A priori error for unilateral contact problems with augmented Lagrange multipliers and IsoGeometric Analysis

TL;DR

This paper extends the a priori error analysis for unilateral contact problems using augmented Lagrangian methods combined with IsoGeometric Analysis, providing optimal error estimates and numerical validation in 2D and 3D scenarios.

Contribution

It introduces a new stable space pairing for NURBS and B-Spline functions and derives optimal a priori error estimates inspired by Nitsche's method.

Findings

Optimal error estimates are established for the proposed method.

Numerical experiments confirm theoretical predictions in 2D and 3D cases.

The approach effectively handles small and large deformations with different element types.

Abstract

The aim of the present work is to extend the a priori error for contact problems with an augmented Lagrangian method. We focus on unilateral contact problem without friction between an elastic body and a rigid one. We consider the pushforward of a NURBS space of degree $p$ for the displacement and the pushforward of a B-Spline space of degree $p-2$ for the Lagrange multipliers. This specific choice of space is a stable couple of spaces. An optimal a priori error estimate inspired from the Nitsche's method theory is provided and compared to the regularity of the solution. We perform a numerical validation with two- and three-dimensions in small and large deformations with $N2/S0$ and $N3/S1$ elements.