The contact problem of punch shape optimization under constraints on the applied forces and moments is considered. The motion of the punch is investigated in a quasi-static formulation, taking into account the friction forces. The optimal punch shape is found as a solution of the optimization problem of determining the pressure distribution acting on an elastic half-space in a 3D formulation. The new decomposition of the optimization problem into a separate solution of the problem of the best possible external action and a solution of the problem of the penetration of the punch into elastic medium with friction is discussed. As an example the solution of a rectangular punch penetration is presented.
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