An interpolatory subdivision for volumetric models over simplicial complexes

Subdivision has gained popularity in computer graphics and shape modeling during the past two decades, yet volumetric subdivision has received much less attention. In this paper, we develop a new subdivision scheme, which can interpolate all of the initial control points in 3D and generate a continuous volume in the limit. We devise a set of solid subdivision rules to facilitate a simple subdivision procedure. The conversion between the subdivided mesh and a simplicial complex is straightforward and effective, which can be directly utilized in solid meshing, finite element simulation, and other numerical processes. In principle, our solid subdivision process is a combination of simple linear interpolations in 3D. Affine operations of neighboring control points produce new control points in the next level, yet inherit the original control points and achieve the interpolatory effect. A parameter is offered to control the tension between control points. The interpolatory property of our solid subdivision offers many benefits, which are desirable in many design applications and physics simulations, including intuitive manipulation on control points and ease of constraint enforcement in numerical procedures. We outline a proof that can guarantee the convergence and C¹ continuity of our volumetric subdivision and limit volumes in regular cases. In addition to solid subdivision, we derive special rules to generate C¹ surfaces as B-reps and to model shapes of non-manifold topology. Several examples demonstrate the ability of our subdivision to handle complex manifolds easily. Numerical experiments and future research suggestions for extraordinary cases are also presented.