Optimal design of the cooling plunger cavity. (Q375445)

The paper describes a mathematical model of the cooling process of glass during the manufacturing process. The objective is to find a shape of the so-called plunger cavity in order to achieve the desired (constant) distribution of the temperature of the moulding device. The geometry of the problem is confined to a fixed planar domain which is divided into four regions describing the mould, the glass, the plunger and the cooling channel. The shape of the region to be optimized is described by a graph of a function which then becomes the control variable. The state problem is given by the stationary linear advection-diffusion equation describing the heat conduction, supplemented with boundary conditions of the Dirichlet, Neumann and Newton type. The vector of advection and the source term depends on a velocity field which is assumed to be uniquely determined for every admissible domain. Due to axisymmetry, the weak formulation is posed in the weighted Sobolev space. Existence and uniqueness of solutions to the state problem is proved using the Lax-Milgram lemma. After defining the cost function of least-square type the shape optimization problem is formulated and existence of at least one solution is proved.