Geometry physics neural operator solver for solid mechanics

Abstract

This study developed Geometry Physics neural Operator (GPO), a novel solver framework to approximate the partial differential equation (PDE) solutions for solid mechanics problems with irregular geometry and achieved a significant speedup in simulation time compared to numerical solvers. GPO leverages a weak form of PDEs based on the principle of least work, incorporates geometry information, and imposes exact Dirichlet boundary conditions within the network architecture to attain accurate and efficient modeling. This study focuses on applying GPO to model the behaviors of complicated bodies without any guided solutions or labeled training data. GPO adopts a modified Fourier neural operator as the backbone to achieve significantly improved convergence speed and to learn the complicated solution field of solid mechanics problems. Numerical experiments involved a two-dimensional plane with a hole and a three-dimensional building structure with Dirichlet boundary constraints. The results indicate that the geometry layer and exact boundary constraints in GPO significantly contribute to the convergence accuracy and speed, outperforming the previous benchmark in simulations of irregular geometry. The comparison results also showed that GPO can converge to solution fields faster than a commercial numerical solver in the structural examples. Furthermore, GPO demonstrates stronger performance than the solvers when the mesh size is smaller, and it achieves over 3$\times$ and 2$\times$ speedup for a large degree of freedom in the two-dimensional and three-dimensional examples, respectively. The limitations of nonlinearity and complicated structures are further discussed for prospective developments. The remarkable results suggest the potential modeling applications of large-scale infrastructures.