Dimensionless Energy Graph Network with Hard Boundary Embedding and Finite Element Interpolation: Unsupervised Variational Solution Framework for Linear Elastic and Hyperelastic Problems

To address the challenges in existing neural solvers for solid mechanics—such as the difficulty of balancing multiple loss terms, limited expressiveness on complex unstructured meshes, the optimization competition introduced by soft constraints for boundary conditions, and the sensitivity of energy-based training to physical scales—this work proposes a dimensionless energy graph network with hard boundary embedding and finite element interpolation (HiDEG‑Net). The method constructs a graph structure based on finite element discretization meshes, predicts nodal dimensionless displacement degrees of freedom using a graph message‑passing network, reconstructs strains, stresses and energy densities at the element level via finite element shape functions and their derivatives, and adopts the minimization of the dimensionless discrete total potential energy as the training objective. Unlike conventional coordinate‑based PINNs and deep energy methods, HiDEG‑Net explicitly leverages mesh adjacency relations, local geometric information, and boundary semantic information, thereby enhancing the representation capability of displacement fields under complex geometries and unstructured mesh conditions. For essential boundary conditions, a node‑masking and output‑replacement mechanism is employed to directly embed them into the network’s output space, avoiding additional boundary penalty terms; for natural boundary conditions, they are incorporated into the variational objective through the external potential energy term. Furthermore, a unified dimensionless treatment of coordinates, displacements, material parameters, loads, and energy improves training stability that would otherwise be compromised by significant differences in physical scales. Two‑dimensional and three‑dimensional linear elasticity examples, problems with holes, and two‑dimensional as well as three‑dimensional hyperelastic examples demonstrate that the proposed method can achieve displacement and stress fields in close agreement with finite element reference solutions under unsupervised training conditions. Ablation studies show that graph topology modeling, hard boundary embedding, and dimensionless treatment all play important roles in improving solution accuracy and training stability. This work provides a neural solution framework for complex boundary‑value problems in solid mechanics that combines the variational structure of finite element discretization with the representation power of graph neural networks.