Optimal insulation problems

Two formulations for the design of the optimal insulation of a domain
have been investigated by computational means. One method is
in the format of a topology design problem of distributing insulating material
in a domain surrounding a non-design domain that is heated by a given heat-source;
this problem is treated in both a relaxed format as well as a penalized material format.
The other approach deals with the optimal distribution of a thin layer of insulation on the
boundary of the non-design domain; this problem is more in the realm of shape design,
or rather, it is similar to optimal design of support conditions for structures.
In both cases mathematical programming is used, but for the shape design case
it is applied to the non-linear analysis problems that arise when the optimal design is
explicitly solved for. The computational results illustrate the similarities
and differences that result from the two approaches.