Subdivision surfaces have been extensively used to model smooth shapes of arbitrary topology. Recursive subdivision on an initial control mesh generates a visually pleasing smooth surface in the limit. However, users have to carefully specify the initial mesh and/or painstakingly manipulate the control vertices at different levels of subdivision hierarchy to satisfy various functional and aesthetic requirements in the limit surface. This modeling drawback results from the lack of direct manipulation tools for the limit surface. This paper integrates physics-based modeling techniques with geometric subdivision methodology and present an unified approach for arbitrary subdivision schemes. Our dynamic framework permits users to directly manipulate the limit surface via "force" tools. The key contribution of this unified approach is to formulate the limit surface of any subdivision scheme as a single type of novel finite elements. The geometric and dynamic features of our subdivision-based finite elements depend on the subdivision scheme involved. We present our finite element method (FEM) for the modified butterfly and Catmull-Clark subdivision schemes, and further generalize our dynamic framework for any subdivision scheme. Our FEM-based approach significantly advances the state-of-the-art of physics-based geometric modeling because (1) our framework provides a universal physics-based solution to any subdivision scheme beyond popular spline-like subdivision techniques; (2) we systematically devise a natural mechanism that allows users to intuitively deform any subdivision surface; (3) we represent the limit surface of any subdivision scheme using a single type of novel subdivision-based finite elements. Our experiments demonstrate that the new unified FEM-based framework promises a greater potential of subdivision techniques for solid modeling, finite element analysis, and engineering design.
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