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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Chenais: On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975), 189–219.
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.
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.
A. Hansbo: Error estimates for the numerical solution of a time-periodic linear parabolic problem. BIT 31 (1991), 664–685.
J. Haslinger, P. Neittaanmäki: Finite Element Approximation for Optimal Shape Design: Theory and Applications. John Wiley & Sons, Chichester, 1988.
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.
A. Kufner: Weighted Sobolev Spaces. A Wiley-Interscience Publication. John Wiley & Sons, New York, 1985.
G. M. Lieberman: Time-periodic solutions of linear parabolic differential equations. Commun. Partial Differ. Equations 24 (1999), 631–663.
I. Matoušek, J. Cibulka: Analýza tvarovacího cyklu na karuselovém lisu NOVA. Technická univerzita v Liberci, Liberec, 1999. (In Czech.)
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.)
K. Rektorys: The Method of Discretization in Time and Partial Differential Equations. Mathematics and Its Applications 4, D. Reidel Publishing Company, Dordrecht, 1982.
T. Roubíček: Nonlinear Partial Differential Equations with Applications. ISNM. International Series of Numerical Mathematics 153, Birkhäuser, Basel, 2013.
P. Salač: Optimal design of the cooling plunger cavity. Appl. Math., Praha 58 (2013), 405–422.
R. Temam: Navier-Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and Its Applications 2, North-Holland Publishing Company, Amsterdam, 1977.
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.
O. Vejvoda: Partial Differential Equations: Time-Periodic Solutions. Martinus Nijhoff Publishers, Hague, 1981.
Author information
Authors and Affiliations
Corresponding authors
Additional information
The research has been supported by TUL.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.21136/AM.2019.0301-18