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

Jerome Droniou, Raman Kumar

TL;DR

This paper introduces a locking-free nodal-based DDR scheme for linear elasticity on general polyhedral meshes by enriching the lowest-order gradient space with scalar face bubble DOFs to accurately capture normal flux and improve divergence approximation. The authors build a comprehensive framework of reconstruction operators, an interpolator, and a stabilized variational formulation, and prove $H^1$-type error bounds that are independent of the second Lamé coefficient $λ$, ensuring robustness from compressible to quasi-incompressible regimes. An abstract error estimate and a concrete convergence result are established, supported by a discrete Korn inequality and a Fortin-type commutation property. The approach extends to frictionless contact mechanics and is validated by numerical experiments showing locking-free performance and optimal convergence on general polytopal meshes, highlighting practical applicability as an alternative to traditional mixed formulations.

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 Lamé parameter $λ$ approaches infinity. We establish $H^1$-error estimates that are independent of $λ\ge 0$, and depend only on the lower bound of $μ$, 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.

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

TL;DR

This paper introduces a locking-free nodal-based DDR scheme for linear elasticity on general polyhedral meshes by enriching the lowest-order gradient space with scalar face bubble DOFs to accurately capture normal flux and improve divergence approximation. The authors build a comprehensive framework of reconstruction operators, an interpolator, and a stabilized variational formulation, and prove -type error bounds that are independent of the second Lamé coefficient , ensuring robustness from compressible to quasi-incompressible regimes. An abstract error estimate and a concrete convergence result are established, supported by a discrete Korn inequality and a Fortin-type commutation property. The approach extends to frictionless contact mechanics and is validated by numerical experiments showing locking-free performance and optimal convergence on general polytopal meshes, highlighting practical applicability as an alternative to traditional mixed formulations.

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 Lamé parameter approaches infinity. We establish -error estimates that are independent of , and depend only on the lower bound of , 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.
Paper Structure (16 sections, 10 theorems, 64 equations, 2 figures, 8 tables)

This paper contains 16 sections, 10 theorems, 64 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Model and scheme
  3. Model
  4. Mesh and spaces
  5. Reconstruction operators in $\mathbf{U}_{0,\mathcal{D}}$
  6. Interpolator
  7. Variational formulation
  8. Main results
  9. Proof of the error estimate
  10. Commutation property
  11. Proof of the abstract error estimate (Theorem \ref{['abstr.err.est']})
  12. Proof of the error estimate (Theorem \ref{['error_estimate']})
  13. Frictionless contact mechanics model
  14. Numerical Experiments
  15. Convergence rate definition
  16. ...and 1 more sections

Key Result

Theorem 3.2

The numerical scheme mixed_discrete has a unique solution $\mathbf{u}_{\mathcal{D}} \in \mathbf{U}_{0,\mathcal{D}}$ and, if $\mathbf{u}$ is the solution of Lagrange_meca_contactfriction_1, we have the following abstract error estimate: where we recall that the hidden constant in $\lesssim$ is independent of the Lamé coefficients $\mu$ and $\lambda$.

Figures (2)

  • Figure 1: Schematic representation of degrees of freedom in the DDR scheme with face-bubble enrichment. The discrete space $\mathbf{U}_\mathcal{D}$ consists of vertex displacements and face-bubble corrections.
  • Figure 2: (Example \ref{['exm2']}). Discrete displacement magnitude $\mathbf{u}_\mathcal{D}$ on the Voronoi (left) and tetrahedral (right) meshes with mesh sizes $h = 0.3130676$ and $h = 0.2213817$, respectively.

Theorems & Definitions (29)

  • Definition 3.1: Discrete $H^1$-like semi-norm on $\mathbf{U}_\mathcal{D}$
  • Theorem 3.2: Abstract error estimate
  • proof
  • Theorem 3.3: Error estimate
  • proof
  • Remark 3.4: Locking-free estimate
  • Remark 3.5: Comparison with existing methods
  • Remark 3.6
  • Lemma 4.1: DOF-based bound on the discrete norm
  • proof
  • ...and 19 more