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

Karl Kunisch; Georg Stadler

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 39, Issue: 4, page 827-854
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Kunisch, Karl, and Stadler, Georg. "Generalized Newton methods for the 2D-Signorini contact problem with friction in function space." ESAIM: Mathematical Modelling and Numerical Analysis 39.4 (2010): 827-854. <http://eudml.org/doc/194288>.

@article{Kunisch2010,
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. },
author = {Kunisch, Karl, Stadler, Georg},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Signorini contact problems; Coulomb and Tresca friction; linear elasticity; semi-smooth Newton method; Fenchel dual; augmented Lagrangians; complementarity system; active sets.; active sets},
language = {eng},
month = {3},
number = {4},
pages = {827-854},
publisher = {EDP Sciences},
title = {Generalized Newton methods for the 2D-Signorini contact problem with friction in function space},
url = {http://eudml.org/doc/194288},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Kunisch, Karl
AU - Stadler, Georg
TI - Generalized Newton methods for the 2D-Signorini contact problem with friction in function space
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 4
SP - 827
EP - 854
AB - 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.
LA - eng
KW - Signorini contact problems; Coulomb and Tresca friction; linear elasticity; semi-smooth Newton method; Fenchel dual; augmented Lagrangians; complementarity system; active sets.; active sets
UR - http://eudml.org/doc/194288
ER -

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