  We design a class of spatially inhomogeneous heat spreaders in the context of
steady-state thermal conduction leading to spatially uniform thermal fields
across a large convective surface. Each spreader has a funnel-shaped design,
either in the form of a trapezoidal prism or truncated cone, and is forced by a
thermal source at its base. We employ transformation-based techniques, commonly
used to study metamaterials, to determine the require thermal conductivity for
the spreaders. The obtained materials, although strongly anisotropic and
inhomogeneous, can be accurately approximated by assembling isotropic,
homogeneous layers, rendering them realisable. An alternative approach is then
considered for the conical and trapezoidal spreaders by dividing them into two
or three isotropic, homogeneous components respectively. We refer to these
simple configurations as neutral layers. All designs are validated numerically
both with and without the effects of thermal contact resistance between
interfaces. Such novel designs pave the way for future materials that can
manipulate and control the flow of heat, helping to solve traditional heat
transfer problems such as controlling the temperature of an object and energy
harvesting.
