  Excision techniques are used in order to deal with black holes in numerical
simulations of Einstein equations and consist in removing a topological sphere
containing the physical singularity from the numerical domain, applying instead
appropriate boundary conditions at the excised surface. In this work we present
recent developments of this technique in the case of constrained formulations
of Einstein equations and for spherically symmetric spacetimes. We present a
new set of boundary conditions to apply to the elliptic system in the
fully-constrained formalism of Bonazzola et al. (2004), at an arbitrary
coordinate sphere inside the apparent horizon. Analytical properties of this
system of boundary conditions are studied and, under some assumptions, an
exponential convergence toward the stationary solution is exhibited for the
vacuum spacetime. This is verified in numerical examples, together with the
applicability in the case of the accretion of a scalar field onto a
Schwarzschild black hole. We also present the successful use of the excision
technique in the collapse of a neutron star to a black hole, when excision is
switched on during the simulation, after the formation of the apparent horizon.
This allows the accretion of matter remaining outside the excision surface and
for the stable long-term evolution of the newly formed black hole.
