Shape optimization for frictional contact problems by a bundle trust-region method for constrained nonsmooth problems

Product development in engineering applications is a highly nontrivial task, especially for complex load scenarios or the interaction between different components. In mostapplications both difficulties occur. Algorithm-based product development may have a significant impact on the product development process in such complicated settings. In the Collaborative Research Center 666 this approach has been formalized in the framework of a manufacturing induced design process. In this work we consider the shape optimization of mechanical connectors, embedded in this design approach. The behavior of interacting components is mainly based on friction phenomena and the nonpenetration condition. Friction is often the key value to guarantee the functionality of mechanical connections. We assume the Coulomb friction model to describe this phenomenon. The mathematical model of the Coulomb friction leads to several theoretical difficulties in the problem formulation, for example nondifferentiability and ill-posedness. We tackle those by different regularization techniques according to well-established literature on this field. The resulting regularized problem is formulated as a semismooth operator equation, which is solved by a classical semismooth Newton method enhanced by a fixed point iteration for the regularization parameter. We show that the solution operator of the regularized Coulomb problem possesses a certain continuity property, namely Lipschitz continuity with respect to the design. This property still holds true, if we neglect the regularization of the nondifferentiability of the Coulomb problem. The numerical treatment of the derived physical model is based on a geometry description of the different components by splines, which is common in computer aided design systems. Following the approach of isogeometric analysis one can also apply spline functions to approximate the solution space of the Coulomb problem. As a result the expensive transformation between the finite element mesh and the geometry description is obsolete, which is known to be one of the most time-consuming working steps in product development. The shape optimization problem governed by the Coulomb problem is a nonsmooth and nonconvex problem. Additionally, we take various constraints into account, for example maximal volume or minimal durability according to fatigue strength. Here, the fatigue strength is determined by a local measure according to a deformation energy density. It is necessary to choose a suitable optimization algorithm for such nonsmooth and constraint problem. In this work we develop a constrained bundle trust-region method, which belongs to the family of cutting plane methods. We show that this leads to an efficient algorithm for the considered shape optimization problem and also to general nonsmooth and nonconvex optimization problems with nonlinear constraints.