Dimensionless energy graph network with hard boundary embedding and finite element interpolation: Unsupervised variational solution framework for linear elastic and hyperelastic problems

Abstract

Neural solvers for solid mechanics still face challenges related to loss-term balancing, representation on complex unstructured meshes, enforcement of essential boundary conditions, and scale sensitivity in energy-based optimization. This work proposes HiDEG-Net, a dimensionless energy graph network with hard boundary embedding and finite-element interpolation, for unsupervised variational solution of linear elastic and hyperelastic boundary-value problems. The finite-element mesh is used as a graph carrier, and a message-passing neural network predicts dimensionless nodal displacement degrees of freedom from nodal, edge, and boundary-semantic features. Strains, stresses, and energy densities are reconstructed at the element level using finite-element shape functions and their derivatives, avoiding spatial automatic differentiation of the network output. Essential boundary conditions are imposed through component-wise degree-of-freedom masks and output replacement, while natural boundary conditions are included through the external potential in the dimensionless total potential energy. A unified scaling of geometry, displacement, material parameters, loads, and energy is introduced to improve the conditioning of variational optimization. The method is assessed through mesh-convergence tests, frozen cross-grid prediction, target-mesh and load-transfer learning, benchmark comparisons with representative neural solvers, architecture sensitivity analysis, and full-factorial ablation studies. Numerical examples, including two-dimensional perforated plates, a three-dimensional cantilever beam, a two-dimensional hyperelastic perforated specimen, and a three-dimensional hyperelastic Cook’s membrane, show that HiDEG-Net provides displacement, stress, and energy predictions that are generally close to finite-element reference solutions under unsupervised training. Larger discrepancies mainly appear near holes, fixed boundaries, and strongly nonlinear local regions. The ablation results indicate that graph topology, hard boundary embedding, and dimensionless scaling each contribute to accuracy and training stability. Frozen cross-grid transfer shows mesh-dependent error degradation, whereas short unsupervised fine-tuning improves target-mesh performance. The proposed framework offers a topology-aware and scale-consistent neural variational strategy for solid-mechanics computation on unstructured meshes.