Induced heating solver

In­duc­tion heat­ing is the process of heat­ing an elec­tri­cal­ly con­duct­ing ob­ject (usu­al­ly a met­al) by elec­tro­mag­net­ic in­duc­tion (usu­al­ly with a mov­ing or non-mov­ing coil), where ed­dy cur­rents are gen­er­at­ed with­in the met­al and re­sis­tance leads to Joule heat­ing of the met­al. Nu­mer­i­cal­ly, in or­der to solve an ed­dy cur­rent prob­lem a EM time step com­pat­i­ble with the fre­quen­cy (i.e. a time step such that there are at least a few dozens of steps in the quar­ter pe­ri­od of the cur­rent) is need­ed. For ex­am­ple, with a fre­quen­cy of 1MHz, a time step around 1.e-8 sec­onds would be need­ed and thus 1.e8 time steps to solve a full prob­lem last­ing 1s. There­fore, an in­duc­tion heat­ing analy­sis would be time con­sum­ing us­ing the clas­sic Ed­dy-cur­rent solver.

The in­duc­tion heat­ing solver works the fol­low­ing way: it as­sumes a cur­rent which os­cil­lates very rapid­ly com­pared to the to­tal time of the process (typ­i­cal­ly, a cur­rent with a fre­quen­cy rang­ing from kHz to Mhz and a to­tal time for the process around a few sec­onds). The fol­low­ing as­sump­tion is done: a full ed­dy-cur­rent prob­lem is solved on a half-pe­ri­od with a “mi­cro” EM time step. An av­er­age of the EM fields dur­ing this half-pe­ri­od as well as the joule heat­ing are com­put­ed. It is then as­sumed that the prop­er­ties of the ma­te­r­i­al (and most­ly the elec­tri­cal con­duc­tiv­i­ty which dri­ves the flow of the cur­rent and the joule heat­ing) do not change for the next pe­ri­ods of the cur­rent. These prop­er­ties de­pend­ing most­ly on the tem­per­a­ture, the as­sump­tion can there­fore be con­sid­ered ac­cu­rate as long as the tem­per­a­ture does­n’t change too much. Dur­ing these pe­ri­ods, no EM com­pu­ta­tion is done, on­ly the av­er­aged joule heat­ing is added to the ther­mal solver. But, as the tem­per­a­ture changes, and thus the elec­tri­cal con­duc­tiv­i­ty, the EM fields need to be up­dat­ed ac­cord­ing­ly, so an­oth­er full ed­dy cur­rent res­o­lu­tion is com­put­ed for a half-pe­ri­od of the cur­rent giv­ing new av­er­aged EM fields. The solver can there­fore ef­fi­cient­ly solve prob­lems in­volv­ing in­duc­tive heat­ing for a mov­ing or non-mov­ing coil.