On the solution of contact problems with Tresca friction by the semismooth* Newton method

TL;DR

This paper applies a semismooth Newton method to solve contact problems with Tresca friction, demonstrating superlinear convergence and mesh independence through numerical tests.

Contribution

It introduces a novel application of the semismooth* Newton method to Tresca friction contact problems, handling multi-valued parts via Morduchovich coderivative.

Findings

Superlinear convergence observed in numerical tests

Method is mesh independent

Effective handling of multi-valued friction conditions

Abstract

An equilibrium of a linear elastic body subject to loading and satisfying the friction and contact conditions can be described by a variational inequality of the second kind and the respective discrete model attains the form of a generalized equation. To its numerical solution we apply the semismooth* Newton method by Gfrerer and Outrata (2019) in which, in contrast to most available Newton-type methods for inclusions, one approximates not only the single-valued but also the multi-valued part. This is performed on the basis of limiting (Morduchovich) coderivative. In our case of the Tresca friction, the multi-valued part amounts to the subdifferential of a convex function generated by the friction and contact conditions. The full 3D discrete problem is then reduced to the contact boundary. Implementation details of the semismooth* Newton method are provided and numerical tests demonstrate its superlinear convergence and mesh independence.