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An Arbitrary-Order Augmented Lagrangian Discrete De Rham Scheme for Contact Mechanics with Tresca Friction

Jérôme Droniou, Raman Kumar, Roland Masson

arXiv:2610.32471v1math.NAcs.NA

TL;DR

Fracture-contact simulation must address nonsmooth Signorini and Tresca conditions on complex networks, while supporting general polytopal meshes. This paper develops an arbitrary-order augmented Lagrangian DDR scheme with face bubbles and proves an energy-norm error estimate; numerical tests support its rates and robustness.

  • Problem

    Fractured-media simulations need discretisations that handle nonsmooth Signorini and Tresca contact conditions and complex fracture geometries on general meshes.

    Source: §1 · pp. 1–2

  • Method

    An arbitrary-order augmented Lagrangian DDR scheme uses face-bubble enrichment to provide a discrete lifting between the multiplier space and displacement jump.

    Source: §1 · p. 3

  • Results

    O(h^(r+1/2)) energy-norm convergence is established, with r=k+1/2 when ℓ=k and r=k when ℓ=k−1; numerical tests confirm theoretical rates.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Tests on intersecting fractures indicate that higher-order schemes are efficient and the Newton method is robust to the Alart–Curnier projection parameter.

    Source: §1 · p. 3

    The model restricts fracture configurations so every connected component of the matrix touches the outer boundary on a set of positive area.

    Source: §2 · p. 4

Abstract

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We design and analyse an arbitrary-order augmented Lagrangian Discrete de Rham polytopal scheme for contact mechanics in fractured media. The fractures are represented as objects of codimension one in the matrix, a linear elastic medium, and a Tresca friction model is adopted when contact between fracture walls occurs. Both the normal Signorini-type/traction constraint and the Tresca friction law are recast as projection identities, and are combined into a single augmented Lagrangian bilinear form. The displacement is discretised by a DDR space of arbitrary polynomial degree $k\ge 0,$ enriched on the fracture faces by bubble degrees of freedom that ensure stability via the existence of a discrete lifting between the Lagrange multiplier space and the discrete jump of the displacement along the fracture. We establish an abstract error estimate yielding optimal order convergence for the displacement and the Lagrange multiplier variables. Extensive numerical experiments confirm the theoretical analysis and demonstrate the practical accuracy and robustness of the scheme.

1 Introduction

Fracture-contact problems combine nonsmooth Signorini and Tresca conditions with complex geometries that require polytopal discretisations. The paper develops and analyses an arbitrary-order augmented Lagrangian DDR scheme, with a discrete lifting supporting stability and error estimates.

1 Introduction

  • The scheme combines normal Signorini and Tresca friction constraints in an augmented Lagrangian formulation for arbitrary-order polytopal DDR discretisation.It targets static, isotropic, linearly elastic media with contact at matrix–fracture interfaces.Source: §1 · p. 3
  • The analysis requires quasi-uniformity only along the fracture network, allowing coarser meshes in fracture-free matrix regions.This addresses the geometric flexibility needed for complex fracture networks and distorted or heterogeneous cells.Source: §1 · pp. 2–3
  • Face-bubble enrichment provides a discrete lifting between the multiplier space and displacement jump, supporting well-posedness and stability.The lifting is constructed directly from DDR degrees of freedom, without the degree-dependent limitations described for a finite-element lifting.Source: §1 · p. 3
  • O(h^(r+1/2)) energy-norm convergence is established, with r=k+1/2 when ℓ=k and r=k when ℓ=k−1.The estimate is for the scheme with the standard discrete jump.Source: §1 · p. 3
  • A stabiliser-free projected-jump variant permits local multiplier elimination but has only a suboptimal proved error bound.For ℓ=0, it coincides with the mixed schemes of [26].Source: §1 · p. 3
  • Tests on tetrahedral and generalised hexahedral meshes confirm theoretical rates, while intersecting-fracture tests show higher-order efficiency and Newton robustness to the Alart–Curnier parameter.The paper also validates the abstract error estimate through extensive numerical tests.Source: §1 · p. 3

2 Model

The model describes linear elastic deformation in a domain containing a codimension-one fracture network. Contact is expressed through normal and tangential tractions, with Signorini and Tresca conditions recast using projections.

2 Model

  • The fracture network consists of planar fractures, and the model assumes every connected component of the matrix touches the external boundary on a set of positive area.Thus the network cannot disconnect a subregion of the matrix from the outer boundary.Source: §2 · p. 4
  • The model uses a displacement space that permits jumps across fractures and imposes homogeneous Dirichlet boundary conditions.The displacement norm is defined using the gradient over the matrix excluding the fracture network.Source: §2 · p. 4
  • Normal Signorini contact and tangential Tresca friction are recast as Alart–Curnier projection identities and combined in an augmented Lagrangian formulation.The Tresca projection uses a ball-radius parameter and the normal reformulation uses a positive-part operation.Source: §2 · p. 5

3 Scheme

The scheme uses a fracture-conforming polytopal mesh and a face-bubble-enriched DDR displacement space. Its fracture-aware degrees of freedom allow displacement jumps, while the multiplier is represented by piecewise polynomials on fracture faces.

3.1 Mesh

  • The mesh conforms to the fracture network and is assumed quasi-uniform along it, while fracture-free matrix regions may use coarser meshes.The mesh is also subject to regularity requirements, and the Tresca threshold is piecewise constant on fracture faces.Source: §3.1 · p. 6

3.2.1 Displacement space

  • Displacement degrees of freedom are shared by cells on the same connected side of a fracture but independent across the fracture, permitting jumps there.Face-bubble enrichment of the DDR space ensures a discrete lifting between the multiplier space and displacement jump, used for stability and a priori estimates.Source: §3.2.1 · p. 7

3.2.2 Lagrange multiplier space

  • The Lagrange multiplier space consists of piecewise polynomial functions on fracture faces.The allowed degree is ℓ∈{max{0,k−1},k}.Source: §3.2.1 · p. 7, §3.2.2 · p. 8

3.3 Interpolator

The section introduces an interpolator into the discrete space and constructs local reconstructions and jump operators that are assembled into global piecewise-polynomial counterparts.

3.3 Interpolator

  • The interpolator maps continuous functions vanishing on the boundary into the discrete displacement space.Its domain is given as C⁰ functions on Ω\Γ that vanish on ∂Ω.Source: §3.3 · p. 8

3.4 Discrete operators

  • Local gradient, displacement-reconstruction, and jump operators are patched together into global piecewise-polynomial counterparts.The local operators include face and enriched cell gradient reconstructions, as well as the discrete displacement jump on fracture faces.Source: §3.4 · pp. 9–10
  • The enriched cell gradient reconstruction includes a correction associated with the positive side of a fracture.The correction is absent when the face is not on the positive side of the fracture.Source: §3.4 · p. 9
  • The discrete symmetric gradient, divergence, and stress tensor are deduced from the preceding operators.Matrix divergences are taken row-wise.Source: §3.4 · pp. 9–10

3.5 Relation to the standard DDR framework

The enriched operators reduce to standard DDR counterparts when the bubble correction is absent; the enrichment otherwise appears as an additional correction.

3.5 Relation to the standard DDR framework

  • When ℓ≠k, the bubble component is absent, and the space coincides with the vector version of the standard discrete gradient space.In this case, δ_{k,ℓ}=0.Source: §3.5 · p. 10
  • When the bubble correction is absent or the face is not on the positive side of Γ, the local enriched cell operators reduce to their standard DDR counterparts.The stated condition is δ_{k,ℓ}=0 or a face not lying on Γ’s positive side.Source: §3.5 · p. 10
  • The enrichment is reflected in an additional correction term in the reconstruction.Source: §3.5 · p. 10

3.6 Augmented Lagrangian DDR scheme

The section defines the augmented Lagrangian DDR scheme and the discrete norms used in its analysis, with multiplier-jump stabilization and a discrete Korn inequality.

3.6 Augmented Lagrangian DDR scheme

  • The stabilization term penalizes multiplier jumps across interior edges of contact faces and is introduced to ensure optimal convergence.The formulation assumes γ_h=γ₀h_Γ.Source: §3.6 · pp. 10–11
  • The augmented Lagrangian DDR scheme has a unique solution.Source: §3.6 · p. 11

3.7 Discrete norms

  • A discrete Korn inequality is established for the discrete space.For k=0 the proof follows arguments analogous to [26, Theorem 7]; for k≥1 it follows [28, Theorem 3.1] with a face unknown replaced on positive-side fracture faces.Source: §3.7 · p. 11
  • The energy inner product is confirmed to be an inner product, with norm equivalence constants depending only on the stated domain, mesh regularity, and elastic parameters.The listed parameters are Ω, Γ, the mesh regularity parameter, Young modulus E, and Poisson ratio ν.Source: §3.7 · p. 12

4 Main results

The paper gives abstract error bounds for the scheme and derives convergence estimates under regularity assumptions. It also analyzes a modified unstabilized variant and notes locking-free behavior as ν approaches 1.

4 Main results

  • The abstract error estimate for the discrete solution holds when u∈U₀∩C⁰₀(Ω\Γ), with γ₀ sufficiently small and δ sufficiently large.Source: §4 · p. 12
  • For u∈U₀∩H^{k+2}(M) and the stated multiplier regularity, the scheme satisfies the error estimate of Theorem 4.2.The theorem requires γ₀ sufficiently small and δ sufficiently large; its estimate is not legible in the supplied passage.Source: §4 · p. 13
  • Without S_Γ and with the projected jump reconstruction, the modified scheme has a suboptimal O(h^{ℓ+1/2}) error estimate under the theorem’s assumptions.The estimate requires sufficiently small γ₀ and sufficiently large δ.Source: §4 · p. 13
  • The hidden constants in the error estimates do not depend on ν, so the schemes are locking-free as ν→1.For k=0 this is attributed to bubble enrichment; for k≥1, to the existing face displacement unknown.Source: §4 · p. 13

5 Numerical experiments

The experiments test predicted convergence rates on manufactured-solution meshes and assess accuracy and robustness on an intersecting-fracture network. Higher-order choices approach common fracture-level values, while stabilisation and polynomial degree affect the computed fields.

5.1 Tresca friction model with manufactured solution

  • The stabiliser-free projected-jump variant exceeds its predicted energy-norm rates: 1.7 versus 0.5 for k=0 and 2.0–2.1 versus 0.5 for k=1, ℓ=k−1.The multiplier error shows the same pattern, including 2.4–2.6 versus 0.5 for k=1, ℓ=k−1.Source: §5.1 · p. 14, Figure 1 · p. 15
  • On hexa-cut meshes, multiplier rates for k=0 and ℓ=k−1 are non-monotonic at coarse refinements before settling into a regime consistent with tetrahedral results.For k=ℓ=0 with the stabilised jump, the reported sequence is −0.1, 0.6, 1.0, 1.6; for k=1, ℓ=0, it is −0.3, 0.6, 1.0, 1.6.Source: §5.1 · pp. 14–15, Figure 2 · p. 16
  • On tetrahedral meshes, the stabilised scheme reaches energy-norm rates of 2.0–2.1 for k=1, ℓ=k and 3.0 for k=2, ℓ=k, matching predictions.For k=2, ℓ=k−1, the observed rate is 2.6 against 2.5 predicted; for k=1, ℓ=k−1, it is still approaching 1.5.Source: §5.1 · p. 14, Figure 1 · p. 15

5.2 Discrete fracture matrix (DFM) model with intersecting fractures

  • In the realistic test, the scheme is assessed on a homogeneous isotropic material with Tresca friction across a conforming tetrahedral mesh of an intersecting fracture network.The test uses E=4 × 10^9, ν=0.2, and a constant friction threshold g=4×10^6.Source: §5.2 · pp. 16–17, Figure 3 · p. 18
  • On fracture F1, increasing polynomial degree smooths the normal multiplier profile and sharpens contact-boundary resolution without spurious oscillations.The reported comparison progresses from (k, ℓ)=(0,0) to (1,0) and then (1,1).Source: §5.2 · p. 17, Figure 4 · p. 18
  • For the standard-jump scheme, increasing δ from 0 to 0.5 reduces multiplier oscillations, while δ=10 smooths the field and lowers its peak from 7.5 × 10^5 to 5.4 × 10^5.At δ=1, the field remains smooth with smaller additional changes and a maximum of 7 × 10^5.Source: §5.2 · p. 17, Figure 5 · p. 19
  • In the intersecting-fracture test, all four projected-jump discretisations approach common mean normal-jump and multiplier limits, but at different rates in degrees of freedom.The lowest-order choice changes more substantially over the tested meshes, while higher-order schemes are already near the limits on the coarsest meshes shown.Source: §5.2 · p. 17, Figure 7 · p. 20, Figure 8 · p. 21
  • For the projected-jump scheme with ℓ=0, γ0 does not change the converged discrete solution, though it affects semismooth Newton convergence; (1,0) uses fewer degrees of freedom and 5–7 Newton iterations versus 6–8 for (0,0).The reported convergence behavior is similar for δ∈{1,10,100,1000}; δ=0 makes the full-jump scheme behave essentially like the projected-jump scheme.Source: §5.2 · p. 22, Table 1 · p. 22

6 Existence and uniqueness of the discrete solution

The analysis establishes a discrete lifting and supporting stability properties, then proves existence and uniqueness for the discrete scheme.

6.1 Preliminary lemmas

  • The discrete lifting is bounded with constants independent of mesh size, augmentation parameter, and Lamé coefficients for the admissible polynomial degrees.Its constants may depend on the domain, fracture, mesh regularity, and degrees.Source: §6.1 · p. 23
  • The DDR-DOF lifting with face enrichment avoids the cutoff and quadrature-positivity conditions of a prior construction, holding for every k ≥ 0 and admissible ℓ.The compared construction requires degree-by-degree verification and, in three dimensions, caps analysis at k ≤ 12.Source: §6.1 · p. 27

6.2 Existence and uniqueness of the solution

  • Existence of a discrete solution is established using a topological-degree argument.The proof constructs a homotopy to an invertible affine map.Source: §6.2 · pp. 30–31
  • The discrete solution is unique in both displacement and Lagrange multiplier.The proof concludes that the differences between two solutions vanish for both variables.Source: §6.2 · p. 31

7 Proof of the error estimates

The error analysis first derives an abstract bound, then controls its consistency and approximation terms to establish degree-dependent estimates.

7.1 Proof of Theorem 4.1

  • The abstract error bound is obtained by estimating error-equation terms and absorbing controlled contributions into the left-hand side.Young’s inequality bounds consistency and approximation products, while the resulting squared error terms are absorbed.Source: §7.1 · pp. 35–36

7.2 Proof of Theorem 4.2

  • Gradient-reconstruction consistency is proved for displacements in H^(k+2)(M)^3, with the face-bubble enrichment preserving polynomial consistency.The proof reduces the consistency error to interpolation of the residual after exact reproduction of the polynomial part.Source: §7.2 · pp. 38–39
  • Theorem 4.2’s degree-dependent error estimates follow by bounding the abstract estimate’s consistency and approximation terms.The proof invokes gradient consistency, adjoint consistency, and the other approximation estimates before applying the abstract bound.Source: §7.2 · pp. 39–40

8 A stabiliser-free SΓ variant: suboptimal bound by projecting the discrete jump

Projecting the discrete jump onto the multiplier degree removes the stabilizer and its parameter, but weakens the analytical convergence estimate.

8 A stabiliser-free SΓ variant: suboptimal bound by projecting the discrete jump

  • Projecting the degree-(k+1) discrete jump onto the degree-ℓ multiplier space removes the degree mismatch and permits a stabilizer-free scheme.The scheme’s jump-consistency term is then controlled by the modified reconstruction estimate.Source: §8 · pp. 40–41
  • Projecting the jump allows local elimination of Lagrange multiplier degrees of freedom and may reduce the global system size.The projected coupling is diagonal on each fracture face.Source: §8 · pp. 40–41
  • The projected jump reproduces only the degree-ℓ part of a polynomial trace, which underlies its modified consistency estimate.The proof uses the approximation properties of the degree-ℓ projector.Source: §8 · p. 41
  • The stabilizer-free variant has a suboptimal analytical bound of order h^(ℓ+1/2), losing half an order for ℓ = k and a full order for ℓ = k−1.The comparison is with the optimal rate h^(r+1/2) for the standard-jump scheme.Source: §8 · p. 41

9 Conclusion

The conclusion contrasts a stabilized standard-jump scheme with a stabilizer-free projected-jump variant, and reports their practical behavior on intersecting-fracture benchmarks.

9 Conclusion

  • The standard-jump formulation uses an edge-based multiplier stabilizer and yields optimal error estimates, while the projected-jump variant avoids the stabilizer and its parameter.The projected variant also enables local multiplier elimination but has a suboptimal analytical bound.Source: §9 · p. 42
  • Intersecting-fracture benchmarks show that increasing δ mitigates localized normal-multiplier fluctuations, while excessive stabilization suppresses peak magnitude.Higher-order projected-jump schemes capture the contact transition without tuning δ.Source: §9 · p. 42
  • The authors identify sharper analytical error estimates for the projected-jump scheme as an open research question.They also suggest adaptive selection of δ for the standard-jump scheme, following for instance [40].Source: §9 · p. 42
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