Abstract
We consider the optimal shape design of a distributor geometry in the context of industrial fiber spinning. In this process a molten polymer is routed from a pipe to a spinneret plate with a larger cross section, where thin fibers, which are then further processed, are spun from the fluid. The residence time or material age of the polymer in the distributor, which is modeled through an additional advection-diffusion-reaction equation, has to be controlled such that fluid stagnation is prevented, since this would cause material degradation and a decrease in the quality of the fibers. In order to optimize the geometry, we formally derive the adjoint equations and the volume formulation of the shape derivative and apply them within a gradient descent method.
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References
Bauschke, H., Combettes, P.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces (Springer, New York, 2017)
Delfour, M.C., Zolésio, J.-P.: Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization. Advances in Design and Control. SIAM, Philadelphia (2011)
Guermond, J.-L., Quartapelle, L.: On sensitive vector Poisson and Stokes problems. Math. Models Methods Appl. Sci. 7(5), 681–698 (1997)
Józsa, J., Krámer, T: Modelling residence time as advection-diffusion with zero-order reaction kinetics. In: Proceedings of the Hydrodynamics 2000 Conference, International Association of Hydraulic Engineering and Research, pp. 23–27 (2000)
Leithäuser, C., Pinnau, R., Feßler, R.: Approximate controllability of linearized shape-dependent operators for flow problems. ESAIM Control Optim. Calc. Var. 23(3), 751–771 (2017)
Leithäuser, C., Pinnau, R., Feßler, R.: Designing polymer spin packs by tailored shape optimization techniques. Optim. Eng. 19(3), 733–764 (2018)
Sokolowski, J., Zolesio, J.-P.: Introduction to Shape Optimization: Shape Sensitivity Analysis (Springer, Berlin, 1992)
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Hohmann, R., Leithäuser, C. (2019). Shape Optimization of Liquid Polymer Distributors. In: Faragó, I., Izsák, F., Simon, P. (eds) Progress in Industrial Mathematics at ECMI 2018. Mathematics in Industry(), vol 30. Springer, Cham. https://doi.org/10.1007/978-3-030-27550-1_54
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DOI: https://doi.org/10.1007/978-3-030-27550-1_54
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