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

David Doyen; Alexandre Ern; Serge Piperno

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

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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  1. P. Alart and A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Engrg.92 (1991) 353–375.  Zbl0825.76353
  2. K.J. Bathe and F. Brezzi, Stability of finite element mixed interpolations for contact problems. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.12 (2001) 167–183.  Zbl1097.74054
  3. D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods. Athena Scientific (1982).  Zbl0572.90067
  4. D.P. Bertsekas, Nonlinear Programming. Athena Scientific (1999).  
  5. B. Bourdin, G.A. Francfort and J.-J. Marigo, The variational approach to fracture. J. Elasticity91 (2008) 5–148.  Zbl1176.74018
  6. L. Champaney, J.-Y. Cognard and P. Ladevèze, Modular analysis of assemblages of three-dimensional structures with unilateral contact conditions. Comput. Struct.73 (1999) 249–266.  Zbl1049.74562
  7. Z. Chen, On the augmented Lagrangian approach to Signorini elastic contact problem. Numer. Math.88 (2001) 641–659.  Zbl1047.74054
  8. P.G. Ciarlet, Mathematical elasticity, Vol.I: Three-dimensional elasticity, Studies in Mathematics and its Applications20. North-Holland Publishing Co., Amsterdam (1988).  Zbl0648.73014
  9. F.H. Clarke, Optimization and nonsmooth analysis, Classics in Applied Mathematics5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA, second edition (1990).  Zbl0696.49002
  10. Z. Denkowski, S. Migórski and N.S. Papageorgiou, An introduction to nonlinear analysis: applications. Kluwer Academic Publishers, Boston, USA (2003).  Zbl1054.47001
  11. I. Ekeland and R. Témam, Convex analysis and variational problems, Classics in Applied Mathematics. 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (1999).  Zbl0939.49002
  12. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences159. Springer-Verlag, New York, USA (2004).  Zbl1059.65103
  13. M. Fortin and R. Glowinski, Augmented Lagrangian methods: Applications to the numerical solution of boundary value problems, Studies in Mathematics and its Applications15. North-Holland Publishing Co., Amsterdam (1983).  Zbl0525.65045
  14. M. Frémond, Contact with adhesion, in Topics in nonsmooth mechanics, Birkhäuser, Basel, Switzerland (1988) 157–185.  
  15. R. Glowinski and P. Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM Studies in Applied Mathematics9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (1989).  Zbl0698.73001
  16. J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, in Handbook of numerical analysisIV, Amsterdam, North-Holland (1996) 313–485.  Zbl0873.73079
  17. P. Hauret and P. Le Tallec, A discontinuous stabilized mortar method for general 3d elastic problems. Comput. Methods Appl. Mech. Engrg.196 (2007) 4881–4900.  Zbl1173.74424
  18. P. Hild and P. Laborde, Quadratic finite element methods for unilateral contact problems. Appl. Numer. Math.41 (2002) 401–421.  Zbl1062.74050
  19. S. Hüeber and B.I. Wohlmuth, An optimal a priori error estimate for nonlinear multibody contact problems. SIAM J. Numer. Anal.43 (2005) 156–173 (electronic).  Zbl1083.74047
  20. N. Kikuchi and J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics8. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (1988).  Zbl0685.73002
  21. D. Kinderlehrer, Remarks about Signorini's problem in linear elasticity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)8 (1981) 605–645.  Zbl0482.73017
  22. K. Kunisch and G. Stadler, Generalized Newton methods for the 2D-Signorini contact problem with friction in function space. ESAIM: M2AN39 (2005) 827–854.  Zbl1330.74132
  23. P. Ladevèze, Nonlinear Computational Structural Mechanics – New Approaches and Non-Incremental Methods of Calculation. Springer-Verlag (1999).  Zbl0912.73003
  24. J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applicationsI, Die Grundlehren der mathematischen Wissenschaften, Band181. Springer-Verlag, New York, USA (1972).  
  25. E. Lorentz, A mixed interface finite element for cohesive zone models. Comput. Methods Appl. Mech. Engrg.198 (2008) 302–317.  Zbl1194.74438
  26. M. Marcus and V.J. Mizel, Every superposition operator mapping one Sobolev space into another is continuous. J. Funct. Anal.33 (1979) 217–229.  Zbl0418.46024
  27. M. Moussaoui and K. Khodja, Régularité des solutions d'un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan. Commun. Partial Differ. Equ.17 (1992) 805–826.  Zbl0806.35049
  28. L. Qi and J. Sun, A nonsmooth version of Newton's method. Math. Program.58 (1993) 353–367.  Zbl0780.90090
  29. L. Slimane, A. Bendali and P. Laborde, Mixed formulations for a class of variational inequalities. ESAIM: M2AN38 (2004) 177–201.  Zbl1100.65059

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