A novel FEM-based dynamic framework for subdivision surfaces

Abstract

Recursive subdivision on an initial control mesh generates a visually pleasing smooth surface in the limit. Nevertheless, users must carefully specify the initial mesh and/or painstakingly manipulate the control vertices at different levels of subdivision hierarchy to satisfy a diverse set of functional requirements and aesthetic criteria in the limit shape. This modeling drawback results from the lack of direct manipulation tools for the limit geometric shape. To improve the efficiency of interactive geometric modeling and engineering design, in this paper we integrate novel physics-based modeling techniques with powerful geometric subdivision principles, and develop a unified finite element method (FEM)-based methodology for arbitrary subdivision schemes. Strongly inspired by the recent research on Dynamic Non-Uniform Rational B-Splines (D-NURBS), we formulate and develop a dynamic framework that permits users to directly manipulate the limit surface obtained from any subdivision procedure via simulated “force” tools. The most significant contribution of our unified approach is the formulation of the limit surface of an arbitrary subdivision scheme as being composed of a single type of novel finite element. The specific geometric and dynamic features of our subdivision-based finite elements depend on the subdivision scheme used. We present our novel FEM for the modified butterfly and Catmull–Clark subdivision schemes, and generalize our dynamic framework to be applicable to other subdivision schemes. Our FEM-based approach significantly advances the 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 via 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 unified FEM-based framework not only promises a greater potential for subdivision techniques in solid modeling, finite element analysis, and engineering design, but that it will further foster the applicability of subdivision geometry in a wide range of visual computing applications such as visualization, virtual reality, computer graphics, computer vision, robotics, and medical imaging as well.