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.

[1]  Ole Sigmund,et al.  On the Design of Compliant Mechanisms Using Topology Optimization , 1997 .

[2]  J. Petersson,et al.  Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima , 1998 .

[3]  J. Petersson,et al.  Slope constrained topology optimization , 1998 .

[4]  Krister Svanberg,et al.  A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations , 2002, SIAM J. Optim..

[5]  M. Bendsøe,et al.  On two formulations of an optimal insulation problem , 2007 .

[6]  Kurt Maute,et al.  Design of phononic materials/structures for surface wave devices using topology optimization , 2007 .

[7]  Kouichi Kanda,et al.  Design impact of positive temperature dependence of drain current in sub 1 V CMOS VLSIs , 1999 .

[8]  A. Majumdar,et al.  Monte Carlo Study of Phonon Transport in Solid Thin Films Including Dispersion and Polarization , 2001 .

[9]  K. Svanberg.  The method of moving asymptotes—a new method for structural optimization , 1987 .

[10]  Sani R. Nassif,et al.  Full chip leakage estimation considering power supply and temperature variations , 2003, ISLPED '03.

[11]  B. Bourdin.  Filters in topology optimization , 2001 .

[12]  C. Kittel.  Introduction to solid state physics , 1954 .

[13]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[14]  M. Bendsøe,et al.  Topology Optimization: "Theory, Methods, And Applications" , 2011 .

[15]  K. Stuben,et al.  Algebraic Multigrid (AMG) : An Introduction With Applications , 2000 .

[16]  K. Maute,et al.  Topology optimization of fluid domains: kinetic theory approach , 2008 .

[17]  Lei He,et al.  Temperature and supply Voltage aware performance and power modeling at microarchitecture level , 2005, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[18]  Yuan Taur,et al.  Simulation of Nanoscale Multidimensional Transient Heat Conduction Problems Using Ballistic-Diffusive Equations and Phonon Boltzmann Equation , 2005 .

[19]  K. Maute,et al.  Topology optimization of flow domains using the lattice Boltzmann method , 2007 .

[20]  Kurt Maute,et al.  Topology Optimization of Fluid Problems by the Lattice Boltzmann Method , 2006 .

[21]  T. E. Bruns,et al.  Topology optimization of convection-dominated, steady-state heat transfer problems , 2007 .

[22]  Gang Chen,et al.  Thermal conductivity and ballistic-phonon transport in the cross-plane direction of superlattices , 1998 .

[23]  M. Bendsøe,et al.  Topology optimization of heat conduction problems using the finite volume method , 2006 .