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A Barrier-Regularized Symmetric Nitsche Method for the Signorini Problem
Peter Hansbo, Mats G. Larson
TL;DR
The paper addresses how to regularize and discretize the nonsmooth Signorini contact problem while controlling the smoothing–mesh interaction. It derives a barrier-regularized symmetric Nitsche method, establishes solvability and central-path properties, and shows that suitable parameter balancing preserves energy-norm rates, although contact-set resolution can remain poor.
Problem
The Signorini problem combines an unknown contact region with nonsmooth complementarity conditions, raising the question of how smoothing should scale with mesh size without degrading finite element accuracy.
Method
A logarithmic barrier is applied to the nonnegative slack variable in an augmented-Lagrangian symmetric Nitsche formulation, then the slack is eliminated to obtain a smooth positive-part operator and central-path contact law.
Results
The method is uniquely solvable with a symmetric positive definite Newton matrix for sufficiently large Nitsche parameter, while the continuous central path has a unique strictly feasible minimizer and the discrete method is s-uniformly quasi-optimal.
Takeaways & Limitations
Balancing the barrier and mesh parameters can preserve the available energy-norm convergence rate, but rate-preserving smoothing may still poorly resolve the contact set.
Takeaways & Limitations
Global uniform Sobolev regularity is conditional, parameter-uniform regularity above H2 near the free boundary is open, and contact-set convergence is not established.
Abstract
from arXiv · showhide
We introduce and analyze a barrier-regularized symmetric Nitsche method for the scalar Signorini problem. Applying a logarithmic barrier to the nonnegative slack variable in an augmented Lagrangian and then eliminating that variable yields a smooth positive-part operator and the perturbed complementarity relation on a primal-dual central path, with barrier parameter $μ=γs$. For every $s>0$, the discrete problem is continuously differentiable, uniquely solvable, and has a symmetric positive definite Newton matrix when the Nitsche parameter is sufficiently large. Under a mild barrier-feasibility condition, the continuous logarithmic energy has a unique minimizer $u_μ$ for every $μ>0$, its gap is positive almost everywhere, and $\|u-u_μ\|_{H^1(Ω)}\lesssimμ^{1/2}$. Under the Sobolev regularity used for finite element approximation, this minimizer satisfies the \(L^2\) central-path boundary law. If the obstacle is locally the trace of an $H^2$-function, we also prove a uniform local $H^2$ bound in the interior of each planar contact face, on neighborhoods that may contain a free-boundary point of the limiting Signorini solution. The method is exactly consistent and quasi-optimal relative to $u_μ$, with constants independent of $s$. If $u_μ$ is uniformly bounded in $H^r(Ω)$, $3/2<r\leq k+1$, then $\|u-u_h\|_{H^1(Ω)}\lesssim h^{r-1}\|u_μ\|_{H^r(Ω)}+μ^{1/2}$; hence the sufficient balance $s\lesssim h^{2r-1}$ preserves the available energy-norm rate. We also derive rates for discrete penetration and the complementarity residual. Numerical experiments with $P_1$ and $P_2$ elements on structured and unstructured meshes support the predicted rates and the local Newton theory, while showing that a rate-preserving smoothing may still resolve the contact set poorly.
1 Introduction
The paper develops a barrier-regularized symmetric Nitsche method for Signorini contact, addressing smoothing, solvability, regularity, discretization, and parameter selection.
- Motivation: The Signorini problem models unilateral contact with an unknown contact region and produces a nonsmooth nonlinear system after finite element discretization.
- Barrier formulation: Logarithmic-barrier elimination of the nonnegative slack variable yields a smooth positive-part operator and a primal-dual central-path complementarity law.The barrier parameter satisfies μ = γs.
- Analysis: For every s > 0, the discrete residual is continuously differentiable and uniquely solvable, while sufficiently large Nitsche parameters make the Newton matrix symmetric positive definite.For fixed mesh and positive s, the local Newton rate is quadratic, although the sufficient convergence radius deteriorates as s → 0.
- Discretization: The method is exactly consistent and quasi-optimal relative to the continuous central path, with regularity-dependent balances preserving the available energy-norm rate.A sufficient balance is s ≲ h^(2r−1); discrete penetration and complementarity residuals satisfy h^(r−1/2) bounds.
- Numerical assessment: Numerical experiments with P1 and P2 elements support predicted energy-norm rates and local Newton theory but show that energy-rate-preserving smoothing can poorly resolve the contact set.
- Contribution: The contribution integrates barrier elimination with symmetric Nitsche contact and provides an s-uniform discretization analysis not present in the cited barrier-contact work.
2 The Model Problem
The model is a scalar Signorini problem on a bounded Lipschitz domain with Dirichlet and contact boundary parts, an obstacle, and unilateral variational-inequality constraints.
- Setting: The domain is bounded, polygonal or polyhedral, and Lipschitz, with relatively open disjoint boundary portions ΓD and ΓC.The analysis allows d ∈ {2,3}.
- Boundary conditions: Homogeneous Dirichlet conditions are imposed on ΓD, while nonhomogeneous data can be treated by standard lifting.An additional Neumann boundary part can be included immediately.
- Admissibility: The obstacle g belongs to H^(1/2)(ΓC), and the admissible convex set is assumed nonempty.
- Variational formulation: The scalar Signorini solution is equivalently characterized as the unique solution of a variational inequality over the admissible set.
- Strong contact notation: When the normal flux has a pointwise representative, the contact reaction is expressed through pointwise unilateral conditions and the positive-part reformulation.The normal flux is p = ∇u·n, and −p is the nonnegative contact pressure.
3 The Finite Element Method
The finite element method uses conforming boundary-fitted spaces and derives a smooth symmetric Nitsche contact form by logarithmic-barrier elimination of a slack variable.
- Finite element setting: The discretization uses continuous piecewise-polynomial conforming spaces on shape-regular, quasiuniform, boundary-fitted meshes.The polynomial degree is k and the mesh parameter is h.
- Augmented-Lagrangian formulation: The augmented-Lagrangian derivation introduces a nonnegative L2 slack variable representing the gap and adopts λ = σn(u) as the multiplier convention.The contact pressure is −λ.
- Barrier regularization: Relaxing the slack constraint with a logarithmic barrier makes the reduced equations replace the nonsmooth positive part by the smooth operator ϕ+,s.The barrier acts through the stationary value of the slack variable.
- Slack elimination: Pointwise elimination produces the unique positive stationary root, whose zero-regularization limit recovers the barrier-free slack z = [−w]+.The regularized max operator is precisely the slack variable generated by the logarithmic barrier.
- Central path: The regularized contact law is equivalent to ρ(v) < 0, p < 0, and ρ(v)p = μ = γs, perturbing complementarity into a central-path relation.For sufficiently regular functions, both gap and contact pressure are strictly positive for s > 0.
4 Newton’s Method
The Newton analysis establishes differentiability, coercivity, unique discrete solvability, positive definiteness, and quadratic local convergence for the regularized finite element method.
- Linearization: For s > 0, the regularized contact operator is smooth, enabling a continuously differentiable finite element form and an explicit Newton linearization.The analysis also derives a Taylor remainder estimate.
- Newton iteration: The Newton iteration updates uh,n = uh,n−1 + δn, with δn obtained by solving the linearized discrete system.
- Coercivity and solvability: Coercivity is uniform in v, s, and h when the Nitsche parameter satisfies γ0 > CI, yielding strong monotonicity and unique solvability.The proof combines the coercivity estimate with the Poincaré inequality.
- Linear algebra: The Newton system is symmetric positive definite at every iterate for every s > 0, so each step is well defined and compatible with conjugate-gradient solution.The underlying discrete functional is strongly convex uniformly in s.
- Local convergence: For fixed h and s > 0, Newton convergence is quadratic, but the sufficient local convergence radius deteriorates as s → 0 and is not h-uniform.Solvability and Newton-matrix definiteness do not deteriorate in the nonsmooth limit.
5 Error Estimates
The analysis separates regularization from finite element error through exact central-path consistency, then derives regularity-dependent energy estimates and diagnostics. It also identifies geometric and regularity limitations, including poor contact-set resolution despite preserved energy rates.
- Central-path consistency: The method is exactly consistent with the continuous central-path problem at μ = γs, so its finite element estimate contains no regularization consistency term.Regularization enters through the continuous estimate between uμ and u.
- Continuous central path: Under barrier feasibility, the continuous logarithmic problem has a unique minimizer for every μ > 0, with positive gap almost everywhere and an L2 central-path boundary law under Sobolev regularity.Barrier feasibility is weaker than uniform separation and can permit the gap to approach zero at the Dirichlet–contact interface.
- Regularity: The central path is uniformly H2 locally inside planar contact faces, even on neighborhoods containing free-boundary points, when the obstacle is locally the trace of an H2 function.The estimate is uniform in μ but excludes Dirichlet–contact interfaces and polyhedral edges.
- Limitations: The local regularity result does not replace the global Sobolev assumption, because corners, edges, and mixed-boundary interfaces can determine the available global exponent.Uniform estimates above H2 near the free boundary require a separate parameter-uniform analysis.
6 Numerical Examples
Numerical experiments with P1 and P2 elements support the predicted regularization, energy-error, Newton, pressure, and mesh-robustness behavior, while revealing that energy-rate-preserving smoothing can poorly resolve contact features.
- Regularization Error: The regularization difference decays faster than the discretization error at the sufficient balances α = 3 for P1 and α = 5 for P2.Observed finest-mesh rates are 1.45 for P1 and 3.06 for P2.
- Newton Performance: Full-balance runs exhibit approximately quadratic local Newton convergence, with residual-based orders of 1.9–2.0 before round-off dominates.Undamped full Newton steps were used in the reported residual histories.
- Exact-Solution Convergence: The exact-solution tests recover near-first-order P1 energy convergence and second-order P2 convergence for the smooth example, while singular Example B limits P2 to a rate near 3/2.For P1, finest-mesh rates are 1.07 and 1.04; P2 gives 2.29 for Example A and decreases to 1.50 for Example B.
- Smoothing-Parameter Effects: At the sufficient P1 balance s = h^3/4, the energy rate is preserved but L2 error is polluted; reducing s restores the L2 rate while leaving energy error discretization-dominated.The L2 rate 2 is recovered for s ∼ h^5/4, whereas further reduction does not materially change the energy error.
- Contact Diagnostics: For the smooth example, contact-pressure errors converge at order h^k, while the singular P2 example shows decreasing preasymptotic rates as the mesh resolves its singularity.Observed smooth-example rates are 1.10–1.13 for P1 and 2.03–2.21 for P2; Example B decreases from 2.06 to 1.32.
- Contact Diagnostics: The sufficient balance preserves energy convergence but leaves contact poorly resolved: three or four additional powers of h are needed for active-set accuracy within one element.At that balance, active-set measures are 0.047 for P1 and 0.62 for P2, with artificial gaps 3.4·10^-2 and 1.1·10^-3.
- Mesh Robustness: On unstructured meshes, Newton counts remain between 7 and 12 and contact-pressure errors are similar to uniform-mesh results.With γ0 = 10 and 20, the method remains away from its coercivity threshold on the tested meshes.
7 Concluding Remarks
The paper interprets smoothing as a logarithmic-barrier central path, establishes well-posedness and local regularity, and separates regularization from discretization. Its computations support energy-rate predictions while showing that contact diagnostics can remain poorly resolved.
- Eliminating the barrier slack variable yields a perturbed complementarity relation with μ = γs and makes the finite element method exactly consistent with a continuous central-path problem.
- Under barrier feasibility, the continuous logarithmic energy has a unique minimizer for every μ > 0, with almost-everywhere positive gap and controlled distance to the Signorini solution.The distance estimate does not require reciprocal-gap or normal-flux regularity.
- Sobolev regularity sufficient for finite element approximation yields the L2(ΓC) central-path Euler equation and normal flux, supporting the discretization analysis.
- The computations support energy-norm estimates and local Newton behavior, but rate-preserving smoothing can leave artificial gaps, degrade L2(Ω) error, or poorly identify contact sets.The smoothing parameter should therefore be chosen according to the quantity of interest.
- Uniform local H2 regularity is proved on interiors of planar contact faces, including neighborhoods that may contain a free-boundary point of the limiting solution.
- Global uniform Sobolev bounds remain conditional near interfaces, corners, and edges, while parameter-uniform regularity above H2 near the limiting free boundary remains open.The analysis also does not provide L2(Ω) estimates or contact-set guarantees.
A Uniform Local Regularity Details
The appendix establishes uniform local H2 control for barrier-regularized problems by combining uniform Moreau estimates, monotone boundary-law difference quotients, and a limiting argument as the Moreau parameter vanishes.
- Uniform bounds: A uniform Moreau-problem bound holds for sufficiently small ε, with ε0 depending on μ0 and trace and Poincaré constants.
- Uniform bounds: The constants in the uniform estimates depend on domain and data quantities but are independent of μ and ε.
- Uniform bounds: The logarithmic lower bound and quadratic minimization provide the uniform parameter estimates used to establish the appendix bounds.
- Local H2 estimate: A local estimate for the monotone boundary law controls derivatives under the geometric and data assumptions of Proposition 5.4.
- Local H2 estimate: Tangential translations and difference quotients control second derivatives containing tangential derivatives on nested flat half-cylinders inside the contact-face neighborhood.
- Local H2 estimate: The boundary term is pointwise nonnegative, so Cauchy–Schwarz and Young’s inequalities complete the local estimate without requiring derivatives of f.
- Limit passage: The Moreau envelopes converge pointwise to the logarithmic barrier, while trace compactness, Fatou’s lemma, and recovery establish Mosco liminf and recovery properties.
- Limit passage: Uniqueness and strong convexity yield weak convergence of the Moreau minimizers to uμ and convergence of minimum values.