# A three-field augmented Lagrangian formulation of unilateral contact problems with cohesive forces

• Volume: 44, Issue: 2, page 323-346
• ISSN: 0764-583X

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## Abstract

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We investigate unilateral contact problems with cohesive forces, leading to the constrained minimization of a possibly nonconvex functional. We analyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmented Lagrangian, and sufficient conditions for the existence of a local saddle-point are derived. Then, we derive and analyze mixed finite element approximations to the stationarity conditions of the three-field augmented Lagrangian. The finite element spaces for the bulk displacement and the Lagrange multiplier must satisfy a discrete inf-sup condition, while discontinuous finite element spaces spanned by nodal basis functions are considered for the unilateral contact variable so as to use collocation methods. Two iterative algorithms are presented and analyzed, namely an Uzawa-type method within a decomposition-coordination approach and a nonsmooth Newton's method. Finally, numerical results illustrating the theoretical analysis are presented.

## How to cite

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Doyen, David, Ern, Alexandre, and Piperno, Serge. "A three-field augmented Lagrangian formulation of unilateral contact problems with cohesive forces." ESAIM: Mathematical Modelling and Numerical Analysis 44.2 (2010): 323-346. <http://eudml.org/doc/250774>.

@article{Doyen2010,
abstract = { We investigate unilateral contact problems with cohesive forces, leading to the constrained minimization of a possibly nonconvex functional. We analyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmented Lagrangian, and sufficient conditions for the existence of a local saddle-point are derived. Then, we derive and analyze mixed finite element approximations to the stationarity conditions of the three-field augmented Lagrangian. The finite element spaces for the bulk displacement and the Lagrange multiplier must satisfy a discrete inf-sup condition, while discontinuous finite element spaces spanned by nodal basis functions are considered for the unilateral contact variable so as to use collocation methods. Two iterative algorithms are presented and analyzed, namely an Uzawa-type method within a decomposition-coordination approach and a nonsmooth Newton's method. Finally, numerical results illustrating the theoretical analysis are presented. },
author = {Doyen, David, Ern, Alexandre, Piperno, Serge},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Unilateral contact; cohesive forces; augmented Lagrangian; mixed finite elements; decomposition-coordination method; Newton's method; mixed finite elements; nonsmooth Newton's method; convergence},
language = {eng},
month = {3},
number = {2},
pages = {323-346},
publisher = {EDP Sciences},
title = {A three-field augmented Lagrangian formulation of unilateral contact problems with cohesive forces},
url = {http://eudml.org/doc/250774},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Doyen, David
AU - Ern, Alexandre
AU - Piperno, Serge
TI - A three-field augmented Lagrangian formulation of unilateral contact problems with cohesive forces
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 44
IS - 2
SP - 323
EP - 346
AB - We investigate unilateral contact problems with cohesive forces, leading to the constrained minimization of a possibly nonconvex functional. We analyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmented Lagrangian, and sufficient conditions for the existence of a local saddle-point are derived. Then, we derive and analyze mixed finite element approximations to the stationarity conditions of the three-field augmented Lagrangian. The finite element spaces for the bulk displacement and the Lagrange multiplier must satisfy a discrete inf-sup condition, while discontinuous finite element spaces spanned by nodal basis functions are considered for the unilateral contact variable so as to use collocation methods. Two iterative algorithms are presented and analyzed, namely an Uzawa-type method within a decomposition-coordination approach and a nonsmooth Newton's method. Finally, numerical results illustrating the theoretical analysis are presented.
LA - eng
KW - Unilateral contact; cohesive forces; augmented Lagrangian; mixed finite elements; decomposition-coordination method; Newton's method; mixed finite elements; nonsmooth Newton's method; convergence
UR - http://eudml.org/doc/250774
ER -

## References

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