Shape optimization in contact problems with Coulomb friction. (English) Zbl 1025.49026

Beremlijski, P.; Haslinger, J.; Kocvara, M.; Outrata, J.

SIAM J. Optim. 13, No. 2, 561-587 (2002).

The very useful and important paper deals with a discretized problem of the shape optimization of elastic bodies in unilateral contact. The aim is to extend existing results to the case of contact problems following the Coulomb-friction law. Mathematical modelling of the Coulomb friction problem leads to a quasivariational inequality. The authors present an implicit programming approach (a class of mathematical programs with equilibrium constraints-MPECs). Its main idea consists of minimizing a nonsmooth composite friction generated by the objective and the (single-valued) control-state mapping. The control-state mapping is much more complicated, than in most MPECs, and the generalization of the relevant results is by no means straightforward. Main result: It is shown that for small coefficients of friction the discretized problem with Coulomb friction has a unique solution and that this solution is Lipschitzian as a function of a control variable describing the shape of the elastic body.
Finally, the numerical examples illustrate the efficiency and reliability of the suggested approach.

Reviewer: J.Lovíšek (Bratislava)

MSC:

49Q10	Optimization of shapes other than minimal surfaces
74M10	Friction in solid mechanics
74S05	Finite element methods applied to problems in solid mechanics

Keywords:

shape optimization; contact problems; Coulomb friction; mathematical programs with equilibrium constraints