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

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.

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 |