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, Tome 23 (2013), p. 263-276 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2013-01-01

Zbl 1282.70031

EUDML-ID : urn:eudml:doc:257113

Bibliographie

[000] Alart, P., Barboteu, M. and Lebon, F. (1997). Solutions of frictional contact problems using an EBE preconditioner, Computational Mechanics 30(4): 370-379. | Zbl 0896.73054

[001] 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. | Zbl 0825.76353

[002] 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. | Zbl 1099.74021

[003] 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. | Zbl 1205.74133

[004] 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. | Zbl 05819255

[005] Clarke, F.H. (1983). Optimization and Nonsmooth Analysis, Wiley Interscience, New York, NY. | Zbl 0582.49001

[006] 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. | Zbl 1054.47001

[007] Duvaut, G. and Lions, J.L. (1976). Inequalities in Mechanics and Physics, Springer-Verlag, Berlin. | Zbl 0331.35002

[008] Franc, V. (2011). Library for quadratic programming, http://cmp.felk.cvut.cz/˜xfrancv/libqp/html.

[009] Han, W. and Sofonea, M. (2002). Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society, Providence, RI/International Press, Sommerville, MA. | Zbl 1013.74001

[010] 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. | Zbl 0949.65069

[011] 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. | Zbl 1146.74050

[012] 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. | Zbl 1023.74034

[013] Ionescu, I.R., Nguyen, Q.L. and Wolf, S. (2003). Slip-dependent friction in dynamic elasticity, Nonlinear Analysis 53(3-4): 375-390. | Zbl 1062.74033

[014] 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. | Zbl 0900.73682

[015] Ionescu, I.R. and Sofonea, M. (1993). Functional and Numerical Methods in Viscoplasticity, Oxford University Press, Oxford. | Zbl 0787.73005

[016] 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. | Zbl 1194.74417

[017] 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. | Zbl 1089.74046

[018] Laursen, T. (2002). Computational Contact and Impact Mechanics, Springer, Berlin/Heidelberg. | Zbl 0996.74003

[019] Mäkelä, M.M. (1990). Nonsmooth Optimization, Theory and Applications with Applications to Optimal Control, Ph.D. thesis, University of Jyväskylä, Jyväskylä. | Zbl 0731.49002

[020] Mäkelä, M.M. (2001). Survey of bundle methods for nonsmooth optimization, Optimization Methods and Software 17(1): 1-29. | Zbl 1050.90027

[021] 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. | Zbl 0947.90125

[022] 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.

[023] Migórski, S. and Ochal, A. (2005). Hemivariational inequality for viscoelastic contact problem with slip-dependent friction, Nonlinear Analysis 61(1-2): 135-161. | Zbl 1190.74020

[024] 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. | Zbl 0918.73363

[025] 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.

[026] Naniewicz, Z. and Panagiotopoulos, P.D. (1995). Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York, NY/Basel/Hong Kong.

[027] Nečas, J. and Hlavaček, I. (1981). Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction, Elsevier, Amsterdam. | Zbl 0448.73009

[028] Panagiotopoulos, P.D. (1985). Inequality Problems in Mechanics and Applications, Birkhauser, Basel. | Zbl 0579.73014

[029] Rabinowicz, E. (1951). The nature of the static and kinetic coefficients of friction, Journal of Applied Physics 22(11): 1373-1379.

[030] Shillor, M., Sofonea, M. and Telega, J.J. (2004). Models and Analysis of Quasistatic Contact, Springer, Berlin. | Zbl 1069.74001

[031] 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. | Zbl 0924.73322

[032] Wriggers, P. (2002). Computational Contact Mechanics, Wiley, Chichester. | Zbl 1104.74002