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Shape optimization for a time-dependent model of a carousel press in glass production

Applications of Mathematics volume 64, pages 195–224 (2019)Cite this article

Abstract

This contribution presents the shape optimization problem of the plunger cooling cavity for the time dependent model of pressing the glass products. The system of the mould, the glass piece, the plunger and the plunger cavity is considered in four consecutive time intervals during which the plunger moves between 6 glass moulds.

The state problem is represented by the steady-state Navier-Stokes equations in the cavity and the doubly periodic energy equation in the whole system, under the assumption of rotational symmetry, supplemented by suitable boundary conditions.

The cost functional is defined as the squared weighted L2 norm of the difference between a prescribed constant and the temperature of the plunger surface layer at the moment before separation of the plunger and the glass piece.

The existence and uniqueness of the solution to the state problem and the existence of a solution to the optimization problem are proved.

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References

  1. D. Chenais: On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975), 189–219.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Fasano (ed.): Mathematical Models in the Manufacturing of Glass. C.I.M.E. Summer School, 2008. Lecture Notes in Mathematics 2010, Springer, Berlin, 2011.

    MATH  Google Scholar 

  3. G. P. Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II: Nonlinear Steady Problems. Springer Tracts in Natural Philosophy 39, Springer, New York, 1994.

    Google Scholar 

  4. A. Hansbo: Error estimates for the numerical solution of a time-periodic linear parabolic problem. BIT 31 (1991), 664–685.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Haslinger, P. Neittaanmäki: Finite Element Approximation for Optimal Shape Design: Theory and Applications. John Wiley & Sons, Chichester, 1988.

    MATH  Google Scholar 

  6. M. Korobkov, K. Pileckas, R. Russo: The existence theorem for steady Navier-Stokes equations in the axially symmetric case. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 14 (2015), 233–262.

    MathSciNet  MATH  Google Scholar 

  7. A. Kufner: Weighted Sobolev Spaces. A Wiley-Interscience Publication. John Wiley & Sons, New York, 1985.

    MATH  Google Scholar 

  8. G. M. Lieberman: Time-periodic solutions of linear parabolic differential equations. Commun. Partial Differ. Equations 24 (1999), 631–663.

    Article  MathSciNet  MATH  Google Scholar 

  9. I. Matoušek, J. Cibulka: Analýza tvarovacího cyklu na karuselovém lisu NOVA. Technická univerzita v Liberci, Liberec, 1999. (In Czech.)

    Google Scholar 

  10. B. Mercier, G. Raugel: Résolution d’un problème aux limites dans un ouvert axisymétrique par éléments finis en r, z et séries de Fourier en θ. RAIRO, Anal. Numér. 16 (1982), 405–461. (In French.)

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Rektorys: The Method of Discretization in Time and Partial Differential Equations. Mathematics and Its Applications 4, D. Reidel Publishing Company, Dordrecht, 1982.

    Google Scholar 

  12. T. Roubíček: Nonlinear Partial Differential Equations with Applications. ISNM. International Series of Numerical Mathematics 153, Birkhäuser, Basel, 2013.

    Google Scholar 

  13. P. Salač: Optimal design of the cooling plunger cavity. Appl. Math., Praha 58 (2013), 405–422.

    MathSciNet  MATH  Google Scholar 

  14. R. Temam: Navier-Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and Its Applications 2, North-Holland Publishing Company, Amsterdam, 1977.

    Google Scholar 

  15. S. Vandewalle, R. Piessens: Efficient parallel algorithms for solving initial-boundary value and time-periodic parabolic partial differential equations. SIAM J. Sci. Stat. Comput. 13 (1992), 1330–1346.

    Article  MathSciNet  MATH  Google Scholar 

  16. O. Vejvoda: Partial Differential Equations: Time-Periodic Solutions. Martinus Nijhoff Publishers, Hague, 1981.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Technical University of Liberec, Studentská 2, 46117, Liberec 1, Czech Republic

    Petr Salač & Jan Stebel

Authors
  1. Petr Salač
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  2. Jan Stebel
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Correspondence to Petr Salač or Jan Stebel.

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The research has been supported by TUL.

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Salač, P., Stebel, J. Shape optimization for a time-dependent model of a carousel press in glass production. Appl Math 64, 195–224 (2019). https://doi.org/10.21136/AM.2019.0301-18

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  • DOI: https://doi.org/10.21136/AM.2019.0301-18

Keywords

  • shape optimization
  • Navier-Stokes equations
  • heat transfer

MSC 2010

  • 49Q10
  • 76D55
  • 93C20