From Elements to Energy: A Controlled Benchmark of Finite Element, Element-Free Galerkin, and Deep Energy Methods in Solid Mechanics

Computational solid mechanics is dominated by mesh-based discretizations, while meshless and physics-informed approaches have emerged as alternatives. This study benchmarks three specified displacement formulations for two-dimensional linear elastostatics: full-integration bilinear quadrilateral finite elements (Q4 FEM), first-order moving-least-squares Element-Free Galerkin (EFG), and a four-hidden-layer tanh Deep Energy Method (DEM). The contribution is a controlled cross-paradigm assessment under one multi-regime protocol. On smooth problems, the three tested implementations achieve comparable accuracy. For the near-incompressible plane-strain cantilever, the tested Q4 FEM implementation underestimates tip displacement by 80.36%, whereas the tested EFG configuration reaches 0.33% error. Six of twenty baseline DEM runs return non-finite values, and the lowest-loss finite run has a 77.54% tip-displacement error. A matched-parameter depth study further confirms optimizer-stage sensitivity. Shared post-processing changes the interpretation of the localized results: FEM and EFG are close on the common near-hole ring, while the converged common-grid crack check preserves the FEM-EFG-DEM ordering for the specified implementations. Across the sampled trials, final DEM loss is informative for some bulk measures but weakly or inversely associated with localized or selected engineering errors. Reporting only the minimum-loss trial can therefore conceal non-finite outcomes and the observed error distribution; seeded errors and failure frequencies should accompany loss-based model selection.