Title: Incorporation of the Joule Heating of highly conducting materials into the Truchas code via an asymptotic approach

The EM module of the Truchas code currently lacks the capability to model the Joule (Ohmic) heating of highly conducting materials that are inserted into induction furnaces from time to time to change the heating profile. This effect is difficult to simulate directly because of the requirement to resolve the extremely thin skin depth of good conductors, which is computationally costly. For example, copper has a skin depth, δ ~ 1 mm, for an oscillation frequency of tens of kHz. The industry is interested in determining what fraction of the heating power is lost to the Joule heating of these good conductors inserted inside the furnaces. The approach presented in this document is one of asymptotics where the leading order (unperturbed) solution is taken as that which emerges from solving the EM problem for a perfectly conducting insert. The conductor is treated as a boundary of the domain. The perturbative correction enters as a series expansion in terms of the dimensionless skin depth δ/L, where L is the characteristic size of the EM system. The correction at each order depends on the previous. This means that the leading order correction only depends on the unperturbed solution, in other words, itmore » does not require Truchas to perform an additional EM field solve. Thus, the Joule heating can be captured by a clever leveraging of the existing tools in Truchas with only slight modifications.« less