Mixed finite element analysis of semi-coercive unilateral contact problems with given friction

Ivan Hlaváček

Applications of Mathematics (2007)

  • Volume: 52, Issue: 1, page 25-58
  • ISSN: 0862-7940

Abstract

top
A unilateral contact 2D-problem is considered provided one of two elastic bodies can shift in a given direction as a rigid body. Using Lagrange multipliers for both normal and tangential constraints on the contact interface, we introduce a saddle point problem and prove its unique solvability. We discretize the problem by a standard finite element method and prove a convergence of approximations. We propose a numerical realization on the basis of an auxiliary “bolted” problem and the algorithm of Uzawa.

How to cite

top

Hlaváček, Ivan. "Mixed finite element analysis of semi-coercive unilateral contact problems with given friction." Applications of Mathematics 52.1 (2007): 25-58. <http://eudml.org/doc/33275>.

@article{Hlaváček2007,
abstract = {A unilateral contact 2D-problem is considered provided one of two elastic bodies can shift in a given direction as a rigid body. Using Lagrange multipliers for both normal and tangential constraints on the contact interface, we introduce a saddle point problem and prove its unique solvability. We discretize the problem by a standard finite element method and prove a convergence of approximations. We propose a numerical realization on the basis of an auxiliary “bolted” problem and the algorithm of Uzawa.},
author = {Hlaváček, Ivan},
journal = {Applications of Mathematics},
keywords = {unilateral contact; Tresca’s model of friction; mixed variational formulation; Uzawa algorithm; unilateral contact; Tresca's model of friction; mixed variational formulation; Uzawa algorithm},
language = {eng},
number = {1},
pages = {25-58},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Mixed finite element analysis of semi-coercive unilateral contact problems with given friction},
url = {http://eudml.org/doc/33275},
volume = {52},
year = {2007},
}

TY - JOUR
AU - Hlaváček, Ivan
TI - Mixed finite element analysis of semi-coercive unilateral contact problems with given friction
JO - Applications of Mathematics
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 1
SP - 25
EP - 58
AB - A unilateral contact 2D-problem is considered provided one of two elastic bodies can shift in a given direction as a rigid body. Using Lagrange multipliers for both normal and tangential constraints on the contact interface, we introduce a saddle point problem and prove its unique solvability. We discretize the problem by a standard finite element method and prove a convergence of approximations. We propose a numerical realization on the basis of an auxiliary “bolted” problem and the algorithm of Uzawa.
LA - eng
KW - unilateral contact; Tresca’s model of friction; mixed variational formulation; Uzawa algorithm; unilateral contact; Tresca's model of friction; mixed variational formulation; Uzawa algorithm
UR - http://eudml.org/doc/33275
ER -

References

top
  1. Approximation of Elliptic Boundary Value Problems, Wiley, New York, 1972. (1972) Zbl0248.65063MR0478662
  2. 10.1016/S1631-073X(02)02356-7, C. R.  Acad. Sci. Paris, Ser. I 334 (2002), 917–922. (2002) MR1909940DOI10.1016/S1631-073X(02)02356-7
  3. 10.1090/S0025-5718-01-01318-7, Math. Comput. 71 (2002), 1–25. (2002) MR1862986DOI10.1090/S0025-5718-01-01318-7
  4. 10.1016/j.matcom.2004.12.007, Math. Comput. Simul. 68 (2005), 271–300. (2005) MR2138931DOI10.1016/j.matcom.2004.12.007
  5. Analyse Convexe et Problèmes Variationnels, Dunod, Paris, 1974. (1974) MR0463993
  6. Special Matrices and Their Applications in Numerical Mathematics, Martinus Nijhoff Publ. (member of Kluwer), Dordrecht-Boston, 1986. (1986) Zbl0677.65019MR1105955
  7. Dual formulations for potentials and complementary energies, In: MAFELAP, J.  R. Whiteman (ed.), Academic Press, London, 1973. (1973) 
  8. 10.1016/0022-247X(82)90257-8, J.  Math. Anal. Appl. 86 (1982), 99–122. (1982) MR0649858DOI10.1016/0022-247X(82)90257-8
  9. Numerical methods for unilateral problems in solid mechanics, In: Handbook of Numerical Analysis, vol. IV, P. G. Ciarlet, J.-L. Lions (eds.), North Holland, Amsterdam, 1996, pp. 313–485. (1996) MR1422506
  10. 10.1051/m2an:2004026, M2AN, Math. Model. Numer. Anal. 38 (2004), 563–578. (2004) MR2075760DOI10.1051/m2an:2004026
  11. An algorithm for numerical realization of 3D  contact problems with Coulomb friction, J.  Comput. Appl. Math. 164–165 (2004), 387–408. (2004) MR2056889
  12. Solution of Variational Inequalities in Mechanics, Springer-Verlag, New York, 1988. (1988) MR0952855
  13. 10.1023/A:1022220711369, Appl. Math. 45 (2000), 357–379. (2000) MR1777018DOI10.1023/A:1022220711369
  14. 10.1016/S0378-4754(01)00433-5, Math. Comput. Simul. 60 (2002), 1–17. (2002) MR1916897DOI10.1016/S0378-4754(01)00433-5
  15. Contact problem for two elastic bodies, Parts I–III, Apl. Mat. 25 (1980), 87–109, 110–136, 137–146. (1980) MR0560325
  16. Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Elsevier, Amsterdam, 1981. (1981) MR0600655
  17. Nonconforming mixed variational formulation of the Signorini problem with a given friction, Preprint of MAPLY No. 365, 2003.http://maply,univ-lyon1.fr/publis/. (2003.http://maply,univ-lyon1.fr/publis/) 

NotesEmbed ?

top

You must be logged in to post comments.