# Article

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Keywords:
shape optimization; Navier-Stokes equations; heat transfer
Summary:
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. \endgraf 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. \endgraf The cost functional is defined as the squared weighted $L^2$ 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. \endgraf The existence and uniqueness of the solution to the state problem and the existence of a solution to the optimization problem are proved.
References:
[1] Chenais, D.: On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975), 189-219. DOI 10.1016/0022-247X(75)90091-8 | MR 0385666 | Zbl 0317.49005
[2] (ed.), A. Fasano: Mathematical Models in the Manufacturing of Glass. C.I.M.E. Summer School, 2008. Lecture Notes in Mathematics 2010, Springer, Berlin (2011). DOI 10.1007/978-3-642-15967-1 | MR 3075369 | Zbl 1203.00024
[3] Galdi, G. P.: 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). MR 1284206 | Zbl 0949.35005
[4] Hansbo, A.: Error estimates for the numerical solution of a time-periodic linear parabolic problem. BIT 31 (1991), 664-685. DOI 10.1007/BF01933180 | MR 1140538 | Zbl 0780.65057
[5] Haslinger, J., Neittaanmäki, P.: Finite Element Approximation for Optimal Shape Design: Theory and Applications. John Wiley & Sons, Chichester (1988). MR 0982710 | Zbl 0713.73062
[6] Korobkov, M., Pileckas, K., Russo, R.: 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. DOI 10.2422/2036-2145.201204_003 | MR 3379492 | Zbl 1321.35137
[7] Kufner, A.: Weighted Sobolev Spaces. A Wiley-Interscience Publication. John Wiley & Sons, New York (1985). MR 0802206 | Zbl 0567.46009
[8] Lieberman, G. M.: Time-periodic solutions of linear parabolic differential equations. Commun. Partial Differ. Equations 24 (1999), 631-663. DOI 10.1080/03605309908821436 | MR 1683052 | Zbl 0928.35012
[9] Matoušek, I., Cibulka, J.: Analýza tvarovacího cyklu na karuselovém lisu NOVA. Technická univerzita v Liberci, Liberec (1999), Czech.
[10] Mercier, B., Raugel, G.: Résolution d'un problème aux limites dans un ouvert axisymétrique par éléments finis en $r, z$ et séries de Fourier en $\theta$. RAIRO, Anal. Numér. 16 (1982), 405-461 French. DOI 10.1051/m2an/1982160404051 | MR 0684832 | Zbl 0531.65054
[11] Rektorys, K.: The Method of Discretization in Time and Partial Differential Equations. Mathematics and Its Applications 4, D. Reidel Publishing Company, Dordrecht (1982). MR 0689712 | Zbl 0505.65029
[12] Roubíček, T.: Nonlinear Partial Differential Equations with Applications. ISNM. International Series of Numerical Mathematics 153, Birkhäuser, Basel (2013). DOI 10.1007/978-3-0348-0513-1 | MR 3014456 | Zbl 1270.35005
[13] Salač, P.: Optimal design of the cooling plunger cavity. Appl. Math., Praha 58 (2013), 405-422. DOI 10.1007/s10492-013-0020-8 | MR 3083521 | Zbl 1289.49043
[14] Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and Its Applications 2, North-Holland Publishing Company, Amsterdam (1977). DOI 10.1016/S0168-2024(13)70303-5 | MR 0609732 | Zbl 0383.35057
[15] Vandewalle, S., Piessens, R.: Efficient parallel algorithms for solving initial-boundary value and time-periodic parabolic partial differential equations. SIAM J. Sci. Stat. Comput. 13 (1992), 1330-1346. DOI 10.1137/0913075 | MR 1185649 | Zbl 0766.65076
[16] Vejvoda, O.: Partial Differential Equations: Time-Periodic Solutions. Martinus Nijhoff Publishers, Hague (1981). MR 0653987 | Zbl 0501.35001

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