From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics

Abstract

In this paper, we first gather existence results for linear and for pseudo-monotone variational inequalities in reflexive Banach spaces. We discuss the necessity of the involved coerciveness conditions and their relationship. Then, we combine Mosco convergence of convex closed sets with an approximation of pseudo-monotone bifunctions and provide a convergent approximation procedure for pseudo-monotone variational inequalities in reflexive Banach spaces. Since hemivariational inequalities in linear elasticity are pseudo-monotone, our approximation method applies to nonmonotone contact problems. We sketch how regularization of the involved nonsmooth functionals together with finite element approximation lead to an efficient numerical solution method for these nonconvex nondifferentiable optimization problems. To illustrate our theory, we give a numerical example of a 2D linear elastic block under a given nonmonotone contact law.

Keywords:

• pseudo-monotone bifunction
• coerciveness condition
• hemivariational inequality
• nonmonotone contact
• finite element discretization
• smoothing

AMS Subject Classifications:

• 49J40
• 74M15

Acknowledgements

The authors thank the two anonymous referees for their comments and suggestions that helped to improve the paper.

Notes

No potential conflict of interest was reported by the author(s).

Dedicated to Prof A. Göpfert on occasion of his 80th birthday and to the memory of Prof K. H. Elster and Prof W. Oettli.