Physics-Informed Graph Neural Networks to Reconstruct Local Fields Considering Finite Strain Hyperelasticity

2025-07-05

AI Summary

Reconstructing microscale local stress fields from macroscale homogenized stresses in multiscale simulations—particularly under finite-strain hyperelastic constitutive laws—remains a fundamental challenge. To address this, we propose a physics-guided periodic graph neural network (P-DivGNN). Our method constructs a periodic graph structure to encode microstructural topology, explicitly embeds local static equilibrium and periodic boundary conditions into the message-passing mechanism, and employs a physics-driven loss function for label-free supervision. Compared to conventional finite element methods, P-DivGNN achieves comparable accuracy in stress field reconstruction while accelerating computation by one to two orders of magnitude. Comprehensive experiments demonstrate its generalizability and robustness across large-deformation, nonlinear hyperelastic regimes. By unifying physical priors with data-efficient learning, P-DivGNN establishes a new paradigm for interpretable, high-fidelity reduced-order modeling in large-scale multiscale mechanical analysis.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

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.

Problem

• Reconstruct local stress fields at micro-scale using physics-informed GNN
• Enforce equilibrium and periodic boundary conditions in stress prediction
• Achieve computational speed-up for hyperelastic materials versus FE simulation

Innovation

• Physics-informed GNN for stress field reconstruction
• Graph representation enforces periodic boundary conditions
• Computational speed-up in nonlinear hyperelastic cases