Shape optimization of unsteady natural convection fields

This paper presents a numerical method for two shape optimization problems, namely, prescribing the temperature history distribution on sub-boundaries and maximizing the discharged heat on sub-boundaries of unsteady natural convection fields. The square error integral between the actual temperature distribution and the target temperature distribution on the sub-boundaries during a specified period of time was used as the objective functional for the prescribed temperature history distribution. The shape gradients of these shape determination problems were derived theoretically using the Lagrange multiplier method, adjoint variable method, and the material derivative formulae. Reshaping was performed by the traction method, which was proposed as an approach for solving shape optimization problems. Numerical programs for the shape determination problems are developed based on FreeFem++ in order to verify the proposed method.