  A systematic investigation on the third dredge up in a 3M$_{\odot}$, solar
metallicity AGB star will be presented. The model evolves from the main
sequence up to the Asymptotic Giant Branch (AGB). Intermediate mass stars are
important because they contribute significantly via the slow neutron capture
nucleosynthesis. The aim of this work is to gain insight on the behaviour of
the AGB star during thermal pulsation. This investigation is based on an
extended numerical simulation of the evolutionary phases and full, consistent
AGB model calculations. In particular, the convective structure during
pulsation will be studied, giving particular emphasis to the analysis of the
stability of the Schwarzschild boundary that will eventually determine the
occurrence of Third Dredge Up (hereafter referred to as TDUP). We provide a
brief description of our updated evolutionary code and focus primarily on the
obtaining the TDUP after 14 thermal pulses. We elaborate on the non-standard
treatment of convection known as "overshooting" and its efficiency in achieving
the TDUP event associated with the thermal pulsations (helium shell flashes).
However, our knowledge of the physics behind the TDUP event, and the
theoretical description of the overshooting mechanism suffer from embarrassing
gaps and are, therefore, not properly treated in stellar modelling. This is not
surprising, particularly with the oversimplified descriptions of important
mixing mechanisms like convection in stellar models. However, since such event
drives the $^{13}$C nucleosynthesis of interest for the following s-process
activation, we present a full assessment of this process and its implications
as calculations will further proceed in different directions.
