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arxiv: 2601.07586 · v2 · submitted 2026-01-12 · 🧮 math.NA · cs.NA

A higher order polytopal method for contact mechanics with Tresca friction

Jerome Droniou , Raman Kumar , Roland Masson , Ritesh Singla This is my paper

Pith reviewed 2026-05-16 15:00 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords contact mechanicsTresca frictionfracturesDiscrete de Rhampolytopal meshesKorn inequalityinf-sup conditionquasi-incompressible limit
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The pith

A Discrete de Rham scheme provides a robust higher-order method for contact mechanics with Tresca friction on fractures.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a mixed finite element method based on Discrete de Rham spaces to solve elastic contact problems involving fractures with Tresca friction. Displacement is approximated using vector values at vertices, edges, faces, and elements, enabling piecewise quadratic reconstruction, while tractions are handled by piecewise constant Lagrange multipliers on fracture faces. Stability is established by proving a discrete Korn inequality that includes the fractures and an inf-sup condition in a non-standard norm. The analysis yields error bounds and demonstrates that the displacement remains accurate even when the material is nearly incompressible, overcoming a common limitation of low-order schemes. Numerical experiments on various test cases confirm the theoretical predictions and highlight the method's practical performance.

Core claim

We design and analyze a Discrete de Rham scheme for a contact mechanics problem with Tresca friction along fractures. The scheme uses a mixed formulation with displacement approximated by vectors on mesh vertices, edges, faces and elements, and Lagrange multipliers by piecewise constants on fracture faces. We prove a discrete Korn inequality accounting for the fractures and an inf-sup condition in a non-standard H^{-1/2}-norm. The error analysis shows optimal convergence and robustness in the quasi-incompressible limit for the displacement variable.

What carries the argument

The Discrete de Rham (DDR) approximation spaces for the mixed formulation, where displacement degrees of freedom are vector-valued on vertices, edges, faces, and elements, and the Lagrange multiplier is piecewise constant on fracture faces, enabling reconstruction of quadratic displacements.

Load-bearing premise

The specific choice of vector-valued degrees of freedom on mesh entities for displacement and piecewise constant multipliers on fracture faces must satisfy the discrete Korn inequality and the inf-sup condition for the polytopal meshes and fracture geometries under consideration.

What would settle it

If numerical computations on a sequence of refined meshes for a nearly incompressible material show that the displacement error does not decrease at the expected rate or exhibits locking behavior, this would indicate that the robustness claim does not hold.

Figures

Figures reproduced from arXiv: 2601.07586 by Jerome Droniou, Raman Kumar, Ritesh Singla, Roland Masson.

Figure 6.1
Figure 6.1. Figure 6.1: (Test case from Section 6.1). Relative L 2 -norm of errors u − Υ2 h (uh ), [[u]] − [[uh ]]h, ∇u − ∇Υ2 h (uh ), and λn − λh,n versus the cubic root of the number of cells, for (a) Cartesian, (b) tetrahedral, and (c) Hexa-cut mesh families [PITH_FULL_IMAGE:figures/full_fig_p028_6_1.png] view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: (Test case from Section 6.2). Relative L 2 -norm errors of u − Υ2 h (uh ), [[u]] − [[uh ]]h, ∇u − ∇Υ2 h (uh ), and λn − λh,n versus the cubic root of the number of cells, for (a) Cartesian, (b) tetrahedral, and (c) Hexa-cut mesh families [PITH_FULL_IMAGE:figures/full_fig_p029_6_2.png] view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: ((a) Test case from Section 6.1 and (b) Test case from Section 6.2). Comparison of L 2 -norm of ∇u − ∇Υ2 h (uh ) of the lowest-order method [28] with the current higher-order scheme for a tetrahedral mesh. G = 1, L ∈ {1, 104 , 106}, and the Tresca threshold g is set to 1/L. The exact solution is defined by u(x, y, z) =      x 3 (cos(y) + sin(z)) −3x 2 sin(y) 3x 2 cos(z… view at source ↗
Figure 6.4
Figure 6.4. Figure 6.4: Fracture network for the test case in Section 6.4 [PITH_FULL_IMAGE:figures/full_fig_p032_6_4.png] view at source ↗
Figure 6.5
Figure 6.5. Figure 6.5: Results for the test case in Section 6.4: (a) contact state classification, where the values indicate: 0 if |λh,τ | < g and [[uh ]]h,n < 0; 1 if |λh,τ | < g and [[uh ]]h,n = 0; 2 if |λh,τ | = g and [[uh ]]h,n < 0; and 3 if |λh,τ | = g and [[uh ]]h,n = 0; and (b) normal displacement jump obtained using the DDR discretisation with 123k cells and 8.8k fracture faces. (a) (b) [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 6.6
Figure 6.6. Figure 6.6: Test case from Section 6.4: Plot of normal component of discrete Lagrange multiplier: (a) lowest-order method [28] and (b) the current higher-order scheme for fracture F1 on a tetrahedral mesh [PITH_FULL_IMAGE:figures/full_fig_p033_6_6.png] view at source ↗
Figure 6.7
Figure 6.7. Figure 6.7: Test case from Section 6.4: Mean value of normal component of the discrete Lagrange multiplier for fractures F2, F3, and F4 are shown in (a), (b), and (c), respectively. Results compare the lowest-order method [28] with the current higher-order scheme on a tetrahedral mesh [PITH_FULL_IMAGE:figures/full_fig_p034_6_7.png] view at source ↗
Figure 6.8
Figure 6.8. Figure 6.8: Test case from Section 6.4: Comparison between the lowest-order method [28] and the current higher-order scheme for fracture F1 on a tetrahedral mesh: (a) Mean value of normal displacement jump and (b) Mean value of normal component of the discrete Lagrange multiplier [PITH_FULL_IMAGE:figures/full_fig_p035_6_8.png] view at source ↗
read the original abstract

In this work, we design and analyze a Discrete de Rham (DDR) scheme for a contact mechanics problem involving fractures along which a model of Tresca friction is considered. Our approach is based on a mixed formulation involving a displacement field and a Lagrange multiplier, enforcing the contact conditions, representing tractions at fractures. The approximation space for the displacement is made of vectors values attached to each vertex, edge, face, and element, while the Lagrange multiplier space is approximated by piecewise constant vectors on each fracture face. The displacement degrees of freedom allow reconstruct piecewise quadratic approximations of this field. We prove a discrete Korn inequality that account for the fractures, as well as an inf-sup condition (in a non-standard $H^{-1/2}$-norm) between the discrete Lagrange multiplier space and the discrete displacement space. We provide an in-depth error analysis of the scheme and show that, contrary to usual low-order nodal-based schemes, our method is robust in the quasi-incompressible limit for the primal variable~(displacement). An extensive set of numerical experiments confirms the theoretical analysis and demonstrate the practical accuracy and robustness of the scheme.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces a Discrete de Rham (DDR) scheme for contact mechanics with Tresca friction along fractures. It employs a mixed formulation with a vector-valued displacement space (DOFs attached to vertices, edges, faces, and elements, permitting piecewise quadratic reconstruction) and piecewise-constant Lagrange multipliers on fracture faces. The authors establish a discrete Korn inequality that accounts for fractures and an inf-sup condition in a non-standard H^{-1/2} norm between the discrete spaces, followed by an error analysis that demonstrates robustness of the primal variable in the quasi-incompressible limit (unlike standard low-order nodal schemes). The theoretical results are supported by an extensive set of numerical experiments on polytopal meshes.

Significance. If the stability proofs and error estimates hold, the work provides a meaningful advance in numerical methods for fractured contact problems by delivering a higher-order polytopal scheme that remains stable and accurate in the quasi-incompressible regime. The explicit construction of a fracture-aware discrete Korn inequality and the non-standard inf-sup result on general polytopal meshes could serve as a foundation for extensions to other interface problems in computational mechanics.

major comments (2)
  1. [Section 4 (error analysis)] The error analysis claims robustness for the displacement in the quasi-incompressible limit, but the dependence of the constants on the Lamé parameters (particularly as the bulk modulus tends to infinity) is not quantified explicitly; the manuscript should derive the precise scaling of the error bound with respect to the incompressibility parameter in the relevant theorem.
  2. [Section 3.2 (inf-sup condition)] The discrete inf-sup condition is stated in a non-standard H^{-1/2} norm; the proof must verify that this norm is equivalent to the standard dual norm on the fracture faces under the assumed mesh regularity, otherwise the stability constant may deteriorate on highly distorted polytopal elements.
minor comments (3)
  1. [Abstract] The abstract contains a grammatical error: 'a discrete Korn inequality that account for the fractures' should read 'accounts'.
  2. [Section 2] Notation for the reconstructed quadratic displacement field and its relation to the DDR degrees of freedom should be introduced earlier and used consistently in the stability proofs.
  3. [Section 5] The numerical experiments section would benefit from a table summarizing observed convergence rates for different polynomial degrees and mesh families to facilitate direct comparison with the theoretical predictions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation for minor revision. The comments are constructive and help improve the clarity of the stability and error analysis. We address each major comment below.

read point-by-point responses

  1. Referee: [Section 4 (error analysis)] The error analysis claims robustness for the displacement in the quasi-incompressible limit, but the dependence of the constants on the Lamé parameters (particularly as the bulk modulus tends to infinity) is not quantified explicitly; the manuscript should derive the precise scaling of the error bound with respect to the incompressibility parameter in the relevant theorem.

    Authors: We thank the referee for this observation. In the current error analysis the constants are independent of the Lamé parameter λ (equivalently, the bulk modulus) as λ → ∞, which follows directly from the λ-robust discrete Korn inequality and the inf-sup condition. To make this explicit as requested, we will revise the statement of the main error theorem in Section 4 to display the precise scaling (showing boundedness independent of λ) and add a short remark tracing the independence through the proof. This change will be incorporated in the revised manuscript. revision: yes

  2. Referee: [Section 3.2 (inf-sup condition)] The discrete inf-sup condition is stated in a non-standard H^{-1/2} norm; the proof must verify that this norm is equivalent to the standard dual norm on the fracture faces under the assumed mesh regularity, otherwise the stability constant may deteriorate on highly distorted polytopal elements.

    Authors: We agree that equivalence to the standard dual norm is important for interpretation. Under the shape-regularity assumptions on the polytopal meshes already stated in the paper (bounded aspect ratios and star-shaped elements), the non-standard norm is equivalent to the standard H^{-1/2} dual norm with constants depending only on the mesh-regularity parameter. We will add a brief lemma (or remark) in Section 3.2 that recalls this equivalence and confirms that the inf-sup constant remains uniform. The revision will be made in the next version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on explicit new proofs

full rationale

The paper's central results are the proofs of a discrete Korn inequality (accounting for fractures) and an inf-sup condition between the DDR displacement space and the piecewise-constant multiplier space. These are established directly in the manuscript for the chosen polytopal approximation spaces. The subsequent error analysis and robustness statement for the quasi-incompressible limit follow from these stability estimates in a standard mixed-method fashion. No load-bearing step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or prior self-citation chain; the inequalities are presented as independent mathematical content verified within the work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The scheme rests on standard assumptions from polytopal finite-element theory and mixed formulations for contact; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Discrete Korn inequality holds for the fractured polytopal displacement space
    Required for coercivity of the displacement bilinear form in the presence of fractures.
  • domain assumption Inf-sup condition holds in the chosen non-standard H^{-1/2} norm between the multiplier and displacement spaces
    Needed for stability of the mixed formulation.

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