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Hybridized Augmented Lagrangian Methods for Contact Problems

Erik Burman, Peter Hansbo, Mats G. Larson

TL;DR

This work addresses friction-free contact between two elastic bodies by developing a hybridized Nitsche finite element method embedded in a Rockafellar-inspired augmented Lagrangian framework that reformulates the contact into a nonlinear variational equality. The key idea is an interstitial hybrid layer carrying a displacement variable that decouples $\Omega_1$ and $\Omega_2$, enabling independent interface discretization and even physical interfacial models such as membranes ($\Omega_0$) or plates. The authors establish coercivity, continuity, stability, and a best-approximation error framework, and validate the method with numerical examples including a 2D Hertz problem and coupled plate/membrane interfaces. The approach offers substantial flexibility for interface modeling, reduces mesh-intersection constraints, and has potential for broader multiphysics applications in contact mechanics.

Abstract

This paper addresses the problem of friction-free contact between two elastic bodies. We develop an augmented Lagrangian method that provides computational convenience by reformulating the contact problem as a nonlinear variational equality. To achieve this, we propose a Nitsche-based method incorporating a hybrid displacement variable defined on an interstitial layer. This approach enables complete decoupling of the contact domains, with interaction occurring exclusively through the interstitial layer. The layer is independently approximated, eliminating the need to handle intersections between unrelated meshes. Additionally, the method supports introducing an independent model on the interface, which we leverage to represent a membrane covering one of the bodies. We present the formulation of the method, establish stability and error estimates, and demonstrate its practical utility through illustrative numerical examples.

Hybridized Augmented Lagrangian Methods for Contact Problems

TL;DR

This work addresses friction-free contact between two elastic bodies by developing a hybridized Nitsche finite element method embedded in a Rockafellar-inspired augmented Lagrangian framework that reformulates the contact into a nonlinear variational equality. The key idea is an interstitial hybrid layer carrying a displacement variable that decouples and , enabling independent interface discretization and even physical interfacial models such as membranes () or plates. The authors establish coercivity, continuity, stability, and a best-approximation error framework, and validate the method with numerical examples including a 2D Hertz problem and coupled plate/membrane interfaces. The approach offers substantial flexibility for interface modeling, reduces mesh-intersection constraints, and has potential for broader multiphysics applications in contact mechanics.

Abstract

This paper addresses the problem of friction-free contact between two elastic bodies. We develop an augmented Lagrangian method that provides computational convenience by reformulating the contact problem as a nonlinear variational equality. To achieve this, we propose a Nitsche-based method incorporating a hybrid displacement variable defined on an interstitial layer. This approach enables complete decoupling of the contact domains, with interaction occurring exclusively through the interstitial layer. The layer is independently approximated, eliminating the need to handle intersections between unrelated meshes. Additionally, the method supports introducing an independent model on the interface, which we leverage to represent a membrane covering one of the bodies. We present the formulation of the method, establish stability and error estimates, and demonstrate its practical utility through illustrative numerical examples.

Paper Structure

This paper contains 20 sections, 2 theorems, 58 equations, 14 figures.

Table of Contents

  1. Introduction
  2. A Contact Model Problem
  3. Hybridized Problem Formulation
  4. Governing Equations for the Elastic Bodies.
  5. The Equality and Inequality Constraints.
  6. A Hybrid Nitsche Finite Element Method
  7. Function and Finite Element Spaces.
  8. Augmented Lagrangian Formulation.
  9. Optimality Equations
  10. Error Estimates
  11. Properties of the Forms
  12. Estimates
  13. Numerical Examples
  14. A 2D Hertz Problem
  15. Contact Problems with Model Coupling
  16. ...and 5 more sections

Key Result

Proposition 3.1

The discrete problem (eq:fem) admits a unique solution $\boldsymbol u_h \in W_h$ such that

Figures (14)

  • Figure 1: Example of the different domains $\Omega_i$.
  • Figure 2: Geometry and meshes used for the Hertz problem.
  • Figure 3: Close up of the solution with 50 constants.
  • Figure 4: Cose up of the solution with 100 constants.
  • Figure 5: Close up of the solution with 1000 constants.
  • ...and 9 more figures

Theorems & Definitions (7)

  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1