Evolutionary design with freedom: Time dependent heat spreading

This paper illustrates the method of evolutionary design with freedom by considering the problem of determining the distribution of the thickness of a high-conductivity layer (heat spreader) installed on a conducting wall. Heating comes from a steady line source imposed at t = 0. Heat spreads in time-dependent fashion in two directions: longitudinally by conduction along the layer and the wall, and transversally by conduction through the wall and then by convection into the flow that cools the other side of the wall. The heat spreading has two regimes, which are dictated by the two diffusion time scales along the layer and across the wall. It is shown that the solution to the optimal heat spreader thickness problem is the parabolic profile, and it is the same in both regimes of time-dependent conduction.