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Optimal control of radiative heat transfer in glass cooling with restrictions on the temperature gradient. (English) Zbl 1258.49066

Summary: This paper is motivated by an optimal boundary control problem for the cooling process of molten and already formed glass down to room temperature. The high temperatures at which glass is processed demand to include radiative heat transfer in the computational model. Since the complete radiative heat transfer equations are too complex for optimization purposes, we use simplified approximations of spherical harmonics coupled with a practically relevant frequency bands model. The optimal control problem is considered as a Partial Differential Algebraic Equation (PDAE)-constrained optimization problem with box constraints on the control. In this paper, we augment the objective by a functional depending on the state gradient, which forces a minimization of thermal stress inside the glass. To guarantee consistent and grid-independent values of the reduced objective gradient at the end of the cooling process, we pursue two approaches. The first includes the temperature gradient with a time-dependent linearly decreasing weight. In the second approach, we augment the objective functional by the final state tracking and final state gradient optimization. To determine an optimal boundary control, we apply a projected gradient method with the Armijo step size rule. The reduced objective gradient is computed by the continuous adjoint approach. The arising time-dependent PDAEs are numerically solved by variable step size one-step methods of Rosenbrock type in time and adaptive multilevel finite elements in space. We present two-dimensional numerical results for an infinitely long glass block and compare the two different approaches derived to ensure consistency at the end of the cooling process.

MSC:

49N90 Applications of optimal control and differential games
49M25 Discrete approximations in optimal control
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

Software:

dsoa; KELLEY; KARDOS; ROS3P
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References:

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