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2013 | 23 | 2 | 263-276
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Tom
23
Numer
2
Strony
263-276
Opis fizyczny
Daty
wydano
2013
otrzymano
2012-07-19
poprawiono
2012-10-30
Twórcy
  • Laboratory of Applied Mathematics and Physics (LAMPS), University of Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan, France
  • Institute of Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland
autor
  • Institute of Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland
Bibliografia
  • Alart, P., Barboteu, M. and Lebon, F. (1997). Solutions of frictional contact problems using an EBE preconditioner, Computational Mechanics 30(4): 370-379.
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  • 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.
  • 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.
  • 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.
  • Clarke, F.H. (1983). Optimization and Nonsmooth Analysis, Wiley Interscience, New York, NY.
  • 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.
  • Duvaut, G. and Lions, J.L. (1976). Inequalities in Mechanics and Physics, Springer-Verlag, Berlin.
  • Franc, V. (2011). Library for quadratic programming, http://cmp.felk.cvut.cz/˜xfrancv/libqp/html.
  • Han, W. and Sofonea, M. (2002). Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society, Providence, RI/International Press, Sommerville, MA.
  • 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.
  • 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.
  • 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.
  • Ionescu, I.R., Nguyen, Q.L. and Wolf, S. (2003). Slip-dependent friction in dynamic elasticity, Nonlinear Analysis 53(3-4): 375-390.
  • 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.
  • Ionescu, I.R. and Sofonea, M. (1993). Functional and Numerical Methods in Viscoplasticity, Oxford University Press, Oxford.
  • 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.
  • 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.
  • Laursen, T. (2002). Computational Contact and Impact Mechanics, Springer, Berlin/Heidelberg.
  • Mäkelä, M.M. (1990). Nonsmooth Optimization, Theory and Applications with Applications to Optimal Control, Ph.D. thesis, University of Jyväskylä, Jyväskylä.
  • Mäkelä, M.M. (2001). Survey of bundle methods for nonsmooth optimization, Optimization Methods and Software 17(1): 1-29.
  • 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.
  • 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.
  • Migórski, S. and Ochal, A. (2005). Hemivariational inequality for viscoelastic contact problem with slip-dependent friction, Nonlinear Analysis 61(1-2): 135-161.
  • 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.
  • 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.
  • Naniewicz, Z. and Panagiotopoulos, P.D. (1995). Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York, NY/Basel/Hong Kong.
  • Nečas, J. and Hlavaček, I. (1981). Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction, Elsevier, Amsterdam.
  • Panagiotopoulos, P.D. (1985). Inequality Problems in Mechanics and Applications, Birkhauser, Basel.
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  • Shillor, M., Sofonea, M. and Telega, J.J. (2004). Models and Analysis of Quasistatic Contact, Springer, Berlin.
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  • Wriggers, P. (2002). Computational Contact Mechanics, Wiley, Chichester.
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Bibliografia
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