感。本工作提出了一种变分、无标签的物理信息图神经网络（PI-GNN），其中非均质性由离散化承载，而非试验场。该求解器在一致的适应性网格图上运行，为每个单元分配本构行为，并将离散总势能最小化为单一无加权目标，其中仅出现一阶导数。分段线性单元上的离散能量与有限元（FE）Ritz泛函一致。Dirichlet条件通过构造强制执行，无惩罚项、无界面权重、无预设过渡宽度。使用固定架构、优化器和损失函数，在小应变弹性和有限应变Neo-Hookean超弹性（二维和三维）中，von Mises误差在刚度对比扫描（$E_{\mathrm{inc}}/E_{\mathrm{mat}}\in[10^{-2},10^{2}]$）中保持在$3.58\\%$以下，而强形式PINN退化至$5.58\\%$，其位移误差达到$7.66\\%$，而PI-GNN为$0.49\\%$。训练后的网络将基于能量的PINN的（$\sigma_{xx}$）误差减半（$5.01\\%$对比$10.94\\%$）。训练成本超过单次有限元求解一个数量级以上，因此该构造是参数化代理和逆识别的一种变分一致、无惩罚的界面表示，而非一次性有限元分析的替代品。
英文摘要：
Stress localization in heterogeneous solids is governed by the bimaterial interface, where the displacement field remains $C^0$-continuous, while in-plane stresses jump due to the stiffness mismatch. Coordinate-based physics-informed neural networks (PINNs) represent this jump via a prescribed regularization width or a weighted interface penalty, making their accuracy sensitive to how phase-contrast changes are handled. This work presents a variational, label-free physics-informed graph neural network (PI-GNN) in which the heterogeneity is carried by the discretization rather than by the trial field. The solver operates on a conforming adaptive mesh graph, assigns constitutive behavior per element, and minimizes the discrete total potential energy as a single unweighted objective in which only first derivatives appear. The discrete energy on piecewise-linear elements coincides with the finite element (FE) Ritz functional. Dirichlet conditions are enforced by construction, with no penalty term, no interface weight, and no prescribed transition width. Using one fixed architecture, optimizer, and loss across small-strain elasticity and finite-strain Neo-Hookean hyperelasticity in two and three dimensions, the von Mises error remains below $3.58\%$ across a stiffness-contrast sweep spanning $(E_{\mathrm{inc}}/E_{\mathrm{mat}}\in[10^{-2},10^{2}])$, where a strong-form PINN degrades to $5.58\%$, and its displacement error reaches $7.66\%$ against $0.49\%$ for the PI-GNN. A trained network halves the ($σ_{xx}$) error of an energy-based PINN ($5.01\%$ versus $10.94\%$). Training cost exceeds a single FE solve by more than an order of magnitude, so the construction is a variationally consistent, penalty-free interface representation for parametric surrogates and inverse identification rather than a replacement for a one-off FE analysis.
