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**Shape optimization in contact problems based on penalization of the state inequality.**
*(English)*
Zbl 0594.73109

The paper deals with the frictionless plane contact problem of a linear- elastic sheet resting on a rigid foundation. The shape optimization is carried out in such a manner that the contact boundary curve \(\alpha\) should be the result of the minimization of the total potential energy with respect to \(\alpha\), the contact problem being described by a variational inequality. This inequality is replaced by a family of penalized state problems. The existence of their solution is proved and the relation between the optimal shapes for the original state inequality and those for the penalized equations is established. Further a proof for the convergence of a finite element approximation is given. This approximation leads to a nonlinear programming problem with box constraints, linear inequality constraints and a linear equality constraint. Instructive examples are worked out numerically showing that the contact region is enlarged and the contact pressure becomes uniform as a result of the optimization.

Reviewer: H.Bufler

### MSC:

74A55 | Theories of friction (tribology) |

74M15 | Contact in solid mechanics |

74P99 | Optimization problems in solid mechanics |

74S05 | Finite element methods applied to problems in solid mechanics |

49J40 | Variational inequalities |

49M30 | Other numerical methods in calculus of variations (MSC2010) |

### Keywords:

frictionless plane contact; linear-elastic sheet; rigid foundation; shape optimization; contact boundary curve; minimization of the total potential energy; family of penalized state problems; existence; convergence; nonlinear programming problem; box constraints; linear inequality constraints; linear equality constraint### Citations:

Zbl 0594.73108
Full Text:
EuDML

### References:

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