Optimal thermal conductivity design for the volume-to-point heat conduction problem based on adjoint analysis

Volume-to-point (VP) problem is a classic prototype of heat conduction optimization and benefits important applications such as electronic devices thermal management. However, many studies employ various optimization principles without verifying their applicability; some other studies rashly apply a single principle on different cases but ignore influences of various objectives and boundary conditions. Here, a novel optimization principle for VP problems is derived by applying adjoint analysis, where the minimization of average temperature is considered as an example. The derived optimization principle reads the optimal thermal conductivity distribution maximizes the synergy between temperature gradient and adjoint temperature gradient fields. Relations among the proposed principle and three common ones are clarified rigorously, and reasons of why the latter cannot give optimal results are revealed. A one- and a two-dimensional cases are numerically optimized to validate the proposed principle, and results show that it gives the lowest average temperature as expected. For instance, the average temperature after optimization by using the proposed principle can be lower than that obtained from entropy generation minimization by around 25 and 10 K in the one- and two-dimensional cases studied, respectively. The presented derivation also applies to other objectives for obtaining corresponding optimization principle, showing its flexibility.