You mentioned element birth and death to simulating an "instantaneous annealing". As you know, EKILL will erase the element solution history, but the elements are still in the model. They are just reduced to negligible stiffness (see ESTIF command) and no results are saved to the rst file. When they are re-introduced (via EALIVE), the ESTIF knockdown factor is removed and results begin to be saved to the rst file.
It is hard to say what is causing the convergence trouble without closer examination of the application, but any abrupt status change or disruption in the stiffness is going to present a potential convergence challenge. Perhaps you need to be more strategic in how many elements are reintroduced (via EALIVE) in one load step. Perhaps you need to use smaller time increments. You should also take care that the elements are sufficiently supported both while killed and after they are reactivated, especially if they are in a primary load path when reactivated.
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