An analytical and numerical approach to a bilateral contact problem with nonmonotone friction

Mikaël Barboteu; Krzysztof Bartosz; Piotr Kalita

International Journal of Applied Mathematics and Computer Science (2013)

  • Volume: 23, Issue: 2, page 263-276
  • ISSN: 1641-876X

Abstract

top
We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is bilateral, i.e., there is no loss of contact. The friction is modeled with a nonmotonone law. The purpose of this work is to provide an error estimate for the Galerkin method as well as to present and compare two numerical methods for solving the resulting nonsmooth and nonconvex frictional contact problem. The first approach is based on the nonconvex proximal bundle method, whereas the second one deals with the approximation of a nonconvex problem by a sequence of nonsmooth convex programming problems. Some numerical experiments are realized to compare the two numerical approaches.

How to cite

top

Mikaël Barboteu, Krzysztof Bartosz, and Piotr Kalita. "An analytical and numerical approach to a bilateral contact problem with nonmonotone friction." International Journal of Applied Mathematics and Computer Science 23.2 (2013): 263-276. <http://eudml.org/doc/257113>.

@article{MikaëlBarboteu2013,
abstract = {We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is bilateral, i.e., there is no loss of contact. The friction is modeled with a nonmotonone law. The purpose of this work is to provide an error estimate for the Galerkin method as well as to present and compare two numerical methods for solving the resulting nonsmooth and nonconvex frictional contact problem. The first approach is based on the nonconvex proximal bundle method, whereas the second one deals with the approximation of a nonconvex problem by a sequence of nonsmooth convex programming problems. Some numerical experiments are realized to compare the two numerical approaches.},
author = {Mikaël Barboteu, Krzysztof Bartosz, Piotr Kalita},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {linearly elastic material; bilateral contact; nonmonotone friction law; hemivariational inequality; finite element method; error estimate; nonconvex proximal bundle method; quasi-augmented Lagrangian method; Newton method},
language = {eng},
number = {2},
pages = {263-276},
title = {An analytical and numerical approach to a bilateral contact problem with nonmonotone friction},
url = {http://eudml.org/doc/257113},
volume = {23},
year = {2013},
}

TY - JOUR
AU - Mikaël Barboteu
AU - Krzysztof Bartosz
AU - Piotr Kalita
TI - An analytical and numerical approach to a bilateral contact problem with nonmonotone friction
JO - International Journal of Applied Mathematics and Computer Science
PY - 2013
VL - 23
IS - 2
SP - 263
EP - 276
AB - We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is bilateral, i.e., there is no loss of contact. The friction is modeled with a nonmotonone law. The purpose of this work is to provide an error estimate for the Galerkin method as well as to present and compare two numerical methods for solving the resulting nonsmooth and nonconvex frictional contact problem. The first approach is based on the nonconvex proximal bundle method, whereas the second one deals with the approximation of a nonconvex problem by a sequence of nonsmooth convex programming problems. Some numerical experiments are realized to compare the two numerical approaches.
LA - eng
KW - linearly elastic material; bilateral contact; nonmonotone friction law; hemivariational inequality; finite element method; error estimate; nonconvex proximal bundle method; quasi-augmented Lagrangian method; Newton method
UR - http://eudml.org/doc/257113
ER -

References

top
  1. Alart, P., Barboteu, M. and Lebon, F. (1997). Solutions of frictional contact problems using an EBE preconditioner, Computational Mechanics 30(4): 370-379. Zbl0896.73054
  2. Alart, P. and Curnier, A. (1991). A mixed formulation for frictional contact problems prone to Newton like solution methods, Computer Methods in Applied Mechanics and Engineering 92(3): 353-375. Zbl0825.76353
  3. Baniotopoulos, C., Haslinger, J. and Moravkova, Z. (2005). Mathematical modeling of delamination and nonmonotone friction problems by hemivariational inequalities, Applications of Mathematics 50(1): 1-25. Zbl1099.74021
  4. Barboteu, M., Han, W. and Sofonea, M. (2002). Numerical analysis of a bilateral frictional contact problem for linearly elastic materials, IMA Journal of Numerical Analysis 22(3): 407-436. Zbl1205.74133
  5. Barboteu, M. and Sofonea, M. (2009). Analysis and numerical approach of a piezoelectric contact problem, Annals of the Academy of Romanian Scientists: Mathematics and Its Applications 1(1): 7-31. Zbl05819255
  6. Clarke, F.H. (1983). Optimization and Nonsmooth Analysis, Wiley Interscience, New York, NY. Zbl0582.49001
  7. Denkowski, Z., Migórski, S. and Papageorgiou, N.S. (2003). An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, MA/Dordrecht/London/New York, NY. Zbl1054.47001
  8. Duvaut, G. and Lions, J.L. (1976). Inequalities in Mechanics and Physics, Springer-Verlag, Berlin. Zbl0331.35002
  9. Franc, V. (2011). Library for quadratic programming, http://cmp.felk.cvut.cz/˜xfrancv/libqp/html. 
  10. Han, W. and Sofonea, M. (2002). Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society, Providence, RI/International Press, Sommerville, MA. Zbl1013.74001
  11. Haslinger, J., Miettinen, M. and Panagiotopoulos, P.D. (1999). Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications, Kluwer Academic Publishers, Boston, MA/Dordrecht/London. Zbl0949.65069
  12. Hild, P. and Renard, Y. (2007). An error estimate for the Signorini problem with Coulomb friction approximated by finite elements, SIAM Journal of Numerical Analysis 45(5): 2012-2031. Zbl1146.74050
  13. Ionescu, I.R. and Nguyen, Q.L. (2002). Dynamic contact problems with slip-dependent friction in viscoelasticity, International Journal of Applied Mathematics and Computer Science 12(1): 71-80. Zbl1023.74034
  14. Ionescu, I.R., Nguyen, Q.L. and Wolf, S. (2003). Slip-dependent friction in dynamic elasticity, Nonlinear Analysis 53(3-4): 375-390. Zbl1062.74033
  15. Ionescu, I.R. and Paumier, J.C. (1996). On the contact problem with slip displacement dependent friction in elastostatics, International Journal of Engineering Sciences 34(4): 471-491. Zbl0900.73682
  16. Ionescu, I.R. and Sofonea, M. (1993). Functional and Numerical Methods in Viscoplasticity, Oxford University Press, Oxford. Zbl0787.73005
  17. Khenous, Y., Laborde, P. and Renard, Y. (2006a). On the discretization of contact problems in elastodynamics, in P. Wriggers and U. Nackenhorst (Eds.), Analysis and Simulation of Contact Problems, Lecture Notes in Applied and Computational Mechanics, Vol. 27, Springer, Berlin/Heidelberg, pp. 31-38. Zbl1194.74417
  18. Khenous, H.B., Pommier, J. and Renard, Y. (2006b). Hybrid discretization of the Signorini problem with coulomb friction. theoretical aspects and comparison of some numerical solvers, Applied Numerical Mathematics 56(2): 163-192. Zbl1089.74046
  19. Laursen, T. (2002). Computational Contact and Impact Mechanics, Springer, Berlin/Heidelberg. Zbl0996.74003
  20. Mäkelä, M.M. (1990). Nonsmooth Optimization, Theory and Applications with Applications to Optimal Control, Ph.D. thesis, University of Jyväskylä, Jyväskylä. Zbl0731.49002
  21. Mäkelä, M.M. (2001). Survey of bundle methods for nonsmooth optimization, Optimization Methods and Software 17(1): 1-29. Zbl1050.90027
  22. Mäkelä, M.M., Miettinen, M., Lukšan, L. and Vlček, J. (1999). Comparing nonsmooth nonconvex bundle methods in solving hemivariational inequalities, Journal of Global Optimization 14(2): 117-135. Zbl0947.90125
  23. Miettinen, M. (1995). On contact problems with nonmonotone contact conditions and their numerical solution, in M.H. Aliabadi and C. Alessandri (Eds.), Contact Mechanics II: Computational Techniques, Transactions on Engineering Sciences, Vol. 7, WIT Press, Southampton/Boston, MA, pp. 167-174. 
  24. Migórski, S. and Ochal, A. (2005). Hemivariational inequality for viscoelastic contact problem with slip-dependent friction, Nonlinear Analysis 61(1-2): 135-161. Zbl1190.74020
  25. Mistakidis, E.S. and Panagiotopulos, P.D. (1997). Numerical treatment of problems involving nonmonotone boundary or stress-strain laws, Computers & Structures 64(1-4): 553-565. Zbl0918.73363
  26. Mistakidis, E.S. and Panagiotopulos, P.D. (1998). The search for substationary points in the unilateral contact problems with nonmonotone friction, Mathematical and Computer Modelling 28(4-8): 341-358. 
  27. Naniewicz, Z. and Panagiotopoulos, P.D. (1995). Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York, NY/Basel/Hong Kong. 
  28. Nečas, J. and Hlavaček, I. (1981). Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction, Elsevier, Amsterdam. Zbl0448.73009
  29. Panagiotopoulos, P.D. (1985). Inequality Problems in Mechanics and Applications, Birkhauser, Basel. Zbl0579.73014
  30. Rabinowicz, E. (1951). The nature of the static and kinetic coefficients of friction, Journal of Applied Physics 22(11): 1373-1379. 
  31. Shillor, M., Sofonea, M. and Telega, J.J. (2004). Models and Analysis of Quasistatic Contact, Springer, Berlin. Zbl1069.74001
  32. Tzaferopoulos, M.A., Mistakidis, E.S., Bisbos, C.D. and Panagiotopulos, P.D. (1995). Comparison of two methods for the solution of a class of nonconvex energy problems using convex minimization algorithms, Computers & Structures 57(6): 959-971. Zbl0924.73322
  33. Wriggers, P. (2002). Computational Contact Mechanics, Wiley, Chichester. Zbl1104.74002

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.