On two formulations of an optimal insulation problem

Two formulations for the design of the optimal insulation of a domain are investigated by computational means. The results illustrate the similarities and differences that result from the two approaches. 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.