Shape Optimization in 3d Contact Problems with Coulomb Friction

Since 1980, a considerable attention of applied mathematicians has been devoted to unilateral contact problems with Coulomb friction, cf. [2] and the references therein. Concerning the static case, our comprehension has reached a fairly satisfactory level. In [1], the authors have developed a numerical approach to a class of optimization problems, where one computes optimal shape of a 2D elastic body in contact with a rigid obstacle which obeys the Coulomb friction law. The problem has been formulated as a mathematical program with equilibrium constraints (MPEC) and solved via the so called implicit programming approach (ImP), cf. [5]. The technique from [1] cannot be, however, extended to the 3D case in a straightforward way. The reason is the nonpolyhedral nature of the subdifferential map of the Euclidean norm in Rn, whenever n ≥ 2. Whereas the stability and sensitivity analysis of variational inequalities/generalized equations over polyhedral constraint sets has been developed quite deeply so far, much less is known about the nonpolyhedral case. This holds in particular for the generalized equation (GE) modeling the investigated 3D contact problem. Further, also the numerical solution of this GE with a fixed shape of the body (which is the state problem in our MPEC) is substantially more demanding. The main aim of this contribution is to extend the ImP technique of [1] to the 3D case, which requires to discretize this MPEC by finite elements, to construct a fast and precise solver for the state problem and, by using tools of sensitivity analysis, to compute a ”subgradient” information, needed in the used nonsmooth optimization method. 2. NUMERICAL APPROACH