In this work the interaction between turbulence and premixed combustion is studied by means of direct numerical simulations using a level set equation based on the scalar G that describes the motion of the name front represented by an iso-surface G = Go. The G-equation contains terms accounting for name propagation, flame curvature and gas expansion effects. The flame front is transported and wrinkled by the turbulent flow field and propagates in a direction normal to itself with its laminar burning velocity. The turbulent now field is based on the constant density Navier-Stokes equations. Results are presented for cases with and without heat release at the flame front. In the former case hear release is modeled by volume sources located on the flame front. For the case without heat release the flame surface area ratio and the source terms appearing in its balance equation accounting for production, kinematic restoration and dissipation are evaluated from the DNS results. The flame surface area ratio shows a linear increase with increasing ratios of v'/s(L) in the corrugated flamelets regime but a bending in the thin reaction zones regime. This is consistent with the evaluation of the source terms where it is found that kinematic restoration is the main sink term in the corrugated flamelets regime, while scalar dissipation becomes the dominant sink term in the thin reaction zones regime, By evaluating the pdf of G-fluctuations the link between the flame surface area ratio and the flame surface density is established. Agreement with existing models for the latter is found, When gas expansion effects are added, the induced velocity at the flame front can be evaluated. This leads to a purely kinematic explanation for counter gradient diffusion, Effects of gas expansion on the flame surface area ratio are found to be significant only if the laminar burning velocity is of the same order of magnitude as the turbulence intensity or larger.