Designing polymer spin packs by tailored shape optimization techniques

We consider optimal shape design problems for polymer spin packs which are widely used in the production of synthetic fibers and nonwoven materials. The design goal is the minimization of the residence time of the polymer, which can be achieved by adjusting the wall shear stress along the boundary. Depending on the specific industrial setting we construct two tailored algorithms. First, we consider the design in three spatial dimensions based on a PDE constrained shape optimization problem. Here, the constraint is given by the Stokes flow. Second, we change the design goal and want to construct shapes in two spatial dimensions which allow for a lower bound on the wall shear stress. This can be incorporated as an additional state constraint. By relaxing this condition and employing the method of mapping we can pull-back the problem onto a fixed reference domain. We get an elegant formulation of this state constrained optimization problem, in which geometric constraints on the boundary can also be included. After discretization we end up with a large-scale NLP which can be handled by existing solvers. Finally, we present numerical results underlining the feasibility of our approach.