A 3D Shape Optimization Problem in Heat Transfer: Analysis and Approximation via BEM

In this paper an optimal shape control problem dealing with heat transfer in enclosures is studied. We have considered an enclosure heated by a flame surface (taking into account radiation, conduction and convection effects), and we look for an optimal flame shape which minimizes a cost functional defined on the temperature field. This kind of problem arises in industrial furnaces optimization, as temperature uniformity is one of the most important aspects in industrial plant analysis and design. Analytical results (smoothness of the control-to-state mapping, existence of an optimal shape in a certain admissible class) as well as numerical optimization results by the boundary element method have been obtained; we have employed the gradient method to optimize the flame shape, exploiting the adjoint equation associated with our state equation and cost functional.