Designing polymer spin packs by tailored shape optimization techniques

Abstract

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.

Appendices

Appendix 1: Wall shear stress

The WSS defined in (3) agrees with the classical definition (Tavoularis 2005, eq. 14.1). To see this, let $\mathbf{x} \in \varGamma^{\text{wall}}$ and assume w.l.o.g that $\mathbf{x}$ is the center of the coordinate system and that $\mathbf{n}(\mathbf{x}) = (0, 0, 1)^\intercal$ holds. Then

$$\begin{aligned} \sigma (\mathbf{x}) = \eta \left| \nabla \times \mathbf{u} \right| (\mathbf{x}) = \eta \left| \begin{pmatrix} \partial _2 u_3 - \partial _3 u_2\\ \partial _3 u_1 - \partial _1 u_3\\ \partial _1 u_2 - \partial _2 u_1 \end{pmatrix} \right| (x) = \eta \left| \begin{pmatrix} - \partial _3 u_2\\ \partial _3 u_1\\ 0 \end{pmatrix} \right| (x), \end{aligned}$$

where $\partial _1 u_2 = \partial _1 u_3 = \partial _2 u_1 = \partial _2 u_3 = 0$ are derivatives in tangential direction at the boundary where $\mathbf{u} = 0$ holds and thus vanish. Furthermore, w.l.o.g the coordinate system can be rotated around the third axis, i.e., the normal, such that $\partial _3 u_2$ vanishes. This yields the classical formulation of the WSS (Tavoularis 2005, eq. 14.1)

$$\begin{aligned} \sigma (\mathbf{x}) = \eta \left| \partial _3 u_1 \right| (\mathbf{x}). \end{aligned}$$

Appendix 2: Weak formulation of the 3D Stokes problem

We provide the weak form of (1) which is used to solve the problem numerically. Our outflow boundary condition agrees with (Ern and Guermond 2004, Eq. 4.11c). The following weak form of (1) relies on the identity $\Delta \mathbf{u} = \nabla ({{\mathrm{div}}}\, \mathbf{u}) - \nabla \times \nabla \times \mathbf{u}$ and is derived in (Ern and Guermond 2004, Eq. 4.12):

$$\begin{aligned} &{\text{Find }}\,\mathbf{u} \in X:= \{\mathbf{u} \in [H^1(\varOmega )]^3; \;\mathbf{u}|_{\varGamma^{\text{in}} \cup \varGamma^{\text{wall}}} = 0;\; \mathbf{n}\times \mathbf{u}|_{\varGamma^{\text{out}}} = 0 \} {\hbox{and }} p \in L^2(\varOmega ):\\&\int _\varOmega \eta (\nabla \times \mathbf{u}) \cdot (\nabla \times \mathbf{v}) - p {{\mathrm{div}}}\, \mathbf{v} - q {{\mathrm{div}}}\,\mathbf{u} \,dx + \int _{\varGamma^{\text{out}}} \eta c_{\text{out}} (\mathbf{n}\cdot \mathbf{u}) (\mathbf{n}\cdot \mathbf{v}) \,ds = \int _\varOmega \mathbf{f} \cdot \mathbf{v}\\&\qquad \forall \mathbf{v} \in X, q \in L^2(\varOmega ). \end{aligned}$$

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Note that the problem has been transformed into the homogeneous form such that the non-homogeneous Dirichlet boundary condition is included in the right-hand side term $\mathbf{f} = \eta \Delta \tilde{\mathbf{u}}$, where $\tilde{\mathbf{u}}$ is an extension of the Dirichlet boundary conditions which satisfies ${{\mathrm{div}}}\tilde{\mathbf{u}} = 0$. With sufficient regularity $\mathbf{u} + \tilde{\mathbf{u}}$ is then a solution of (1).