A locking-free nodal-based polytopal method for linear elasticity

This work presents a Discrete de Rham (DDR) numerical scheme for solving linear elasticity problems on general polyhedral meshes, with a focus on preventing volumetric locking in the quasi-incompressible regime. The method is formulated as a nodal-based approach using the lowest-order gradient space of the DDR complex, enriched with scalar face bubble degrees of freedom that effectively capture the normal flux across element faces. This face-bubble enrichment is crucial for ensuring sufficient approximation flexibility of the divergence field, thereby eliminating the {volumetric locking} phenomenon that typically occurs as the Lam\'e parameter $\lambda$ approaches infinity. We establish $H^1$-error estimates that are independent of $\lambda\ge 0$, and depend only on the lower bound of $\mu$, guaranteeing robustness across the entire range from compressible to nearly incompressible regimes. We also show how to adapt our scheme to the frictionless contact mechanics model, maintaining a locking-free estimate for the primal variable (displacement). Numerical experiments confirm that the proposed {locking-free} method delivers accurate and stable approximations on general polytopal discretizations, even when the material behaves as an incompressible medium. The flexibility and robustness of this approach make it a practical alternative to mixed formulations for engineering applications involving nearly incompressible elastic materials.