[Paper Review] A stabilized finite element formulation of non-smooth contact

Ghadir Haikal|ArXiv.org|Jun 2, 2009

Contact Mechanics and Variational Inequalities86 references3 citations

TL;DR

This paper presents a stabilized finite element formulation for non-smooth contact problems using a node-to-surface gap function and discontinuous Galerkin-based stabilization to ensure accurate pressure transfer across interfaces. The method effectively handles sharp corners and nonlinear surfaces, guarantees energy conservation via a Lagrange multiplier approach, and enables robust coupling of non-conforming meshes in hyperelastic and contact problems.

ABSTRACT

The computational modeling of many engineering problems using the Finite Element method involves the modeling of two or more bodies that meet through an interface. The interface can be physical, as in multi-physics and contact problems, or purely numerical, as in the coupling of non-conforming meshes. The most critical part of the modeling process is to ensure geometric compatibility and a complete transfer of surface tractions between the different components at the connecting interfaces. Popular contact modeling techniques rely on geometric projections to detect and resolve overlapping or mass interpenetration between two or more contacting bodies. Such approaches have been shown to have two major drawbacks: they are not suitable for contact at highly nonlinear surfaces and sharp corners where smooth normal projections are not feasible, and they fail to guarantee a complete and accurate transfer of pressure across the interface. This dissertation presents a novel formulation for the modeling of contact problems that possesses the ability to resolve complicated contact scenarios effectively, while being simpler to implement and more widely applicable than currently available methods. We show that the formulation boils down to a node-to-surface gap function that works effectively for non-smooth contact. The numerical implementation using the midpoint rule shows the need to guarantee the conservation of the total energy during impact, for which a Lagrange multiplier method is used. We propose a local enrichment of the interface and a simple stabilization procedure based on the discontinuous Galerkin method to guarantee an accurate transfer of the pressure field. The result is a robust interface formulation for contact problems and the coupling of non-conforming meshes.

Motivation & Objective

• Address the limitations of traditional contact methods that rely on smooth normal projections, which fail at sharp corners and highly nonlinear surfaces.
• Develop a robust interface formulation capable of accurately transferring tractions across non-smooth and non-conforming interfaces.
• Ensure energy conservation during impact by employing a Lagrange multiplier method in time integration.
• Provide a stable and consistent formulation for coupling non-conforming meshes in hyperelastic problems.
• Enable effective simulation of complex contact scenarios, including sliding and separation, with improved numerical stability and accuracy.

Proposed method

• Formulates contact constraints using an oriented volume approach based on a node-to-surface gap function, avoiding reliance on smooth normal projections.
• Employs the midpoint rule for time integration with a Lagrange multiplier method to enforce energy conservation during impact events.
• Introduces local enrichment of the interface to improve approximation quality at contact boundaries.
• Applies a discontinuous Galerkin (DG)-based stabilization procedure to ensure consistent and accurate transfer of pressure across the interface.
• Uses a Jacobian and Hessian formulation derived from the contact constraint function to enable consistent linearization in Newton-Raphson solution schemes.
• Applies the Multi-Point-Constraint (MPC) method for coupling non-conforming meshes, with validation via patch tests and convergence studies.

Experimental results

Research questions

• RQ1 How can contact constraints be formulated to remain effective at sharp corners and highly nonlinear surfaces where traditional normal projections fail?
• RQ2 What stabilization technique ensures accurate and consistent transfer of surface tractions across non-conforming or non-smooth interfaces?
• RQ3 How can energy conservation be guaranteed during impact in dynamic contact simulations?
• RQ4 Can a node-to-surface gap function be effectively combined with DG stabilization to improve robustness in contact and coupling problems?
• RQ5 How does the proposed formulation perform in terms of convergence and consistency for non-conforming mesh coupling in hyperelastic materials?

Key findings

• The proposed node-to-surface gap function successfully resolves contact at non-smooth geometries, including sharp corners, where conventional projection-based methods fail.
• The DG-based stabilization ensures accurate and consistent transfer of pressure across the interface, preventing spurious oscillations and improving solution quality.
• Energy conservation is maintained during impact through the use of a Lagrange multiplier method in the midpoint rule time integration scheme.
• The formulation passes the patch test and demonstrates optimal convergence rates in numerical examples, confirming consistency and stability.
• The method effectively handles sliding contact and complex 3D contact scenarios, such as the double-cantilever beam test, with robust convergence and accurate stress transfer.
• The linearization of the contact constraint yields a symmetric Hessian, which supports efficient and robust Newton-Raphson solution procedures.