Solution to shape design problems of unsteady forced heat-convection fields

This paper presents numerical solution to two shape design problems of unsteady forced heat-convection fields to control temperature to a prescribed distribution. In the first problem, the square error integral between the actual temperature distributions and the prescribed temperature distributions on the prescribed sub-domains during the specified period of time is used as the objective functional. In the second problem, a multi-objective shape optimization problem using normalized objective functional is formulated for the temperature distribution prescribed problem and the total dissipated energy minimization problem in the unsteady forced heat-convection fields. Shape gradient of these shape design problems is derived theoretically using the Lagrange multiplier method, adjoint variable method, and the formulae of the material derivative. Reshaping is carried out by the traction method proposed as an approach to solving shape optimization problems. Numerical analyses program for the shape design is developed based on FreeFem++, and the validity of proposed method is confirmed by results of 2D numerical analyses.