Discrete Morse theory for the barycentric subdivision

Let $F$ be a discrete Morse function on a simplicial complex $L$. We construct a discrete Morse function $\Delta(F)$ on the barycentric subdivision $\Delta(L)$. The constructed function $\Delta(F)$ “behaves the same way” as $F$, i.e., has the same number of critical simplices and the same gradient path structure.