An energy-based differentiable finite element framework for elastic and path-dependent elastoplastic solids

Abstract

This paper proposes a finite-element-integrated deep energy method (DEM), hereafter referred to as FE-integrated DEM, for linear elasticity and path-dependent J 2 /von Mises plasticity. The method targets a central difficulty in neural elastoplastic solvers: path-dependent internal variables must be evolved consistently at fixed integration points, while the optimization objective must remain stable across elastic–plastic transitions and load-step changes. To address this issue, the proposed framework combines the incremental potential-energy objective of DEM with a complete finite element discretization scaffold in a single differentiable computational graph. The neural network is used only to approximate the displacement field through its nodal values; strain evaluation, Gauss quadrature, standard assembly, radial-return mapping, and stepwise history-variable updates are all retained from the classical finite element and computational-plasticity pipeline. The training loss is the Simo–Hughes incremental potential-energy functional, coupled with Gauss-point return mapping, so that the stationarity of the current-step energy is equivalent to the discrete equilibrium equation and the Kuhn–Tucker conditions are satisfied consistently during plastic evolution. No field-supervision data are required. The method is evaluated on five linear-elastic benchmarks involving bending, stress concentration, and material interfaces, two monotonic J 2 plasticity problems, and three non-monotonic loading histories with unloading, reverse loading, and direction reversal. Across these tests, FE-integrated DEM matches the standard finite element method (FEM) reference solutions step by step in displacements, stresses, and accumulated plastic strain. A same-scaffold discrete residual method (DRM) baseline is additionally implemented to isolate the effect of the physics objective; the comparisons show that, at comparable equilibrium-residual levels, the incremental energy objective provides substantially more accurate local fields, especially in stress-concentration regions, at material interfaces and within plastic-localization bands. These results support the incremental potential energy as a robust default physics objective for path-dependent neural plasticity. All source code is publicly released to facilitate reproducibility and to support the broader development of physics-driven machine-learning solvers for computational mechanics.