Finite Element Methods for Elastic Contact: Penalty and Nitsche

TL;DR

This paper compares penalty and Nitsche's methods for elastic contact problems using finite element analysis, demonstrating that Nitsche's method achieves optimal convergence while penalty methods do not.

Contribution

It provides a simple derivation of Nitsche's method from the penalty method and analyzes their convergence properties in elastic contact simulations.

Findings

Nitsche's method is optimally convergent.

Penalty method cannot achieve optimal convergence.

A simple derivation links Nitsche's method to a consistency correction of the penalty method.

Abstract

We consider two methods for treating elastic contact problems with the finite element method; the penalty method and Nitsche's method. For the penalty method we discuss how the penalty parameter should be chosen. Both the theoretical analysis and numerical examples show that an optimal convergence rate cannot be achieved. The method is contrasted to that of Nitsche which is optimally convergent. We also give the derivation of Nitsche's method by a very simple consistency correction of the penalty method.