Physics informed graph neural network models stress in complex solids
A variational physics-informed graph neural network for heterogeneous solid mechanics
Machine Learning
Summary
Modeling stress in materials with different properties is tricky because stress behaves differently at their boundaries. The authors developed a new type of neural network that uses a mesh to represent the material, assigning behavior to each part, which allows it to model stress without complicated penalty terms. This method shows improved accuracy over previous approaches across a range of material stiffness differences. However, training takes much longer than traditional methods, so it’s best suited for repeated analyses or identifying unknown material properties, not one-time calculations.
What this means in practice
- •For mechanical engineers: Produce accurate parametric surrogate models for simulating stress in composite materials over various stiffness contrasts.
- •For materials design teams: Perform inverse identification to estimate unknown material properties from observed stress patterns without manual tuning of interface parameters.
Authors
Aashay Rajan Yadav, Amiya Prakash Das, Ratna Kumar Annabattula
Abstract
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.