On the Objective Function for Topology Optimization of Heat Sinks

In topology optimization, the optimal layout of the structure is obtained through an iterative update of the geometry. Since this update procedure utilizes the gradient of the objective function, it is important to select an appropriate objective function in order to obtain a meaningful geometry. From this point of view, the mathematical definition and physical meaning of the objective functions for topology optimization of heat sinks are reviewed in this article. For this, studies from conductive heat transfer problems to recent computational fluid dynamics (CFD)-based conjugate heat transfer problems are covered. In most of the studies, the objective functions are defined in order to minimize the thermal resistance of the heat sink or to minimize the maximum temperature of the heat source. In place of thermal resistance, thermal compliance is commonly used as the objective function. However, it is worth noting that minimization of the thermal compliance may lead to considerable deviation of the design from that based on the minimum thermal resistance. This is because the thermal compliance is linearly proportional to the average temperature of the heat source in the computational domain, and the average temperature of the heat source is usually different from the maximum temperature of the heat source. In heat conduction problems, the difference between the maximum temperature and the average temperature becomes large when the thermal conductivity of the solid material becomes low. In convection problems, the difference becomes large when the streamwise fluid temperature variation becomes large due to the low heat capacity rate of the fluid.