  Based on the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric
settings, we study expansions of models of a supersimple theory with a new
predicate distiguishing a set of forking-independent elements that is dense
inside a partial type $\mathcal{G}(x)$, which we call $H$-structures. We show
that any two such expansions have the same theory and that under some technical
conditions, the saturated models of this common theory are again
$H$-structures. We prove that under these assumptions the expansion is
supersimple and characterize forking and canonical bases of types in the
expansion. We also analyze the effect these expansions have on one-basedness
and CM-triviality. In the one-based case, when $T$ has $SU$-rank
$\omega^\alpha$ and the $SU$-rank is continuous, we take $\mathcal{G}(x)$ to be
the type of elements of $SU$-rank $\omega^\alpha$ and we describe a natural
"geometry of generics modulo $H$" associated with such expansions and show it
is modular.
