Shape optimal design problem with convective and radiative heat transfer: Analysis and implementation

We present a study of an optimal design problem for a coupled system, gover ned by a steady-state potential flow equation and a thermal equation taking into account radiative phenomena with multiple reflections. The state equa tion is a nonlinear integro-differential system. We seek to minimize a cost function, depending on the temperature, with respect to the domain of the equations. First, we consider an optimal design problem in an abstract fram ework and, with the help of the classical adjoint state method, give an exp ression of the cost function differential. Then, we apply this result in th e two-dimensional case to the nonlinear integro-differential system conside red. We prove the differentiability of the cost function, introduce the adj oint state equation, and give an expression of its exact differential. Then , we discretize the equations by a finite-element method and use a gradient -type algorithm to decrease the cost function. We present numerical results as applied to the automotive industry.