This paper proposes a new grid-based method for adaptive anisotropic meshing of implicit surfaces. Grid-based methods are considered the major technique for implicit surface meshing, mainly due to their efficiency and simplicity. However, these methods suffer from a number of inherent drawbacks, resulting from the fact that the imposed Cartesian grid is generally not well adapted to the iso-surface, either in size or in orientation. To overcome the above obstacles, we propose a new implicit surface meshing method. The main idea of this method is first to construct a geometric field which is induced by the shape of the surface. This geometric field represents the natural directions and grid cell size for each point in R3. Then, the imposed volumetric grid is deformed toward the objects shape according to the produced geometric field. The iso-surface meshes can be extracted from the resulting adaptive grid by any conventional grid-based contouring technique, since the topology of the grid is not modified during the adaptation process. These meshes provide better approximation of the unknown surface and exhibit anisotropy, which is present in this surface.