  We present the COLIBRI code for computing the evolution of stars along the
TP-AGB phase. Compared to purely synthetic TP-AGB codes, COLIBRI relaxes a
significant part of their analytic formalism in favour of a detailed physics
applied to a complete envelope model, in which the stellar structure equations
are integrated from the atmosphere down to the bottom of the hydrogen-burning
shell. This allows to predict self-consistently: (i) the effective temperature,
and more generally the convective envelope and atmosphere structures, correctly
coupled to the changes in the surface chemical abundances and gas opacities;
(ii) sphericity effects in the atmospheres; (iii) the core mass-luminosity
relation and its break-down due to hot bottom burning (HBB) in the most massive
AGB stars, (iv) the HBB nucleosynthesis via the solution of a complete nuclear
network (pp chains, and the CNO, NeNa, MgAl cycles), including also the
production of 7Li via the Cameron-Fowler beryllium transport mechanism; (v) the
chemical composition of the pulse-driven convective zone; (vi) the onset and
quenching of the third dredge-up, with a suitable temperature criterion. At the
same time COLIBRI pioneers new techniques in the treatment of the physics of
stellar interiors. It is the first evolutionary code ever to use accurate
on-the-fly computation of the equation of state for roughly 800 atoms, ions,
molecules, and of the Rosseland mean opacities throughout the deep envelope.
Another distinguishing aspect of COLIBRI is its high computational speed. This
feature is necessary for calibrating the uncertain parameters and processes
that characterize the TP-AGB phase, a step of paramount importance for
producing reliable stellar population synthesis models of galaxies up to high
redshift. (abridged)
