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

TL;DR

This paper introduces a locking-free nodal-based polytopal method for linear elasticity that remains accurate and stable across all regimes, including nearly incompressible materials, on general polyhedral meshes.

Contribution

It develops a novel Discrete de Rham scheme with face bubble enrichment to prevent volumetric locking in linear elasticity problems on arbitrary polyhedral meshes.

Findings

The method is robust for all Lamé parameters, including the incompressible limit.

Numerical experiments demonstrate accurate, locking-free solutions on complex meshes.

The approach extends to frictionless contact mechanics without losing stability.

Abstract

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.