  Our previous molecular dynamic simulation studies of simple two-dimensional
(2D) systems \cite{matt_big} suggested that both geometrical defects
(localized, large-amplitude deviations from hexagonal ordering) and topological
defects (dislocations and disclinations) play a role in 2D melting. To capture
the main features of the 2D melting transition and investigate the respective
roles of these two classes of defects, we study a discrete elastic model
consisting of an array of nodes connected by springs, for which the relative
number of geometrical defects (modeled as broken springs) and topological
defects (nodes having a coordination number different from 6) may be precisely
controlled. We preform Monte Carlo simulations of this model in the
isobaric-isothermal ensemble, and present the phase diagram as well as various
thermodynamic, statistical and structural quantities as a function of the
relative populations of geometrical and topological defects. The model exhibits
a rich phase behavior including hexagonal and square crystals, expanded
crystal, dodecagonal quasicrystal, and liquid structure. We found that the
solid--liquid transition temperature is lowered by a factor 3 to 4, with
respect to the case when only topological defects are allowed, when both
topological and geometrical defects are permitted, supporting our hypothesis
concerning the central role played by the geometrical defects in the melting.
The microscopic structure of the dense liquid has been investigated and the
results are compared to those from simulations of 2D particle systems.
