Skip to main content
SpringerLink
Log in

The Production of Filaments and Non-woven Materials: The Design of the Polymer Distributor

  • Chapter
  • First Online:
Math for the Digital Factory

Part of the book series: Mathematics in Industry ((TECMI,volume 27))

Abstract

We present results from the joint research project ProFil (Stochastic Processes for the Production of Filaments and Non-wovens), which were derived for the optimal design of the polymer distributor. In particular, one is interested in designs which prevent the cooling and degeneration of the polymer due to long occupation times. Since this is directly related to the wall shear stress distribution the questions arise, which wall shear stresses are attainable and how the corresponding design can be computed numerically. Employing the concept of approximate controllability we can answer the first one and characterize the set of attainable wall shear stresses. Further, we present a new numerical approach based on conformal mappings which allows for an optimization in the supremum norm and for an additional incorporation of state constraints. Finally, we show how the real industrial problem is solved by a least-squares optimization using shape gradients.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Anderson, J., Wendt, J.: Computational Fluid Dynamics, vol. 206. McGraw-Hill, New York (1995)

    Google Scholar 

  2. Chenais, D., Zuazua, E.: Controllability of an elliptic equation and its finite difference approximation by the shape of the domain. Numer. Math. 95, 63–99 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Grund, T., Rösch, A.: Optimal control of a linear elliptic equation with a supremum norm functional. Optim. Methods Softw. 15, 299–329 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints, vol. 23. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  5. Leithäuser, C.: Controllability of shape-dependent operators and constrained shape optimization for polymer distributors. PhD Thesis, TU Kaiserslautern (2013)

    Google Scholar 

  6. Leithäuser, C., Feßler, R.: Characterizing the image space of a shape-dependent operator for a potential flow problem. Appl. Math. Lett. 25(11), 1959–1963 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Leithäuser, C., Gramsch, S., Hietel, D., Wegener, R.: Modellierung und Simulation entlang der gesamten Vliesstoff-Prozesskette. In: Proceedings, vol. 28. Hofer Vliesstofftage (2013)

    Google Scholar 

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Leithäuser, C., Pinnau, R., Feßler, R.: A numerical approach to shape optimization with state constraints. arXiv:1412.4350 (2014)

    Google Scholar 

  10. Mohammadi, B., Pironneau, O.: Shape optimization in fluid mechanics. Annu. Rev. Fluid Mech. 36, 255–279 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  12. Quarteroni, A., Rozza, G.: Optimal control and shape optimization of aorto-coronaric bypass anastomoses. Math. Models Methods Appl. Sci. 13, 1801–1824 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rozza, G.: On optimization, control and shape design of an arterial bypass. Int. J. Numer. Methods Fluids 47, 1411–1419 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Schinzinger, R., Laura, P.: Conformal Mapping: Methods and Applications. Dover Publications, New York (2003)

    MATH  Google Scholar 

  15. Simon, J.: Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim. 2, 649–687 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sokolowski, J., Zolesio, J.: Introduction to Shape Optimization: Shape Sensitivity Analysis, vol. 16. Springer, Berlin (1992)

    MATH  Google Scholar 

  17. Vanderbei, R., Shanno, D.: An interior-point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl. 13, 231–252 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wegener, R., Marheineke, N., Hietel, D.: Virtuelle Produktion von Filamenten und Vliesstoffen. In: Neunzert, N., Prätzel-Wolters, D. (eds.) Mathematik im Fraunhofer-Institut Problemgetrieben - Modellbezogen - Lösungsorientiert, pp. 105–165. Springer, Berlin (2014)

    Google Scholar 

  19. Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the German Federal Ministry of Education and Research (BMBF) grant no. 03MS606F and by the German Federation of Industrial Research Associations (AiF) grant no. 17629N.

Author information

Authors and Affiliations

  1. Fraunhofer ITWM, Transport Processes, Fraunhofer-Platz 1, 67663, Kaiserslautern, Germany

    Christian Leithäuser

  2. Department of Mathematics, TU Kaiserslautern, Gottlieb-Daimler-Straße, 67663, Kaiserslautern, Germany

    René Pinnau

Authors
  1. Christian Leithäuser
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. René Pinnau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to René Pinnau .

Editor information

Editors and Affiliations

  1. ABB Italy S.p.A., Mathematical Modeling & Numerical Simulations, Vittuone, Milano, Italy

    Luca Ghezzi

  2. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Dietmar Hömberg

  3. Zurich University of Applied Sciences, Winterthur, Switzerland

    Chantal Landry

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Leithäuser, C., Pinnau, R. (2017). The Production of Filaments and Non-woven Materials: The Design of the Polymer Distributor. In: Ghezzi, L., Hömberg, D., Landry, C. (eds) Math for the Digital Factory. Mathematics in Industry(), vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-63957-4_15

Download citation

Publish with us

Policies and ethics

Search

Navigation