A novel FEM-based dynamic framework for subdivision surfaces

Recursive subdivision on an initial control mesh generates a visually pleas ing smooth surface in the limit. Nevertheless, users must carefully specify the initial mesh and/or painstakingly manipulate the control vertices at d ifferent levels of subdivision hierarchy to satisfy a diverse set of functi onal requirements and aesthetic criteria in the limit shape. This modeling drawback results from the lack of direct manipulation tools for the limit g eometric shape. To improve the efficiency of interactive geometric modeling and engineering design, in this paper we integrate novel physics-based mod eling techniques with powerful geometric subdivision principles, and develo p a unified finite element method (FEM)-based methodology for arbitrary sub division schemes. Strongly inspired by the recent research on Dynamic Non-U niform Rational B-Splines (D-NURBS), we formulate and develop a dynamic fra mework that permits users to directly manipulate the limit surface obtained from any subdivision procedure via simulated "force" tools. The most signi ficant contribution of our unified approach is the formulation of the limit surface of an arbitrary subdivision scheme as being composed of a single t ype of novel finite element. The specific geometric and dynamic features of our subdivision-based finite elements depend on the subdivision scheme use d. We present our novel FEM for the modified butterfly and Catmull-Clark su bdivision schemes, and generalize our dynamic framework to be applicable to other subdivision schemes. Our FEM-based approach significantly advances t he state-of-the-art in physics-based geometric modeling since it provides a universal physics-based framework for any subdivision scheme. In addition, we systematically devise a mechanism that allows users to directly (not vi a control meshes) deform any subdivision surface; finally, we represent the limit surface of any subdivision scheme using a collection of subdivision- based novel finite elements. Our experiments demonstrate that the new unifi ed FEM-based framework not only promises a greater potential for subdivisio n techniques in solid modeling, finite element analysis, and engineering de sign, but that it will further foster the applicability of subdivision geom etry in a wide range of visual computing applications such as visualization , virtual reality, computer graphics, computer vision, robotics, and medica l imaging as well. (C) 2000 Published by Elsevier Science Ltd. All rights r eserved.