This paper discusses a new approach to controlling 2D local sizing in a bicameral anisotropic mesh. We define bicameral anisotropy as a mesh size variation of two distinctly different types in two separate chambers or subdomains. The first chamber is controlled by constant to linear local size functions. The second subdomain is governed by a nonlinear sizing function leading to transitioning meshes. A controlled advancing front approach is proposed for both triangular and quadrangular meshes with the singular goal of ensuring a high local quality metric in the first chamber. An H-shock sizing scheme governs the second chamber. Virtual mesh topology is constructed at the face boundary both at geometry and nodeloop levels to facilitate this type of bicameral meshing. Results clearly indicate the efficacy of the proposed approach leading to both a well controlled desired size field and high local element quality.