Topology optimization for nano‐scale heat transfer

We consider the problem of optimal design of nano‐scale heat conducting systems using topology optimization techniques. At such small scales the empirical Fourier's law of heat conduction no longer captures the underlying physical phenomena because the mean‐free path of the heat carriers, phonons in our case, becomes comparable with, or even larger than, the feature sizes of considered material distributions. A more accurate model at nano‐scales is given by kinetic theory, which provides a compromise between the inaccurate Fourier's law and precise, but too computationally expensive, atomistic simulations. We analyze the resulting optimal control problem in a continuous setting, briefly describing the computational approach to the problem based on discontinuous Galerkin methods, algebraic multigrid preconditioned generalized minimal residual method, and a gradient‐based mathematical programming algorithm. Numerical experiments with our implementation of the proposed numerical scheme are reported. Copyright © 2008 John Wiley & Sons, Ltd.