Based on the subdivision generation analysis of B-spline curves and surfaces, a novel construction non-stationary subdivision scheme is proposed for free form curve and surface modeling. This proposed subdivision rules are designed as a succession of a simple linear subdivision operator followed by several averaging steps, and an optional moving back step is added to handle the interpolation requirement in some application cases. The main advantage of the new scheme is that this framework can work well for both triangular meshes and quadrilateral meshes, and build a natural transition between the interpolating and approximating schemes. In particular, many existing subdivision schemes can be represented in the same framework by choosing suitable parameters