The Penalty Free Nitsche Method and Nonconforming Finite Elements for the Signorini Problem

TL;DR

This paper introduces a novel Nitsche method for the Signorini contact problem that avoids penalty terms by reformulating contact conditions, using nonconforming finite elements, and providing optimal error estimates.

Contribution

It presents a new Nitsche-based approach for contact problems that expresses contact conditions via a nonlinear function, eliminating the need for penalty parameters.

Findings

Achieved optimal error estimates in the energy norm.

Developed a penalty-free Nitsche method for unilateral contact problems.

Utilized nonconforming finite elements for discretization.

Abstract

We design and analyse a Nitsche method for contact problems. Compared to the seminal work of Chouly and Hild (A Nitsche-based method for unilateral contact problems: numerical analysis. SIAM J. Numer. Anal. 51 (2013), no. 2) our method is constructed by expressing the contact conditions in a nonlinear function for the displacement variable instead of the lateral forces. The contact condition is then imposed using the nonsymmetric variant of Nitsche's method that does not require a penalty term for stability. Nonconforming piecewise affine elements are considered for the bulk discretization. We prove optimal error estimates in the energy norm.