The term "triangulation" tends to be ambiguous, as they appear both in a geometric and topological context. In this case, what you want is called a dissection, at least in the discrete geometry literature (see e.g. here and there).

The term "triangulation" tends to be ambiguous, as they appear both in a geometric and topological context. In this case, what you want is called a dissection, at least in the discrete geometry literature (see e.g. here and there).

P.S. Oh, yes, plenty of research. See e.g. here for whether dissections of convex polytopes can be smaller than triangulations.