Shape Optimization in Three-Dimensional Contact Problems with Coulomb Friction

We study the discretized problem of the shape optimization of three-dimensional (3D) elastic bodies in unilateral contact. The aim is to extend existing results to the case of contact problems obeying the Coulomb friction law. Mathematical modeling of the Coulomb friction problem leads to an implicit variational inequality. 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. The 2D case of this problem was studied by the authors in [P. Beremlijski, J. Haslinger, M. Kocvara, and J. V. Outrata, SIAM J. Optim., 13 (2002), pp. 561-587]; there we used the so-called implicit programming approach combined with the generalized differential calculus of Clarke. The extension of this technique to the 3D situation is by no means straightforward. The main source of difficulties is the nonpolyhedral character of the second-order (Lorentz) cone, arising in the 3D model. To facilitate the computation of the subgradient information, needed in the used numerical method, we exploit the substantially richer generalized differential calculus of Mordukhovich. Numerical examples illustrate the efficiency and reliability of the suggested approach.