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Generalized Newton methods for the 2D-Signorini contact problem with friction in function space

Published online by Cambridge University Press:  15 August 2005

Karl Kunisch  and
Georg Stadler
Karl Kunisch
Affiliation:
Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010 Graz, Austria. karl.kunisch@uni-graz.at; ge.stadler@uni-graz.at
Georg Stadler
Affiliation:
Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010 Graz, Austria. karl.kunisch@uni-graz.at; ge.stadler@uni-graz.at
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Abstract

The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented Lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas.

Keywords

Signorini contact problemsCoulomb and Tresca frictionlinear elasticitysemi-smooth Newton methodFenchel dualaugmented Lagrangianscomplementarity systemactive sets.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 4 , July 2005 , pp. 827 - 854
Copyright
© EDP Sciences, SMAI, 2005

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