  This paper treats topology optimization of natural convection problems. A
simplified model is suggested to describe the flow of an incompressible fluid
in steady state conditions, similar to Darcy's law for fluid flow in porous
media. The equations for the fluid flow are coupled to the thermal
convection-diffusion equation through the Boussinesq approximation. The coupled
non-linear system of equations is discretized with stabilized finite elements
and solved in a parallel framework that allows for the optimization of high
resolution three-dimensional problems. A density-based topology optimization
approach is used, where a two-material interpolation scheme is applied to both
the permeability and conductivity of the distributed material. Due to the
simplified model, the proposed methodology allows for a significant reduction
of the computational effort required in the optimization. At the same time, it
is significantly more accurate than even simpler models that rely on convection
boundary conditions based on Newton's law of cooling. The methodology discussed
herein is applied to the optimization-based design of three-dimensional heat
sinks. The final designs are formally compared with results of previous work
obtained from solving the full set of Navier-Stokes equations. The results are
compared in terms of performance of the optimized designs and computational
cost. The computational time is shown to be decreased to around 5-20% in terms
of core-hours, allowing for the possibility of generating an optimized design
during the workday on a small computational cluster and overnight on a high-end
desktop.
