Analysis of a VEM-fully discrete polytopal scheme with bubble stabilisation for contact mechanics with Tresca friction

TL;DR

This work addresses the challenge of simulating contact mechanics with Tresca friction on fracture networks within poroelastic media using a robust fully discrete discretisation on polytopal meshes. It combines a bubble-stabilised, first-order nodal displacement space with a face-wise constant Lagrange multiplier space, analyzed through a fracture-aware $H^{-1/2}_{0,\mathrm{j}}(\Gamma)$-type norm and a discrete Korn inequality to establish stability and convergence. The authors prove existence and uniqueness of the discrete solution, derive an abstract and a concrete first-order error estimate, and construct a fracture-specific averaged interpolator to secure the inf-sup condition. Numerical experiments on 2D and 3D manufactured solutions confirm the predicted first-order convergence for the displacement gradient and the Lagrange multiplier, validating the theoretical results and demonstrating robustness on complex fracture geometries.

Abstract

This work performs the convergence analysis of the polytopal nodal discretisation of contact-mechanics (with Tresca friction) recently introduced in [18] in the framework of poro-elastic models in fractured porous media. The scheme is based on a mixed formulation, using face-wise constant approximations of the Lagrange multipliers along the fracture network and a fully discrete first order nodal approximation of the displacement field. The displacement field is enriched with additional bubble degrees of freedom along the fractures to ensure the inf-sup stability with the Lagrange multiplier space. It is presented in a fully discrete formulation, which makes its study more straightforward, but also has a Virtual Element interpretation. The analysis establishes an abstract error estimate accounting for the fully discrete framework and the non-conformity of the discretisation. A first order error estimate is deduced for sufficiently smooth solutions both for the gradient of the displacement field and the Lagrange multiplier. A key difficulty of the numerical analysis is the proof of a discrete inf-sup condition, which is based on a non-standard $H^{-1/2}$-norm (to deal with fracture networks) and involves the jump of the displacements, not their traces. The analysis also requires the proof of a discrete Korn inequality for the discrete displacement field which takes into account fracture networks. Numerical experiments based on analytical solutions confirm our theoretical findings

Figures (10)

• Figure 1: Nodal degrees of freedom.
• Figure 2: Splitting of the fracture network and construction of sub-domains used to define the norm on $\mathbf{M}_\mathcal{D}$.
• Figure 3: Domains for local fracture-compatible averages.
• Figure 4: Unbounded domain containing a single fracture under uniform compression (a) and mesh including nodes for boundary conditions ($\blacklozenge$: $u_x = 0$, $\blacksquare$: $u_y=0$), for the example of Section \ref{['tchelepi_meca']}.
• Figure 5: Comparison between the numerical and analytical solutions on the finest mesh (with 800 fracture faces), in terms of $\lambda_{\mathbf{n}}$ (a) and $\llbracket \mathbf{u} \rrbracket_{{\boldsymbol{\tau}}}$ (b), example of Section \ref{['tchelepi_meca']}.

Theorems & Definitions (35)

• Remark 3.1: Non planar faces
• Remark 3.2
• Remark 3.3: Virtual element interpretation
• Definition 4.1: Discrete $H^1$-like semi-norm on $\mathbf{U}_\mathcal{D}$
• Definition 4.2: $H^{-1/2}_{0,{\rm j}}(\Gamma)$-like norm on $\mathbf{M}_\mathcal{D}$
• Remark 4.3: Norm on the fracture network
• Proposition 4.4: Existence and uniqueness result
• proof
• Theorem 4.5: Abstract error estimate
• proof