A variational physics-informed neural network framework for upper-bound plastic limit analysis of Mindlin plates

This paper presents a physics-informed neural network (PINN) framework based on the kinematic upper-bound formulation for rigid-perfectly plastic Mindlin plates. While PINNs have been increasingly adopted in computational mechanics, their capability to directly minimize the non-smooth plastic dissipation functional for evaluating structural collapse mechanisms has remained largely unexplored. By recasting limit analysis as a sampling-based neural variational optimization problem over admissible velocity fields, the proposed approach provides a mesh-free alternative to traditional finite element and second-order cone programming (SOCP) techniques. In this sampling-based variational framework, the transverse velocity and two independent rotation fields are approximated by neural networks, including both standard multilayer perceptrons and residual networks enriched with learnable Fourier feature embeddings. The von Mises plastic dissipation functional is integrated through the plate thickness by Gauss–Legendre quadrature and over the mid-plane by quasi-Monte Carlo integration, and is minimized subject to a unit external-work normalization. Essential kinematic boundary conditions are embedded through geometry-dependent hard constraints implemented by analytical masking functions and local projection operators. The unit-work condition is treated through work-normalization strategies, with penalty enforcement and explicit work normalization compared in the numerical study. Numerical studies are conducted for square, skew rhombic, circular, triangular, and hexagonal plates under clamped and simply supported boundary conditions. The results show that the proposed PINN formulation can reproduce the main collapse-load trends and dissipation patterns reported in reference upper-bound solutions, while the Fourier-enriched architecture generally improves the resolution of localized plastic dissipation. Furthermore, the thickness studies demonstrate that the continuous neural representation naturally mitigates the severe shear-locking phenomenon commonly associated with low-order Mindlin finite elements, maintaining robust collapse-load predictions even in the thin-plate limit. Finally, remaining sensitivities to network architecture, optimization effort, and geometric boundary treatments are identified and discussed.