The total space $T$ of an embedded into $\mathbb{R}^n$ pure $n$-dimensional simplicial complex (in other words, the union of finitely many $n$-dimensional compact convex polytopes) sometimes admits an "almost partition" into $n$-simplices (i.e. $T$ equals the union of these simplices, and the interiors of these simplices do not, pairwise, intersect) which is coarser than a triangulation.


For instance, consider the union $T$ of two tetrahedra $T\_1$, $T\_2$ in $\mathbb{R}^3$ which intersect at the mid-point $M=E\_1\cap E\_2$ of edges $E\_i\in T\_i$, $1\leq i\leq 2$. Then any triangulation of $T$ will have to include $M$, and have at least 4 simplices. On the other hand, there exist an obvious "almost partition" of $T$ into just 2 simplices, $T\_1$ and $T\_2$.


We wonder whether there exists any research on these kinds of "almost partitions", and whether there is any well-established terminology here.



coarser than triangulations "almost partitions" into simplices
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](http://www.math.ucla.edu/~pak/book.htm) and [there](http://www.springer.com/mathematics/geometry/book/978-3-642-12970-4)).


P.S. Oh, yes, plenty of research. See e.g. [here](http://www-m10.ma.tum.de/foswiki/pub/Lehrstuhl/PublikationenJRG/30_SimplicialDissecions.pdf) for whether dissections of convex polytopes can be smaller than triangulations. 


