Engineering application of physics-informed neural networks for Saint-Venant torsion

TL;DR

The paper addresses the computational challenge of Saint-Venant torsion analysis for complex cross-sections by developing mesh-free physics-informed neural network (PINN) solvers. It derives the governing Poisson equation for the Prandtl stress function and demonstrates three PINN variants—vanilla PINN, variable-scaling PINN (VS-PINN), and a Parametric PINN—each embedding the physics and boundary conditions into the training objective. Through 2D Poisson experiments across circular, square, triangular, and irregular sections, the authors validate accuracy against analytical and FEM/ANSYS benchmarks, showing VS-PINN markedly improves convergence in cases with sharp geometric transitions and that Parametric PINN can deliver real-time surrogates across varying loading scenarios. The work highlights a flexible, efficient workflow for torsional analysis that can be extended to broader elasticity problems and integrated into design optimization and digital-twin pipelines.

Abstract

The Saint-Venant torsion theory is a classical theory for analyzing the torsional behavior of structural components, and it remains critically important in modern computational design workflows. Conventional numerical methods, including the finite element method (FEM), typically rely on mesh-based approaches to obtain approximate solutions. However, these methods often require complex and computationally intensive techniques to overcome the limitations of approximation, leading to significant increases in computational cost. The objective of this study is to develop a series of novel numerical methods based on physics-informed neural networks (PINN) for solving the Saint-Venant torsion equations. Utilizing the expressive power and the automatic differentiation capability of neural networks, the PINN can solve partial differential equations (PDEs) along with boundary conditions without the need for intricate computational techniques. First, a PINN solver was developed to compute the torsional constant for bars with arbitrary cross-sectional geometries. This was followed by the development of a solver capable of handling cases with sharp geometric transitions; variable-scaling PINN (VS-PINN). Finally, a parametric PINN was constructed to address the limitations of conventional single-instance PINN. The results from all three solvers showed good agreement with reference solutions, demonstrating their accuracy and robustness. Each solver can be selectively utilized depending on the specific requirements of torsional behavior analysis.

Figures (11)

• Figure 1: Comparison of schematic diagrams of PINN, VS-PINN and Parametric PINN.
• Figure 2: Case study of four shapes. The Poisson equation with the Dirichlet boundary condition is solved.
• Figure 3: Sensitivity study for grid size determination. The unified grid size was determined to be 0.005 m.
• Figure 4: 0.005 m grid modeling for each shape. The spatial coordinates of the grid generated in ANSYS were used identically as the sampling points of PINN.
• Figure 5: Training loss by epoch. Training was terminated when the loss converged in all shapes.